On a mixed local-nonlocal evolution equation with singular nonlinearity

Kaushik Bal and Stuti Das [email protected] and [email protected] Department of Mathematics and Statistics,
Indian Institute of Technology Kanpur, Uttar Pradesh, 208016, India

Abstract

We will prove several existence and regularity results for the mixed local-nonlocal parabolic equation of the form

\displaystyle\displaystyle\begin{split}u_{t}-\Delta u+(-\Delta)^{s}u&=\frac{f(x,t)}{u^{\gamma(x,t)}}\text{ in }\Omega_{T}:=\Omega\times(0,T),\\ u&=0\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u(x,0)&=u_{0}(x)\text{ in }\Omega;\end{split}

(0.1)

where

(-\Delta)^{s}u=c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^{n}}\frac{u(x,t)-u(y,t)}{|x-y|^{n+2s}}dy.

Under the assumptions that $\displaystyle\gamma$ is a positive continuous function on $\displaystyle\overline{\Omega}_{T}$ and $\displaystyle\Omega$ is a bounded domain with Lipschitz boundary in $\displaystyle\mathbb{R}^{n}$ , $\displaystyle n>2$ , $\displaystyle s\in(0,1)$ , $\displaystyle 0<T<+\infty$ , $\displaystyle f\geq 0$ , $\displaystyle u_{0}\geq 0$ , $\displaystyle f$ and $\displaystyle u_{0}$ belongs to suitable Lebesgue spaces. Here $\displaystyle c_{n,s}$ is a suitable normalization constant, and $\displaystyle\operatorname{P.V.}$ stands for Cauchy Principal Value.

1 Introduction

In this article, we study the evolution of a mixed local-nonlocal operator under the effect of a singular nonlinearity given by:

{}\begin{split}u_{t}-\Delta u+(-\Delta)^{s}u&=\frac{f(x,t)}{u^{\gamma(x,t)}}\;\mbox{ in }\Omega_{T}:=\Omega\times(0,T),\\ u&=0\;\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u(x,0)&=u_{0}(x)\mbox{ in }\Omega;\end{split}

(1.1)

where

(-\Delta)^{s}u:=c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^{n}}\frac{u(x,t)-u(y,t)}{|x-y|^{n+2s}}dy,

$\displaystyle\gamma$ is a positive continuous function on $\displaystyle\overline{\Omega}_{T}$ with $\displaystyle\Omega$ being a bounded domain with Lipschitz boundary in $\displaystyle\mathbb{R}^{n}$ , $\displaystyle n>2$ , $\displaystyle s\in(0,1)$ and $\displaystyle 0<T<+\infty$ . Assuming that $\displaystyle f\geq 0$ and $\displaystyle u_{0}\geq 0$ with variable summability, in this paper we show several existence and regularity results. To this aim, we start by reviewing the literature concerning our problem.

Singular elliptic problems have been extensively studied in the literature for the past few decades starting with the now classical work of Crandall-Rabinowitz-Tartar [15], who showed that the stationary state of (1.1), under Dirichlet boundary conditions given by

\displaystyle\displaystyle{}\begin{split}-\Delta u&=\frac{f}{u^{\gamma(x)}}\text{ in }\Omega,\\ u&>0\;\text{ in }\Omega,\\ u&=0\;\text{ in }\partial\Omega.\end{split}

(1.2)

admits a unique solution $\displaystyle u\in C^{2}(\Omega)\cap C(\bar{\Omega})$ for any $\displaystyle\gamma>0$ constant along with the fact that the solution must behave like a distance function near the boundary provided $\displaystyle f$ is Hölder Continuous. Interestingly enough Lazer-Mckenna [28] showed that the unique solution obtained by [15] is indeed in $\displaystyle H_{0}^{1}(\Omega)$ iff $\displaystyle 0<\gamma<3$ . Boccardo-Orsina [9] in a beautiful paper showed that the followings regarding solutions of (1.2)

\begin{cases}u\in W_{0}^{1,\frac{nr(1+\gamma)}{n-r(1-\gamma)}}(\Omega)&\text{ if }0<\gamma<1\text{ and }f\in L^{r}(\Omega)\text{ with }r\in\left[1,\left(2^{*}/(1-\gamma)\right)^{\prime}\right),\\ u\in H_{0}^{1}(\Omega)&\text{ if }0<\gamma<1\text{ and }f\in L^{r}(\Omega)\text{ with }r=\left(2^{*}/(1-\gamma)\right)^{\prime},\\ u\in H_{0}^{1}(\Omega)&\text{ if }\gamma=1\text{ and }f\in L^{1}(\Omega),\\ u^{\frac{1+\gamma}{2}}\in H_{0}^{1}(\Omega)&\text{ if }\gamma>1\text{ and }f\in L^{1}(\Omega),\end{cases}

hold, which was extended for variable $\displaystyle\gamma$ , introducing certain conditions on its behaviour near the boundary in [14]. The nonlocal variant given by

\displaystyle\displaystyle\begin{split}(-\Delta)_{p}^{s}u&=\frac{f}{u^{\gamma(x)}}\text{ in }\Omega,\\ u&>0\text{ in }\Omega,\\ u&=0\text{ in }\partial\Omega.\end{split}

was studied in [7] for $\displaystyle p=2$ and $\displaystyle\gamma(x)=\gamma\in\mathbb{R}_{*}^{+}$ , the authors proved the existence and uniqueness of positive solutions, according to the range of $\displaystyle\gamma$ and summability of $\displaystyle f$ . For the quasilinear case, we refer [13] for constant $\displaystyle\gamma>0$ and for variable singular exponent, the existence results have been obtained in [25]. As for the Mixed local-nonlocal elliptic problem given by

\displaystyle\displaystyle\begin{split}-\Delta_{p}u+(-\Delta)_{p}^{s}u&=\frac{f}{u^{\gamma(x)}}\text{ in }\Omega,\\ u&>0\text{ in }\Omega,\\ u&=0\text{ in }\partial\Omega;\end{split}

Arora [3] for $\displaystyle p=2$ and $\displaystyle\gamma(x)=\gamma\in\mathbb{R}_{*}^{+}$ , obtained the existence, uniqueness and regularity properties of the weak solutions by deriving uniform a priori estimates and using the approximation technique. They also obtained some existence and nonexistence results when $\displaystyle f$ behaves like a distance function. For the case $\displaystyle p>1$ , the constant exponent $\displaystyle\gamma$ case has been considered in [26], and the variable exponent can be found in [8]. If one considers the parabolic counterpart i.e, the equation given by:

\displaystyle\displaystyle\begin{split}u_{t}-\Delta_{p}u&=\frac{f(x,t)}{u^{\gamma}}\text{ in }\Omega_{T},\\ u&=0\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u(x,0)&=u_{0}(x)\text{ in }\Omega.\end{split}

For such an equation with $\displaystyle p\geq 2,\;0\leq f\in L^{m}\left(\Omega_{T}\right)$ with $\displaystyle m\geq 1$ and assuming that $\displaystyle\forall\omega\subset\subset\Omega,\exists\;d_{\omega}>0$ , such that $\displaystyle u_{0}\geq d_{\omega}$ , the authors [17] proved the existence of a solution $\displaystyle u$ of the above problem such that

\displaystyle\displaystyle\begin{cases}u\in L^{q_{0}}(0,T;W_{0}^{1,q_{0}}(\Omega))&\text{ if }0<\gamma<1\text{ and }f\in L^{r}(\Omega)\text{ with }r\in\left[1,\frac{p(n+2)}{p(n+2)-n(1-\gamma)}\right)\\ &\text{ and }q_{0}=\frac{m[n(p+\gamma-1)+p(\gamma+1)]}{n+2-m(1-\gamma)},\\ u\in L^{p}(0,T;W^{1,p}_{0}(\Omega))&\text{ if }0<\gamma\leq 1\text{ and }f\in L^{m_{0}}\left(\Omega_{T}\right)\text{ with }m_{0}=\frac{p(n+2)}{p(n+2)-n(1-\gamma)},\\ u\in L^{p}(0,T;W_{\operatorname{loc}}^{1,p}(\Omega)),&\text{ if }\gamma>1\text{ and }f\in L^{1}(\Omega_{T}).\end{cases}

The Nonlocal case $\displaystyle(s\in(0,1))$ for the parabolic problem was handled by [1] for $\displaystyle\gamma>0$ constant to show existence and uniqueness results along similar lines. If one restricts the range of $\displaystyle\gamma$ then various existence, uniqueness and regularity results can be found in Bal-Badra-Giacomoni [4, 5, 6] and Giacomoni-Bougherera [11]. We would also like to mention that the regularity theory of mixed local and non-local operators plays a major role in our problem and we cite the following papers [16, 18, 21, 22, 23, 24, 30, 31] and the references therein.
As for the boundedness of our solutions, we refer Aronson-Serrin [2], where the summability requirement of initial data for boundedness was introduced by Aronson and Serrin, for the case of second-order differential equations without singularity. Outside of the Aronson-Serrin domain, the optimal summability of solutions for the local case without singularity was obtained in Boccardo-Porzio-Primo [10]. These results for the nonlocal case have been obtained in Peral [29], this too for the nonsingular case. For the mixed local-nonlocal operator with singularity, we will be able to get similar types of results here depending on the choice of $\displaystyle\gamma$ . We will use suitable approximating problems to get the existence and other summability properties of weak solutions.

Organization of the article

In the next section, we will describe some basic notations and fix some preliminary function spaces to define our solutions, followed by embedding results and other properties regarding those spaces.
Then we will introduce the notion of weak solution for the case $\displaystyle\gamma=0$ and show its existence, uniqueness, positivity and other properties. After that, we write about the existence of weak solutions for approximating problems and give definitions of weak solutions for the singular cases, both for constant and variable exponents. We end this section by stating our main theorems regarding the existence and summability of weak solutions and appropriate comments.
The next section contains the proofs of our main results, and we end with another section that gives the asymptotic behaviour of the solutions in a suitable sense.

2 Preliminaries

2.1 Notations

We gather here all the standard notations that will be used throughout the paper.
$\displaystyle\bullet$ We will take $\displaystyle n$ to be the space dimension and denote by $\displaystyle z=(x,t)$ to be a point in $\displaystyle\mathbb{R}^{n}\times(0,T)$ , where $\displaystyle(0,T)\subset\mathbb{R}$ for some $\displaystyle 0<T<\infty$ .
$\displaystyle\bullet$ Let $\displaystyle\Omega$ be an open bounded domain in $\displaystyle\mathbb{R}^{n}$ with boundary $\displaystyle\partial\Omega$ and for $\displaystyle 0<T<\infty$ , let $\displaystyle\Omega_{T}:=\Omega\times(0,T)$ .
$\displaystyle\bullet$ We denote the parabolic boundary $\displaystyle\Gamma_{T}$ by $\displaystyle\Gamma_{T}=(\Omega\times\{t=0\})\cup(\partial\Omega\times(0,T))$ .
$\displaystyle\bullet$ We define the set $\displaystyle(\Omega_{T})_{\delta}=\{(x,t)\in\Omega_{T}:\operatorname{dist}((x,t),\Gamma_{T})<\delta\}$ for $\displaystyle\delta>0$ fixed.
$\displaystyle\bullet$ We shall alternately use $\displaystyle\partial_{t}g$ or $\displaystyle\frac{\partial g}{\partial t}\text{ or }g_{t}$ to denote the time derivative (partial) of a function $\displaystyle g$ .
$\displaystyle\bullet$ For $\displaystyle r>1$ , the Hölder conjugate exponent of $\displaystyle r$ will be denoted by $\displaystyle r^{\prime}=\frac{r}{r-1}$ .
$\displaystyle\bullet$ The Lebesgue measure of a measurable subset $\displaystyle\mathrm{S}\subset\mathbb{R}^{n}$ will be denoted by $\displaystyle|\mathrm{S}|$ .
$\displaystyle\bullet$ For any open subset $\displaystyle\Omega$ of $\displaystyle\mathbb{R}^{n}$ , $\displaystyle K\subset\subset\Omega$ will imply $\displaystyle K$ is compactly contained in $\displaystyle\Omega.$
$\displaystyle\bullet$ $\displaystyle\int$ will denote integration concerning either space or time only, and integration on $\displaystyle\Omega\times\Omega$ or $\displaystyle\mathbb{R}^{n}\times\mathbb{R}^{n}$ will be denoted by a double integral $\displaystyle\iint$ .
$\displaystyle\bullet$ We will use $\displaystyle\iiint$ to denote integral over $\displaystyle\mathbb{R}^{n}\times\mathbb{R}^{n}\times(0,T)$ .
$\displaystyle\bullet$ Average integral will be denoted by $\displaystyle\fint$ .
$\displaystyle\bullet$ The notation $\displaystyle a\lesssim b$ will be used for $\displaystyle a\leq Cb$ , where $\displaystyle C$ is a universal constant which only depends on the dimension $\displaystyle n$ and sometimes on $\displaystyle s$ too. $\displaystyle C$ may vary from line to line or even in the same line.
$\displaystyle\bullet$ $\displaystyle\langle.,.\rangle$ will denote the usual inner product in some associated Hilbert space.
$\displaystyle\bullet$ For any function $\displaystyle h$ , we denote the positive and negative parts of it by $\displaystyle h_{+}=\operatorname{max}\{h,0\}$ and $\displaystyle h_{-}=\operatorname{max}\{-h,0\}$ respectively.
$\displaystyle\bullet$ For $\displaystyle k\in\mathbb{N}$ , we denote $\displaystyle T_{k}(\sigma)=\max\{-k,\min\{k,\sigma\}\}$ , for $\displaystyle\sigma\in\mathbb{R}$ .

2.2 Function Spaces

In this section, we present definitions and properties of some function spaces that will be useful for our work. We recall that for $\displaystyle E\subset\mathbb{R}^{n}$ , the Lebesgue space $\displaystyle L^{p}(E),1\leq p<\infty$ , is defined to be the space of $\displaystyle p$ -integrable functions $\displaystyle u:E\rightarrow\mathbb{R}$ with the finite norm

\|u\|_{L^{p}(E)}=\left(\int_{E}|u(x)|^{p}dx\right)^{1/p}.

By $\displaystyle L_{\operatorname{loc}}^{p}(E)$ we denote the space of locally $\displaystyle p$ -integrable functions, which means, $\displaystyle u\in L_{\operatorname{loc}}^{p}(E)$ if and only if $\displaystyle u\in L^{p}(F)$ for every $\displaystyle F\subset\subset E$ . In the case $\displaystyle 0<p<1$ , we denote by $\displaystyle L^{p}(E)$ a set of measurable functions such that $\displaystyle\int_{E}|u(x)|^{p}dx<\infty$ .

Definition 2.1.

The Sobolev space $\displaystyle W^{1,p}(\Omega)$ , for $\displaystyle 1\leq p<\infty$ , is defined as the Banach space of locally integrable weakly differentiable functions $\displaystyle u:\Omega\rightarrow\mathbb{R}$ equipped with the following norm

\|u\|_{W^{1,p}(\Omega)}=\|u\|_{L^{p}(\Omega)}+\|\nabla u\|_{L^{p}(\Omega)}.

The space $\displaystyle W_{0}^{1,p}(\Omega)$ is defined as the closure of the space $\displaystyle\mathcal{C}_{0}^{\infty}(\Omega)$ , in the norm of the Sobolev space $\displaystyle W^{1,p}(\Omega)$ , where $\displaystyle\mathcal{C}^{\infty}_{0}(\Omega)$ is the set of all smooth functions whose supports are compactly contained in $\displaystyle\Omega$ .

Definition 2.2.

Let $\displaystyle 0<s<1$ and $\displaystyle\Omega$ be a open connected subset of $\displaystyle\mathbb{R}^{n}$ . The fractional Sobolev space $\displaystyle W^{s,q}(\Omega)$ for any $\displaystyle 1\leq q<+\infty$ is defined by

W^{s,q}(\Omega)=\left\{u\in L^{q}(\Omega):\frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{q}+s}}\in L^{q}(\Omega\times\Omega)\right\},

and it is endowed with the norm

{}\|u\|_{W^{s,q}(\Omega)}=\left(\int_{\Omega}|u(x)|^{q}dx+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{q}}{|x-y|^{n+qs}}dxdy\right)^{1/q}.

(2.1)

It can be treated as an intermediate space between $\displaystyle W^{1,q}(\Omega)$ and $\displaystyle L^{q}(\Omega)$ . For $\displaystyle 0<s\leq s^{\prime}<1$ , $\displaystyle W^{s^{\prime},q}(\Omega)$ is continuously embedded in $\displaystyle W^{s,q}(\Omega)$ , see [19, Proposition 2.1]. The fractional Sobolev space with zero boundary values is defined by

W_{0}^{s,q}(\Omega)=\left\{u\in W^{s,q}(\mathbb{R}^{n}):u=0\text{ in }\mathbb{R}^{n}\backslash\Omega\right\}.

However $\displaystyle W_{0}^{s,q}(\Omega)$ can be treated as the closure of $\displaystyle\mathcal{C}^{\infty}_{0}(\Omega)$ in $\displaystyle W^{s,q}(\Omega)$ with respect to the fractional Sobolev norm defined in Eq. 2.1. Both $\displaystyle W^{s,q}(\Omega)$ and $\displaystyle W_{0}^{s,q}(\Omega)$ are reflexive Banach spaces, for $\displaystyle q>1$ , for details we refer to the readers [19, Section 2].
The following result asserts that the classical Sobolev space is continuously embedded in the fractional Sobolev space; see [19, Proposition 2.2]. The idea applies an extension property of $\displaystyle\Omega$ so that we can extend functions from $\displaystyle W^{1,q}(\Omega)$ to $\displaystyle W^{1,q}(\mathbb{R}^{n})$ and that the extension operator is bounded.

Lemma 2.3.

Let $\displaystyle\Omega$ be a smooth bounded domain in $\displaystyle\mathbb{R}^{n}$ and $\displaystyle 0<s<1$ . There exists a positive constant $\displaystyle C=C(\Omega,n,s)$ such that

\|u\|_{W^{s,q}(\Omega)}\leq C\|u\|_{W^{1,q}(\Omega)},

for every $\displaystyle u\in W^{1,q}(\Omega)$ .

For the fractional Sobolev spaces with zero boundary value, the next embedding result follows from [12, Lemma 2.1]. The fundamental difference of it compared to Lemma 2.3 is that the result holds for any bounded domain (without any condition of smoothness of the boundary), since for the Sobolev spaces with zero boundary value, we always have a zero extension to the complement.

Lemma 2.4.

Let $\displaystyle\Omega$ be a bounded domain in $\displaystyle\mathbb{R}^{n}$ and $\displaystyle 0<s<1$ . There exists a positive constant $\displaystyle C=C(n,s,\Omega)$ such that

\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{q}}{|x-y|^{n+qs}}dxdy\leq C\int_{\Omega}|\nabla u|^{q}dx

for every $\displaystyle u\in W_{0}^{1,q}(\Omega)$ . Here, we consider the zero extension of $\displaystyle u$ to the complement of $\displaystyle\Omega$ .

We now proceed with the basic Poincaré inequality, which can be found in [20, Chapter 5, Section 5.8.1].

Lemma 2.5.

Let $\displaystyle\Omega\subset\mathbb{R}^{n}$ be a bounded domain with $\displaystyle\mathcal{C}^{1}$ boundary and $\displaystyle q\geq 1$ . Then there exist a positive constant $\displaystyle C>0$ depending only on $\displaystyle n$ and $\displaystyle\Omega$ , such that

\int_{\Omega}|u|^{q}dx\leq C\int_{\Omega}|\nabla u|^{q}dx,\qquad\forall u\in W^{1,q}_{0}(\Omega).

Specifically if we take $\displaystyle\Omega=B_{r}$ , then we will get for all $\displaystyle u\in W^{1,q}(B_{r})$ ,

\fint_{B_{r}}\left|u-(u)_{B_{r}}\right|^{q}dx\leq cr^{q}\fint_{B_{r}}|\nabla u|^{q}dx,

where $\displaystyle c$ is a constant depending only on $\displaystyle n$ , and $\displaystyle(u)_{B_{r}}$ denotes the average of $\displaystyle u$ in $\displaystyle B_{r}$ , and $\displaystyle B_{r}$ denotes a ball of radius $\displaystyle r$ centered at $\displaystyle x_{0}\in\mathbb{R}^{n}$ . Here, $\displaystyle\fint$ denotes the average integration.

Using Lemma 2.4, and the above Poincaré inequality, we observe that the following norm on the space $\displaystyle W^{1,q}_{0}(\Omega)$ defined by

\|u\|_{W^{1,q}_{0}(\Omega)}=\left(\int_{\Omega}|\nabla u|^{q}dx+\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{q}}{|x-y|^{n+qs}}dxdy\right)^{\frac{1}{q}},

is equivalent to the norm

\|u\|_{W^{1,q}_{0}(\Omega)}=\left(\int_{\Omega}|\nabla u|^{q}dx\right)^{\frac{1}{q}}.

The following is a version of fractional Poincaré.

Lemma 2.6.

Let $\displaystyle\Omega\subset\mathbb{R}^{n}$ be a bounded domain with $\displaystyle\mathcal{C}^{1}$ boundary and let $\displaystyle s\in(0,1)$ and $\displaystyle q\geq 1$ . If $\displaystyle u\in W^{s,q}_{0}(\Omega)$ , then

\int_{\Omega}|u|^{q}dx\leq c\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{q}}{|x-y|^{n+qs}}dxdy,

holds with $\displaystyle c\equiv c(n,s,\Omega)$ .

In view of Lemma 2.6, we observe that the Banach space $\displaystyle W_{0}^{s,q}(\Omega)$ can be endowed with the norm

\|u\|_{W_{0}^{s,q}(\Omega)}=\left(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{q}}{|x-y|^{n+qs}}dxdy\right)^{\frac{1}{q}},

which is equivalent to that of $\displaystyle\|u\|_{W^{s,q}(\Omega)}$ . For $\displaystyle q=2$ , the space $\displaystyle W^{s,2}(\Omega)$ enjoys certain special properties and we denote $\displaystyle W^{s,2}(\Omega)=H^{s}(\Omega)$ and $\displaystyle W_{0}^{s,2}(\Omega)=H_{0}^{s}(\Omega)$ . Endowed with the inner product

\langle u,v\rangle_{H_{0}^{s}(\Omega)}=\int_{\Omega}\int_{\Omega}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}dxdy,

we note that $\displaystyle(H_{0}^{s}(\Omega),\|\cdot\|_{H_{0}^{s}(\Omega)})$ is a Hilbert space. Similar thing holds for the space $\displaystyle W^{1,2}_{0}(\Omega)=H^{1}_{0}(\Omega).$

Definition 2.7.

The space $\displaystyle X_{0}^{s}(\Omega)$ is defined as

X_{0}^{s}(\Omega)=\left\{f\in H^{s}(\mathbb{R}^{n})\text{ s.t. }f=0\text{ a.e. in }\mathcal{C}\Omega\right\},

where $\displaystyle\mathcal{C}\Omega=\mathbb{R}^{n}\backslash\Omega$ , and is endowed with the norm

\|u\|_{X_{0}^{s}(\Omega)}=\left(\int_{Q}\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}dxdy\right)^{\frac{1}{2}},

where $\displaystyle Q:=\mathbb{R}^{2n}\backslash(\mathcal{C}\Omega\times\mathcal{C}\Omega)$ .

Moreover $\displaystyle W^{-1,q^{\prime}}(\Omega)$ , $\displaystyle W^{-s,q^{\prime}}(\Omega)$ and $\displaystyle X^{-s}(\Omega)$ are defined to be the dual spaces of $\displaystyle W_{0}^{1,q}(\Omega)$ , $\displaystyle W_{0}^{s,q}(\Omega)$ and $\displaystyle X_{0}^{s}(\Omega)$ respectively, where $\displaystyle q^{\prime}:=\frac{q}{q-1}$ . Now, we define the local spaces as

W_{\operatorname{loc}}^{1,q}(\Omega)=\left\{u:\Omega\rightarrow\mathbb{R}:u\in L^{q}(K),\int_{K}|\nabla u|^{q}dx<\infty,\text{ for every }K\subset\subset\Omega\right\},

and

W_{\operatorname{loc}}^{s,q}(\Omega)=\left\{u:\Omega\rightarrow\mathbb{R}:u\in L^{q}(K),\int_{K}\int_{K}\frac{|u(x)-u(y)|^{q}}{|x-y|^{n+qs}}dxdy<\infty,\text{ for every }K\subset\subset\Omega\right\}.

Now for $\displaystyle n>2$ , we define the critical Sobolev exponent as $\displaystyle 2^{*}=\frac{2n}{n-2}$ , then we get the following embedding result for any open subset $\displaystyle\Omega$ of $\displaystyle\mathbb{R}^{n}$ , see for details [20, Chapter 5],

Theorem 2.8.

Let $\displaystyle n>2$ . Then, there exists a constant $\displaystyle C$ depending only on $\displaystyle n$ and $\displaystyle\Omega$ , such that for all $\displaystyle u\in\mathcal{C}_{0}^{\infty}(\Omega)$

\|u\|_{L^{2^{*}}(\Omega)}^{2}\leq C\int_{\Omega}|\nabla u|^{2}dx.

Similarly, for $\displaystyle n>2s$ , we define the fractional Sobolev critical exponent as $\displaystyle 2_{s}^{*}=\frac{2n}{n-2s}$ . The following result is a fractional version of the Sobolev inequality(Theorem 2.8) which also implies a continuous embedding of $\displaystyle H_{0}^{s}(\Omega)$ in the critical Lebesgue space $\displaystyle L^{2_{s}^{*}}(\Omega)$ . One can see the proof in [19].

Theorem 2.9.

Let $\displaystyle 0<s<1$ be such that $\displaystyle n>2s$ . Then, there exists a constant $\displaystyle S(n,s)$ depending only on $\displaystyle n$ and $\displaystyle s$ , such that for all $\displaystyle u\in\mathcal{C}_{0}^{\infty}(\Omega)$

\|u\|_{L^{2_{s}^{*}}(\Omega)}^{2}\leq S(n,s)\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}dxdy.

We now recall the Gagliardo-Nirenberg interpolation inequality that will be useful for proving the boundedness of weak solutions.

Theorem 2.10.

(Gagliardo-Nirenberg) Let $\displaystyle 1\leq q<+\infty$ be a positive real number. Let $\displaystyle j$ and $\displaystyle m$ be non-negative integers such that $\displaystyle j<m$ . Furthermore, let $\displaystyle 1\leq r\leq+\infty$ be a positive extended real quantity, $\displaystyle p\geq 1$ be real and $\displaystyle\theta\in[0,1]$ such that the relations

\frac{1}{p}=\frac{j}{n}+\theta\left(\frac{1}{r}-\frac{m}{n}\right)+\frac{1-\theta}{q},\quad\frac{j}{m}\leq\theta\leq 1

hold. Then,

\left\|D^{j}u\right\|_{L^{p}\left(\mathbb{R}^{n}\right)}\leq C\left\|D^{m}u\right\|_{L^{r}\left(\mathbb{R}^{n}\right)}^{\theta}\|u\|_{L^{q}\left(\mathbb{R}^{n}\right)}^{1-\theta}

for any $\displaystyle u\in L^{q}(\mathbb{R}^{n})$ such that $\displaystyle D^{m}u\in L^{r}(\mathbb{R}^{n})$ . Here, the constant $\displaystyle C>0$ depends on the parameters $\displaystyle j,m,n,q,r,\theta$ , but not on $\displaystyle u$ .

The article will extensively use the embedding results and corresponding inequalities. Now, we need to deal with spaces involving time for the parabolic equations, so we introduce them here. As in the classical case, we define the corresponding Bochner spaces as the following

\begin{array}[]{c}L^{q}(0,T;W_{0}^{1,q}(\Omega))=\left\{u\in L^{q}(\Omega\times(0,T)),\|u\|_{L^{q}(0,T;W_{0}^{1,q}(\Omega))}<\infty\right\},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ L^{q}(0,T;W_{0}^{s,q}(\Omega))=\left\{u\in L^{q}(\Omega\times(0,T)),\|u\|_{L^{q}(0,T;W_{0}^{s,q}(\Omega))}<\infty\right\},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ L^{2}(0,T;X_{0}^{s}(\Omega))=\left\{u\in L^{2}(\mathbb{R}^{n}\times(0,T)),\|u\|_{L^{2}(0,T;X^{s}_{0}(\Omega))}<\infty\right\},\end{array}

where

\begin{array}[]{c}\|u\|_{L^{q}(0,T;W_{0}^{1,q}(\Omega))}=\left(\int_{0}^{T}\int_{\Omega}|\nabla u|^{q}dxdt\right)^{\frac{1}{q}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \|u\|_{L^{q}(0,T;W_{0}^{s,q}(\Omega))}=\left(\int_{0}^{T}\int_{\Omega}\int_{\Omega}\frac{|u(x,t)-u(y,t)|^{q}}{|x-y|^{n+qs}}dxdydt\right)^{\frac{1}{q}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \|u\|_{L^{2}(0,T;X_{0}^{s}(\Omega))}=\left(\int_{0}^{T}\int_{Q}\frac{|u(x,t)-u(y,t)|^{2}}{|x-y|^{n+2s}}dxdydt\right)^{\frac{1}{2}},\end{array}

with their dual spaces $\displaystyle L^{q^{\prime}}(0,T;W^{-1,q^{\prime}}(\Omega))$ , $\displaystyle L^{q^{\prime}}(0,T;W^{-s,q^{\prime}}(\Omega))$ and $\displaystyle L^{2}(0,T;X^{-s}(\Omega))$ respectively. Again, the local spaces are defined as

L^{2}(0,T;H_{loc}^{1}(\Omega))=\left\{u\in L^{2}(K\times(0,T)):\int_{0}^{T}\int_{K}|\nabla u|^{q}dxdt<\infty,\mbox{ for every }K\subset\subset\Omega\right\},

and

L^{2}(0,T;H_{loc}^{s}(\Omega))=\left\{u\in L^{2}(K\times(0,T)):\int_{0}^{T}\int_{K}\int_{K}\frac{|u(x,t)-u(y,t)|^{2}}{|x-y|^{n+2s}}dxdydt<\infty,\mbox{ for every }K\subset\subset\Omega\right\}.

We now recall the following algebraic inequality that can be found in [1, Lemma 2.22].

Lemma 2.11.

i)

Let $\displaystyle\alpha>0$ . For every $\displaystyle x,y\geq 0$ one has

$(x-y)(x^{\alpha}-y^{\alpha})\geq\frac{4\alpha}{(\alpha+1)^{2}}\left(x^{\frac{\alpha+1}{2}}-y^{\frac{\alpha+1}{2}}\right)^{2}.$
ii)

Let $\displaystyle 0<\alpha\leq 1$ . For every $\displaystyle x,y\geq 0$ with $\displaystyle x\neq y$ one has

$\frac{x-y}{x^{\alpha}-y^{\alpha}}\leq\frac{1}{\alpha}(x^{1-\alpha}+y^{1-\alpha}).$
iii)

Let $\displaystyle\alpha\geq 1$ . Then there exists a constant $\displaystyle C_{\alpha}$ depending only on $\displaystyle\alpha$ such that

$|x+y|^{\alpha-1}|x-y|\leq C_{\alpha}\left|x^{\alpha}-y^{\alpha}\right|.$

2.3 Weak Solutions

In this subsection, along with the next subsection, we will introduce notions of very weak solutions to our problem and state the main results that we are going to prove. We begin with the definitions of weak solutions for the nonsingular case. We first take $\displaystyle f$ and $\displaystyle u_{0}$ to be in $\displaystyle L^{2}$ spaces and then relax the condition. We also state some important properties of the weak solutions that we need to use in the rest of the article. Further, we will introduce suitable approximating problems and properties of their solutions.

Definition 2.12.

Assume $\displaystyle(f,u_{0})\in L^{2}(\Omega_{T})\times L^{2}(\Omega)$ , then we say that $\displaystyle u$ is an energy solution to problem

{}\begin{array}[]{lcr}\begin{cases}u_{t}-\Delta u+(-\Delta)^{s}u={f(x,t)}&\mbox{ in }\Omega_{T},\\ u=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u(x,0)=u_{0}(x)&\mbox{ in }\Omega;\end{cases}\end{array}

(2.2)

if $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))\cap\mathcal{C}([0,T],L^{2}(\Omega)),u_{t}\in L^{2}(0,T;H^{-1}(\Omega))$ , and for all $\displaystyle\phi\in L^{2}(0,T;H_{0}^{1}(\Omega))$ we have

\int_{0}^{T}\left\langle u_{t},\phi\right\rangle dt+\int_{0}^{T}\int_{\Omega}\nabla u\cdot\nabla\phi dxdt+\frac{1}{2}\int_{0}^{T}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{(u(x,t)-u(y,t))(\phi(x,t)-\phi(y,t))}{|x-y|^{n+2s}}dxdydt=\int_{0}^{T}\int_{\Omega}f\phi dxdt

and $\displaystyle u(\cdot,t)\rightarrow u_{0}$ strongly in $\displaystyle L^{2}(\Omega)$ , as $\displaystyle t\rightarrow 0$ .

We denote

\mathcal{E}(u(x,t),\phi(x,t)):=\frac{1}{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}(u(x,t)-u(y,t))(\phi(x,t)-\phi(y,t))\times K(x,y,t)dxdy,

where $\displaystyle K(x,y,t)=\frac{1}{{{\left|x-y\right|}^{n+2s}}}.$ Following the way for fractional Laplacian in [29], we give the proof of existence for mixed local-nonlocal case for the sake of completeness.

Theorem 2.13.

There exists a solution to problem Eq. 2.2 in the sense of Definition 2.12. Moreover, if $\displaystyle f$ is also a nonnegative function and $\displaystyle u_{0}\geq 0$ , then the solution is also nonnegative.

Proof.

Let us denote $\displaystyle\mathcal{C}_{*}^{\infty}(\Omega\times[0,T])$ as the $\displaystyle\mathcal{C}^{\infty}(\Omega\times[0,T])$ functions that vanish in $\displaystyle(\mathbb{R}^{n}\backslash\Omega)\times[0,T]$ and in $\displaystyle\Omega\times\{T\}$ . Choosing $\displaystyle\phi\in\mathcal{C}_{*}^{\infty}(\Omega\times[0,T])$ , for $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))$ , we define the operator

L_{\phi}(u):=\int_{0}^{T}\int_{\Omega}-u\phi_{t}dxdt+\int_{0}^{T}\int_{\Omega}\nabla u\cdot\nabla\phi dxdt+\int_{0}^{T}\mathcal{E}(u(x,t),\phi(x,t))dt.

We now observe that $\displaystyle u$ is an energy solution to Eq. 2.2 with $\displaystyle f\in L^{2}(\Omega_{T})\subset L^{2}(0,T;H^{-1}(\Omega))$ if and only if

L_{\phi}(u)=\int_{0}^{T}\langle f,\phi\rangle dt+\int_{\Omega}u(x,0)\phi(x,0)dx,

where $\displaystyle\langle f,\phi\rangle$ denote the usual inner product of $\displaystyle f$ and $\displaystyle\phi$ in $\displaystyle L^{2}(\Omega)$ or the pairing of $\displaystyle f$ and $\displaystyle\phi$ between $\displaystyle H^{-1}(\Omega)$ and $\displaystyle H^{1}_{0}(\Omega)$ .
We also define the following inner product, for $\displaystyle\phi,\varphi\in\mathcal{C}_{*}^{\infty}(\Omega\times[0,T])$

{}\langle\varphi,\phi\rangle_{*}=\frac{1}{2}\langle\varphi(x,0),\phi(x,0)\rangle_{L^{2}(\Omega)}+\int_{0}^{T}\int_{\Omega}\nabla\phi\cdot\nabla\varphi~{}dxdt+\int_{0}^{T}\mathcal{E}(\varphi(x,t),\phi(x,t))dt,

(2.3)

and denote by $\displaystyle H^{*}(\Omega\times[0,T])$ the Hilbert space built as the completion of $\displaystyle\mathcal{C}_{*}^{\infty}(\Omega\times[0,T])$ with the norm $\displaystyle\|\phi\|_{*}$ induced by the inner product Eq. 2.3.
Now $\displaystyle L_{\phi}$ is clearly a linear functional in $\displaystyle H^{*}(\Omega\times[0,T])$ and for any $\displaystyle\varphi\in L^{2}(0,T;H_{0}^{1}(\Omega))$ , by Hölder and Sobolev inequalities, we have

\left|L_{\phi}(\varphi)\right|\leq c_{\phi}\left(\|\varphi\|_{L^{2}(0,T;L^{2}(\Omega))}+\|\varphi\|_{L^{2}(0,T,H_{0}^{1}(\Omega))}+\|\varphi\|_{L^{2}(0,T,H_{0}^{s}(\Omega))}\right)\leq\tilde{c}_{\phi}\|\varphi\|_{*}.

Therefore, $\displaystyle L_{\phi}$ is a bounded linear functional in $\displaystyle H^{*}(\Omega\times[0,T])$ , and hence by the Fréchet-Riesz Theorem, there exists $\displaystyle\mathcal{T}\phi\in H^{*}(\Omega\times[0,T])$ such that

L_{\phi}(\varphi)=\langle\varphi,\mathcal{T}\phi\rangle_{*}\quad\text{ for all }\varphi\in H^{*}(\Omega\times[0,T]).

It is trivial to show that $\displaystyle\mathcal{T}$ is a linear operator in $\displaystyle H^{*}(\Omega\times[0,T])$ . Moreover

L_{\phi}(\phi)=\frac{1}{2}\int_{\Omega}\phi^{2}(x,0)dx+\int_{0}^{T}\int_{\Omega}|\nabla\phi|^{2}dxdt+\int_{0}^{T}\mathcal{E}(\phi(x,t),\phi(x,t))dt=\|\phi\|_{*}^{2},

and consequently, $\displaystyle\langle\phi,\mathcal{T}\phi\rangle_{*}=\|\phi\|_{*}^{2}$ . Then we get by the Cauchy-Schwartz inequality,

\|\phi\|_{*}^{2}\leq\|\phi\|_{*}\|\mathcal{T}\phi\|_{*},\quad\text{ i.e., }\quad\|\phi\|_{*}\leq\|\mathcal{T}\phi\|_{*}.

Therefore, this implies that $\displaystyle\mathcal{T}$ is injective and hence bijective on its range, and its inverse $\displaystyle\mathcal{T}^{-1}$ has a norm less than or equal to $\displaystyle 1$ and can be extended to the $\displaystyle\operatorname{closure}M$ of $\displaystyle\operatorname{Range}(\mathcal{T})$ .
Now, on the other hand, we define

B_{u_{0},f}(\phi):=\int_{\Omega}u_{0}\phi(x,0)dx+\int_{0}^{T}\int_{\Omega}\phi fdxdt.

Denoting $\displaystyle\phi_{0}:=\phi(x,0)$ , we get by Hölder inequality

\left|B_{u_{0},f}(\phi)\right|\leq\left\|u_{0}\right\|_{L^{2}(\Omega)}\left\|\phi_{0}\right\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(0,T;L^{2}(\Omega))}\|\phi\|_{L^{2}(0,T;L^{2}(\Omega))}\leq c_{u_{0},f}\|\phi\|_{*},

and thus,

\left|B_{u_{0},f}(\mathcal{T}^{-1}\psi)\right|\leq c\left\|\mathcal{T}^{-1}\psi\right\|_{*}\leq c\|\psi\|_{*}.

Since $\displaystyle B_{u_{0},f}$ and $\displaystyle\mathcal{T}^{-1}$ both are linear, therefore their composition is also so, and by above line, $\displaystyle B_{u_{0},f}\circ\mathcal{T}^{-1}$ is bounded too, therefore, by applying the Fréchet-Riesz Theorem again, there exists a unique $\displaystyle u\in M$ such that $\displaystyle B_{u_{0},f}(\mathcal{T}^{-1}\psi)=\langle\psi,u\rangle_{*}$ for every $\displaystyle\psi\in M$ . We denote $\displaystyle\phi=\mathcal{T}^{-1}\psi$ and so

B_{u_{0},f}(\phi)=\langle\mathcal{T}\phi,u\rangle_{*}=L_{\phi}(u)

that is,

\begin{array}[]{c}\int_{0}^{T}\int_{\Omega}-u\phi_{t}dxdt+\int_{0}^{T}\int_{\Omega}\nabla u\cdot\nabla\phi dxdt+\int_{0}^{T}\mathcal{E}(u(x,t),\phi(x,t))dt=\int_{0}^{T}\int_{\Omega}f\phi dxdt+\int_{\Omega}u(x,0)\phi(x,0)dx,\end{array}

where $\displaystyle\phi\in L^{2}(0,T;H_{0}^{s}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))\equiv L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle\phi_{t}\in L^{2}(0,T;H^{-1}(\Omega))$ . Finally, by a density argument, one can conclude, integrating by parts, that $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))$ , $\displaystyle u_{t}\in L^{2}(0,T;H^{-1}(\Omega))$ , and

\int_{0}^{T}\int_{\Omega}u_{t}\phi dxdt+\int_{0}^{T}\int_{\Omega}\nabla u\cdot\nabla\phi dxdt+\int_{0}^{T}\mathcal{E}(u(x,t),\phi(x,t))dt=\int_{0}^{T}\int_{\Omega}f\phi dxdt.

Since $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))$ , $\displaystyle u_{t}\in L^{2}(0,T;H^{-1}(\Omega))$ implies that $\displaystyle u\in\mathcal{C}([0,T],L^{2}(\Omega))$ and so we have $\displaystyle u(\cdot,t)\rightarrow u_{0}$ strongly in $\displaystyle L^{2}(\Omega)$ , as $\displaystyle t\rightarrow 0$ . Thus $\displaystyle u(x,t)$ is an energy solution of Eq. 2.2.
Now we show that $\displaystyle u\geq 0$ provided that $\displaystyle f$ and $\displaystyle u_{0}$ are nonnegative. We write $\displaystyle u=u_{+}-u_{-}$ , where $\displaystyle u_{+}=\operatorname{max}{\{u,0}\}\chi_{\Omega}$ and $\displaystyle u_{-}=\operatorname{max}{\{-u,0}\}\chi_{\Omega}$ . We take $\displaystyle\phi=u_{-}\chi_{(0,\tilde{t})},\tilde{t}>0$ , as a test function. Since $\displaystyle f\geq 0$ , and $\displaystyle\phi\geq 0$ , we have

{}0\leq\int_{0}^{T}\int_{\Omega}f\phi dxdt=\int_{0}^{\tilde{t}}\int_{\Omega}fu_{-}dxdt.

(2.4)

On the other hand, since $\displaystyle u$ is $\displaystyle 0$ in $\displaystyle(\mathbb{R}^{n}\backslash\Omega)\times(0,T)$ , we have that

\begin{array}[]{c}\int_{0}^{T}\iint_{\mathbb{R}^{2n}}(u(x,t)-u(y,t))(\phi(x,t)-\phi(y,t))K(x,y,t)dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ =\int_{0}^{\tilde{t}}\iint_{\mathbb{R}^{2n}\backslash(\Omega^{c}\times\Omega^{c})}(u(x,t)-u(y,t))(u_{-}(x,t)-u_{-}(y,t))K(x,y,t)dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ =\int_{0}^{\tilde{t}}\int_{\Omega}\int_{\Omega}(u(x,t)-u(y,t))(u_{-}(x,t)-u_{-}(y,t))K(x,y,t)dxdydt+2\int_{0}^{\tilde{t}}\int_{\Omega}\int_{\Omega^{c}}u(x,t)u_{-}(x,t)K(x,y,t)dydxdt.\end{array}

Moreover, $\displaystyle(u_{+}(x,t)-u_{+}(y,t))(u_{-}(x,t)-u_{-}(y,t))\leq 0$ , and thus

\begin{array}[]{l}\int_{0}^{\tilde{t}}\int_{\Omega}\int_{\Omega}(u(x,t)-u(y,t))(u_{-}(x,t)-u_{-}(y,t))K(x,y,t)dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq-\int_{0}^{\tilde{t}}\int_{\Omega}\int_{\Omega}(u_{-}(x,t)-u_{-}(y,t))^{2}K(x,y,t)dxdydt\leq 0.\end{array}

Further, we have

\int_{0}^{\tilde{t}}\int_{\Omega}\int_{\Omega^{c}}u(x,t)u_{-}(x,t)K(x,y,t)dydxdt=-\int_{0}^{\tilde{t}}\int_{\Omega}\int_{\Omega^{c}}u_{-}^{2}(x,t)K(x,y,t)dydxdt\leq 0.

Therefore, we have shown that

\int_{0}^{T}\iint_{\mathbb{R}^{2n}}(u(x,t)-u(y,t))(\phi(x,t)-\phi(y,t))K(x,y,t)dxdydt\leq 0,

Similarly since $\displaystyle\nabla u_{+}\cdot\nabla u_{-}=0$ , we get

\int_{0}^{T}\int_{\Omega}\nabla u\cdot\nabla\phi dxdt=\int_{0}^{\tilde{t}}\int_{\Omega}\nabla u_{+}\cdot\nabla u_{-}dxdt-\int_{0}^{\tilde{t}}\int_{\Omega}|\nabla u_{-}|^{2}dxdt\leq 0.

Now since $\displaystyle u_{0}\geq 0$ , so $\displaystyle u_{-}(\cdot,0)\equiv 0$ , and we get

\begin{array}[]{rcl}\int_{0}^{T}\left\langle u_{t},\phi\right\rangle dt&=&\int_{0}^{\tilde{t}}\int_{\Omega}\frac{\partial u}{\partial t}u_{-}dxdt=\int_{\Omega}\int_{u(x,0)}^{u(x,\tilde{t})}\sigma_{-}d\sigma dx=-\frac{1}{2}\int_{\Omega}(u_{-}(x,{\tilde{t}}))^{2}dx.\end{array}

Combining the above three inequalities, we get from Eq. 2.4 that

\begin{array}[]{rcl}0\leq\int_{0}^{T}\int_{\Omega}f\phi dxdt&=&\int_{0}^{T}\left\langle u_{t},\phi\right\rangle dt+\int_{0}^{T}\int_{\Omega}\nabla u\cdot\nabla\phi dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &&+\frac{1}{2}\int_{0}^{T}\iint_{\mathbb{R}^{2n}}\frac{(u(x,t)-u(y,t))(\phi(x,t)-\phi(y,t))}{|x-y|^{n+2s}}dxdydt\leq-\frac{1}{2}\int_{\Omega}(u_{-}(x,{\tilde{t}}))^{2}dx\leq 0,\end{array}

and this gives that $\displaystyle||u_{-}(\cdot,\tilde{t})||_{L^{2}(\Omega)}=0$ for each $\displaystyle\tilde{t}>0$ . Therefore $\displaystyle u_{-}\equiv 0$ . So we conclude that $\displaystyle u\geq 0$ . We observe that this comparison result also guarantees the uniqueness of energy solution to Eq. 2.2. ∎

Remark 2.14.

Observing that $\displaystyle(u(x,t)-u(y,t))((u(x,t)-k)_{+}-(u(y,t)-k)_{+})\geq 0$ , for each $\displaystyle k$ , we can show that for $\displaystyle(f,u_{0})\in L^{\infty}(\Omega_{T})\times L^{\infty}(\Omega)$ , the weak solution $\displaystyle u\in L^{\infty}(\Omega_{T})$ . The proof will follow exactly similar to that of [32, Theorem 4.2.1].

Now we relax the spaces where $\displaystyle f$ and $\displaystyle u_{0}$ lie. For the case of $\displaystyle L^{1}$ data, we consider the set

\begin{array}[]{rl}\mathcal{T}:=&\{\phi:\Omega\times[0,T]\rightarrow\mathbb{R},\text{ s.t. }-\phi_{t}-\Delta\phi+(-\Delta)^{s}\phi=\varphi\text{ in }\Omega_{T},\\ &\varphi\in\mathcal{C}_{0}^{\infty}(\Omega_{T}),\phi\in L^{\infty}(\Omega\times(0,T))\cap\mathcal{C}_{\operatorname{loc}}^{\alpha,\beta}(\Omega\times(0,T)),\\ &\phi(x,0)\in L^{\infty}(\Omega),\phi=0\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T],\phi(x,T)=0\text{ in }\Omega\},\end{array}

where $\displaystyle\alpha,\beta\in(0,1)$ .

Definition 2.15.

Let $\displaystyle(f,u_{0})\in L^{1}(\Omega_{T})\times L^{1}(\Omega)$ be nonnegative functions. Then $\displaystyle u\in L^{1}(\Omega_{T})$ is a very weak solution to Eq. 2.2 if we have

\iint_{\Omega_{T}}u\left(-\phi_{t}-\Delta\phi+(-\Delta)^{s}\phi\right)dxdt=\iint_{\Omega_{T}}f\phi dxdt+\int_{\Omega}u_{0}(x)\phi(x,0)dx,\quad\forall\phi\in\mathcal{T}.

The next existence result is following the lines of [29].

Theorem 2.16.

For $\displaystyle(f,u_{0})\in L^{1}(\Omega_{T})\times L^{1}(\Omega)$ being nonnegative, Eq. 2.2 has a unique nonnegative very weak solution $\displaystyle u$ in the sense of Definition 2.15.

Proof.

Firstly, we observe that the existence of valid test functions is guaranteed by the result in [16], [21], [30]. We will now obtain the solution as a limit of solutions to approximated problems. Let $\displaystyle u_{m}\in L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(\Omega_{T})$ be the solution (exists by Theorem 2.13) to the approximated problem

\displaystyle\displaystyle\begin{cases}(u_{m})_{t}-\Delta u_{m}+(-\Delta)^{s}u_{m}=f_{m}&\text{ in }\Omega_{T},\\ u_{m}(x,t)=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u_{m}(x,0)=u_{0m}(x)&\text{ in }\Omega;\end{cases}

where $\displaystyle f_{m}=T_{m}(f(x,t))$ and $\displaystyle u_{0m}=T_{m}(u_{0}(x))$ are $\displaystyle L^{\infty}$ functions. Using $\displaystyle(T_{k}(u_{m}))\chi_{(0,t)}$ , for $\displaystyle t>0$ (admissible by [29, Proposition 3]) as a test function in the approximated problem, it holds that,

{}\begin{array}[]{l}\int_{\Omega}L_{k}(u_{m}(x,t))dx+\int_{0}^{t}\int_{\Omega}\nabla u_{m}\cdot\nabla T_{k}(u_{m})dxd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{\left(T_{k}\left(u_{m}(x,\theta)\right)-T_{k}\left(u_{m}(y,\theta)\right)\right)\left(u_{m}(x,\theta)-u_{m}(y,\theta)\right)}{|x-y|^{n+2s}}dxdyd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq k\|f\|_{L^{1}(\Omega_{T})}+C_{3}(k)\left\|u_{0}\right\|_{L^{1}(\Omega)}+C_{4}(k)|\Omega|,\end{array}

(2.5)

where $\displaystyle L_{k}(\rho)=\int_{0}^{\rho}\left(T_{k}(\xi)\right)d\xi.$ Notice that

\left(T_{k}\left(u_{m}(x,\theta)\right)-T_{k}\left(u_{m}(y,\theta)\right)\right)\left(u_{m}(x,\theta)-u_{m}(y,\theta)\right)\geq\left(T_{k}\left(u_{m}(x,\theta)\right)-T_{k}\left(u_{m}(y,\theta)\right)\right)^{2}

and so

\nabla u_{m}\cdot\nabla T_{k}(u_{m})\geq|\nabla T_{k}(u_{m})|^{2}

and for $\displaystyle\rho>0$ we have

C_{1}(k)\rho-C_{2}(k)\leq L_{k}(\rho)\leq C_{3}(k)\rho+C_{4}(k)

and

L_{k}(\rho)\geq C\left(T_{k}(\rho)\right)^{2},

where $\displaystyle C_{1},C_{2},C_{3},C_{4}$ are constants depending only on $\displaystyle k$ and independent of $\displaystyle m$ . Therefore taking supremum over $\displaystyle t\in(0,T]$ in Eq. 2.5, we get that $\displaystyle\left\{T_{k}(u_{m})\right\}_{m}$ is bounded in $\displaystyle L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H_{0}^{s}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle\left\{u_{m}\right\}_{m}$ is bounded in $\displaystyle L^{\infty}(0,T;L^{1}(\Omega))\subset L^{1}(\Omega_{T})$ .
Now since by comparison principle proved in Theorem 2.13, $\displaystyle\left\{u_{m}\right\}_{m}$ is increasing in $\displaystyle m$ , we get the existence of a measurable function $\displaystyle u$ such that $\displaystyle T_{k}(u)\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H_{0}^{s}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))$ , $\displaystyle u_{m}\uparrow u$ strongly in $\displaystyle L^{1}(\Omega_{T})$ and $\displaystyle u_{m}\uparrow u$ a.e in $\displaystyle\Omega_{T}$ . As each $\displaystyle u_{m}$ is an energy solution, therefore, $\displaystyle u_{m}\in\mathcal{C}([0,T],L^{2}(\Omega))\subset\mathcal{C}([0,T],L^{1}(\Omega))$ and at each time level $\displaystyle t\in[0,T]$ , we have $\displaystyle u_{m}(\cdot,t)\in L^{1}(\Omega)$ , this along with the monotonicity of $\displaystyle\{u_{m}\}_{m}$ in $\displaystyle m$ allows us to define the pointwise limit (a.e.) $\displaystyle u$ of $\displaystyle\{u_{m}\}_{m}$ in $\displaystyle\Omega$ for each time $\displaystyle t\in[0,T]$ . Also $\displaystyle u$ satisfies $\displaystyle u(\cdot,0)=u_{0}(\cdot)$ in $\displaystyle L^{1}$ sense. Again as each $\displaystyle u_{m}=0$ in $\displaystyle(\mathbb{R}^{n}\backslash\Omega)\times(0,T)$ , therefore $\displaystyle u$ also satisfies the same. We now prove that $\displaystyle u$ is a weak solution to Eq. 2.2 in the sense of Definition 2.15. Let $\displaystyle\phi\in\mathcal{T}$ , then as $\displaystyle u_{m}$ is the energy solution to the approximated problem, we have

\iint_{\Omega_{T}}\left((u_{m})_{t}-\Delta u_{m}+(-\Delta)^{s}u_{m}\right)\phi dxdt=\iint_{\Omega_{T}}f_{m}\phi dxdt.

Using the fact that $\displaystyle u_{m}\rightarrow u$ strongly in $\displaystyle L^{1}(\Omega_{T})$ and $\displaystyle u_{m}(x,0)=u_{m0}(x)\rightarrow u_{0}(x)=u(x,0)$ in $\displaystyle L^{1}(\Omega)$ , we have

\begin{array}[]{rcl}\iint_{\Omega_{T}}\left((u_{m})_{t}-\Delta u_{m}+(-\Delta)^{s}u_{m}\right)\phi dxdt&=&\iint_{\Omega_{T}}u_{m}\left(-(\phi)_{t}-\Delta\phi+(-\Delta)^{s}\phi\right)dxdt-\int_{\Omega}u_{m}(x,0)\phi(x,0)dx\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &=&\iint_{\Omega_{T}}u_{m}\varphi dxdt-\int_{\Omega}u_{m}(x,0)\phi(x,0)dx\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\rightarrow&\iint_{\Omega_{T}}u\varphi dxdt-\int_{\Omega}u(x,0)\phi(x,0)dx\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &=&\iint_{\Omega_{T}}u\left(-(\phi)_{t}-\Delta\phi+(-\Delta)^{s}\phi\right)dxdt-\int_{\Omega}u(x,0)\phi(x,0)dx.\end{array}

Notice that in the second last line, in order to pass the limit, we have used the facts that $\displaystyle\phi(x,0)\in L^{\infty}(\Omega)$ and $\displaystyle\varphi\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ . Also, since $\displaystyle\phi\in L^{\infty}(\Omega\times(0,T))$ , and $\displaystyle f_{m}\rightarrow f$ in $\displaystyle L^{1}(\Omega_{T})$ , we get

\iint_{\Omega_{T}}f_{m}\phi dxdt\rightarrow\iint_{\Omega_{T}}f\phi dxdt\text{ as }m\rightarrow\infty.

Thus

\iint_{\Omega_{T}}u\left(-(\phi)_{t}-\Delta\phi+(-\Delta)^{s}\phi\right)dxdt=\iint_{\Omega_{T}}f\phi dxdt+\int_{\Omega}u_{0}(x)\phi(x,0)dx,

and $\displaystyle u$ is a weak solution to Eq. 2.2. For the uniqueness let $\displaystyle w$ be a weak solution of Eq. 2.2 with $\displaystyle(f,u_{0})=(0,0)$ , i.e.

\displaystyle\displaystyle\begin{cases}w_{t}-\Delta w+(-\Delta)^{s}w=0&\text{ in }\Omega_{T},\\ w=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ w(x,0)=0&\text{ in }\Omega;\end{cases}

we want to prove that $\displaystyle w\equiv 0$ . For that we take $\displaystyle F\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ , and let $\displaystyle\phi_{F}$ be the solution of the backward problem

\displaystyle\displaystyle\begin{cases}-(\phi_{F})_{t}-\Delta\phi_{F}+(-\Delta)^{s}\phi_{F}=F&\text{ in }\Omega_{T},\\ \phi_{F}(x,t)=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T],\\ \phi_{F}(x,T)=0&\text{ in }\Omega.\end{cases}

Taking $\displaystyle\phi_{F}$ as a test function we deduce that for any $\displaystyle F\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ ,

\int_{0}^{T}\int_{\Omega}wFdxdt=0,

that means, $\displaystyle w=0$ in $\displaystyle\mathcal{D}^{\prime}(\Omega_{T})$ . ∎

Next, we state some important facts regarding the solution of Eq. 2.2.

Proposition 2.17.

Let $\displaystyle\left(f,w_{0}\right)$ are non-negative functions such that $\displaystyle\left(f,w_{0}\right)\in L^{\infty}(\Omega_{T})\times L^{\infty}(\Omega)$ . Assume that $\displaystyle w$ be the weak solution to the problem

\displaystyle\displaystyle\begin{cases}w_{t}-\Delta w+(-\Delta)^{s}w=f&\text{ in }\Omega_{T},\\ w=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ w(x,0)=w_{0}(x)&\text{ in }\Omega;\end{cases}

then for all $\displaystyle 0<t_{1}<T$ and for each $\displaystyle\omega\subset\subset\Omega$ fixed, there exists $\displaystyle C:=C\left(t_{1},n,s,\omega\right)>0$ such that

w(x,t)\geq C\left(t_{1},n,s,\omega\right)\text{ in }\omega\times[t_{1},T).

Proof.

We will use the weak Harnack inequality as in [23]. First, we show the result for any arbitrary $\displaystyle t_{2}<T$ . As $\displaystyle\bar{\omega}\times[t_{1},t_{2}]\subset\Omega_{T}$ contains $\displaystyle\omega\times[t_{1},t_{2}]$ , so it is enough to show the result for $\displaystyle\bar{\omega}\times[t_{1},t_{2}]$ . Now since $\displaystyle\bar{\omega}\times[t_{1},t_{2}]$ is compact and hence every open cover admits a finite subcover, it suffices to show the positivity of $\displaystyle w$ in a uniform neighbourhood of any arbitrary point $\displaystyle(\tilde{x},\tilde{t})\in\bar{\omega}\times[t_{1},t_{2}]$ .
As $\displaystyle\bar{\omega}\times[t_{1},t_{2}]$ is compact, it has a finite and positive distance from the boundary of $\displaystyle\Omega_{T}$ ; let us denote this by $\displaystyle D$ . We choose $\displaystyle 0<r<\operatorname{min}\{\frac{D}{2},1\}$ , and $\displaystyle(x_{0},t_{0})\in\Omega_{T}$ such that $\displaystyle t_{0}=\tilde{t}-\frac{13}{16}r^{2}\text{ and }x_{0}=\tilde{x}.$ Now for this $\displaystyle(x_{0},t_{0})$ , we can choose $\displaystyle 0<R=D$ such that $\displaystyle B_{R}(x_{0})\times(t_{0}-r^{2},t_{0}+r^{2})\subset\Omega_{T}$ , and then $\displaystyle r$ satisfies $\displaystyle r<\frac{R}{2}$ . Clearly, as $\displaystyle f\geq 0$ , $\displaystyle w$ is a supersolution to the homogeneous problem. Now by using the facts that $\displaystyle w\geq 0$ in $\displaystyle\Omega\times(0,T)$ and $\displaystyle w=0$ in $\displaystyle(\mathbb{R}^{n}\backslash\Omega)\times(0,T)$ , we get $\displaystyle w\geq 0$ in $\displaystyle\mathbb{R}^{n}\times(0,T)$ . Then using [23, Theorem 2.8 and Corollary 2.9], and $\displaystyle f>0$ a.e. in $\displaystyle\Omega_{T}$ , we get the existence of a positive constant $\displaystyle C$ depending only on $\displaystyle\omega,[t_{1},t_{2}],n$ and $\displaystyle s$ such that

0<C=\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-7.18916pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-6.15741pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-4.34457pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle\rotatebox[origin={c}]{18.0}{$\displaystyle-\mkern-9.5mu-$}$}}\kern-3.59457pt}}\!\iint_{V^{-}\left(\frac{r}{2}\right)}w(x,t)dxdt\leq\underset{V^{+}\left(\frac{r}{2}\right)}{\operatorname{{ess}\operatorname{inf}}}w,

where $\displaystyle V^{-}\left(\frac{r}{2}\right)=B_{\frac{r}{2}(x_{0})}\times(t_{0}-r^{2},t_{0}-\frac{3}{4}r^{2})$ and $\displaystyle V^{+}\left(\frac{r}{2}\right)=B_{\frac{r}{2}(x_{0})}\times(t_{0}+\frac{3}{4}r^{2},t_{0}+r^{2})$ . As $\displaystyle(\tilde{t}-\frac{1}{16}r^{2},\tilde{t}+\frac{1}{16}r^{2})\subset(t_{0}+\frac{3}{4}r^{2},t_{0}+r^{2})$ , so the result follows.
Now, we will extend this up to $\displaystyle T$ . For this we denote $\displaystyle D_{0}=\operatorname{dist}\{\omega,\partial\Omega\}$ . As $\displaystyle T>0$ , so $\displaystyle\exists\tilde{r}\in(0,1]$ and $\displaystyle\tilde{r}<\frac{D_{0}}{2}$ such that $\displaystyle\omega\times(T-2\tilde{r}^{2},T)\subset\Omega_{T}$ with $\displaystyle T-\frac{1}{4}\tilde{r}^{2}>t_{1}$ . By the above argument for compact time intervals, we get

w(x,t)\geq C\left(t_{1},n,s,\omega\right)>0\text{ in }\omega\times[t_{1},T-\frac{1}{4}\tilde{r}^{2}].

Therefore it just remains to show the positivity of $\displaystyle w$ in $\displaystyle\omega\times(T-\frac{1}{4}\tilde{r}^{2},T)$ . For $\displaystyle x_{0}\in\bar{\omega}$ , $\displaystyle B_{\frac{\tilde{r}}{2}}(x_{0})$ will form an open cover of $\displaystyle\bar{\omega}$ , and by compactness, it will have a finite subcover. So it is enough to show the positivity of $\displaystyle w$ in $\displaystyle B_{\frac{\tilde{r}}{2}}(x_{0})\times(T-\frac{1}{4}\tilde{r}^{2},T)$ . Now as $\displaystyle w$ is nonnegative in $\displaystyle\mathbb{R}^{n}\times(0,T)$ and $\displaystyle B_{D_{0}}(x_{0})\times(T-2\tilde{r}^{2},T)\subset\Omega_{T}$ , so by [23, Theorem 2.8 and Corollary 2.9], we have

0<C=\fint_{T-2\tilde{r}^{2}}^{T-\frac{7}{8}\tilde{r}^{2}}\fint_{B_{\frac{\tilde{r}}{2}}(x_{0})}w(x,t)dxdt\leq\underset{B_{\frac{\tilde{r}}{2}}(x_{0})\times(T-\frac{1}{4}\tilde{r}^{2},T)}{\operatorname{{ess}\operatorname{inf}}}w,

and hence we conclude.∎

We will use the next parabolic Kato-type inequality to prove comparison results or a priori estimates.

Proposition 2.18.

Let $\displaystyle u\in L^{2}(0,T;H^{1}(\Omega))$ be a weak solution of

{}u_{t}-\Delta u+(-\Delta)^{s}u=f\text{ in }\Omega_{T},

(2.6)

with $\displaystyle f\in L^{1}(\Omega_{T})$ and let $\displaystyle\Phi\in\mathcal{C}^{2}(\mathbb{R})$ be a convex function such that $\displaystyle\Phi^{\prime}$ is bounded. Then

(\Phi(u))_{t}-\Delta\Phi(u)+(-\Delta)^{s}\Phi(u)\leq\Phi^{\prime}(u)\left(u_{t}-\Delta u+(-\Delta)^{s}u\right)

in the sense that for all $\displaystyle\psi\in\mathcal{C}^{2}(\Omega_{T})\cap\mathcal{C}([0,T],L^{2}(\Omega))$ , with the property that $\displaystyle\psi$ has spatial support compactly contained in $\displaystyle\Omega$ , and $\displaystyle\psi\geq 0$ in $\displaystyle\Omega_{T}$ and $\displaystyle\psi(\cdot,T)=0$ and $\displaystyle\psi(\cdot,0)=0$ , we have

\iint_{\Omega_{T}}\Phi(u)\left(-\psi_{t}-\Delta\psi+(-\Delta)^{s}\psi\right)dxdt\leq\iint_{\Omega_{T}}\Phi^{\prime}(u)f\psi dxdt.

Proof.

Assume that $\displaystyle u$ is smooth enough, otherwise one can use approximation argument. Since $\displaystyle\Phi$ is convex so

\Phi(u(x,t))-\Phi(u(y,t))\leq\Phi^{\prime}(u(x,t))(u(x,t)-u(y,t)),

and we get

\begin{array}[]{rcl}(-\Delta)^{s}(\Phi(u(x,t)))&=&\int_{\mathbb{R}^{n}}\frac{\left(\Phi(u(x,t))-\Phi(u(y,t))\right)}{|x-y|^{n+2s}}dy\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&\int_{\mathbb{R}^{n}}\Phi^{\prime}(u(x,t))\frac{(u(x,t)-u(y,t))}{|x-y|^{n+2s}}dy\vskip 3.0pt plus 1.0pt minus 1.0pt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &=&\Phi^{\prime}(u(x,t))(-\Delta)^{s}u(x,t).\end{array}

Again, using the fact that the double derivative of a convex function is nonnegative, we get

-\Delta(\Phi(u))=-\Phi^{\prime}(u)\Delta u-\Phi^{\prime\prime}(u)\sum_{i=1}^{n}\left(\frac{\partial u}{\partial x_{i}}\right)^{2}\leq-\Phi^{\prime}(u)\Delta u.

Now if $\displaystyle\psi\in\mathcal{C}_{0}^{\infty}\left(\Omega_{T}\right)$ , with $\displaystyle\psi\geq 0$ in $\displaystyle\Omega_{T}$ , we have

\begin{array}[]{rcl}\iint_{\Omega_{T}}\Phi(u)\left(-\psi_{t}-\Delta\psi+(-\Delta)^{s}\psi\right)dxdt&=&\iint_{\Omega_{T}}\left(\Phi^{\prime}(u)u_{t}-\Delta(\Phi(u))+(-\Delta)^{s}(\Phi(u))\right)\psi(x,t)dxdt\\ &\leq&\iint_{\Omega_{T}}\Phi^{\prime}(u)(u_{t}-\Delta u+(-\Delta)^{s}u)\psi dxdt=\iint_{\Omega_{T}}\Phi^{\prime}(u)f\psi dxdt.\end{array}

Since $\displaystyle\psi\in\mathcal{C}_{0}^{\infty}\left(\Omega_{T}\right)$ , with $\displaystyle\psi\geq 0$ in $\displaystyle\Omega_{T}$ , can approximate the choice of test functions of our hypothesis, we get the desired result. ∎

Remark 2.19.

We choose the regularization of $\displaystyle|u|$ by

\Phi_{\epsilon}(u)=\left(|u|^{2}+\epsilon^{2}\right)^{\frac{1}{2}}-\epsilon.

for $\displaystyle\epsilon\in(0,1)$ . It is then easy to verify that,

1.

$\displaystyle\Phi_{\epsilon}(u)\rightarrow|u|$ uniformly as $\displaystyle\epsilon\rightarrow 0$ ,
2.

$\displaystyle\Phi_{\epsilon}^{\prime}(u)$ is bounded uniformly with respect to $\displaystyle\epsilon$ .
3.

$\displaystyle\Phi_{\epsilon}(u)$ is convex.

So, using the above theorem we get

\begin{array}[]{rcl}\iint_{\Omega_{T}}\Phi_{\epsilon}(u)\left(-\psi_{t}-\Delta\psi+(-\Delta)^{s}\psi\right)dxdt&\leq&\iint_{\Omega_{T}}\Phi^{\prime}_{\epsilon}(u)(u_{t}-\Delta u+(-\Delta)^{s}u)\psi dxdt=\iint_{\Omega_{T}}\Phi^{\prime}_{\epsilon}(u)f\psi dxdt.\end{array}

Now letting $\displaystyle\epsilon\rightarrow 0$ , we get using Eq. 2.6

\begin{array}[]{rcl}\iint_{\Omega_{T}}|u|\left(-\psi_{t}-\Delta\psi+(-\Delta)^{s}\psi\right)dxdt&\leq&\iint_{\Omega_{T}}sign(u)f\psi dxdt.\end{array}

Adding $\displaystyle\iint_{\Omega_{T}}u(-\psi_{t}-\Delta\psi+(-\Delta)^{s}\psi)dxdt$ to both side, we get

\iint_{\Omega_{T}}u_{+}\left(-\psi_{t}-\Delta\psi+(-\Delta)^{s}\psi\right)dxdt\leq\iint_{\Omega_{T}}\chi_{\{u\geq 0\}}f\psi dxdt.

Therefore

\frac{\partial u_{+}}{\partial t}-\Delta u_{+}+(-\Delta)^{s}u_{+}\leq\left(\frac{\partial u}{\partial t}-\Delta u+(-\Delta)^{s}u\right)\operatorname{sign}^{+}u

in the weak sense.

We now take $\displaystyle\gamma$ to be a positive continuous function over $\displaystyle\overline{\Omega}_{T}$ and proceed with the following comparison principle.

Lemma 2.20.

Let $\displaystyle s\in(0,1)$ and $\displaystyle a\geq 0$ . Consider $\displaystyle\left(f,u_{0}\right)\in L^{\infty}(\Omega_{T})\times L^{\infty}(\Omega)$ to be non-negative bounded functions such that $\displaystyle\left(f,u_{0}\right)\neq(0,0)$ . Assume that $\displaystyle v_{1},v_{2}$ are two non-negative functions with finite energy such that $\displaystyle v_{1},v_{2}\in L^{2}((0,T);H_{0}^{1}(\Omega))\cap L^{\infty}(\Omega_{T})$ with

\displaystyle\displaystyle\begin{cases}(v_{1})_{t}-\Delta v_{1}+(-\Delta)^{s}v_{1}\leq\frac{f}{(v_{1}+a)^{\gamma(x,t)}}&\text{ in }\Omega_{T},\\ v_{1}(x,t)=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ v_{1}(x,0)\leq u_{0}(x)&\text{ in }\Omega;\end{cases}

and

\displaystyle\displaystyle\begin{cases}(v_{2})_{t}-\Delta v_{2}+(-\Delta)^{s}v_{2}\geq\frac{f}{(v_{2}+a)^{\gamma(x,t)}}&\text{ in }\Omega_{T},\\ v_{2}(x,t)=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ v_{2}(x,0)\geq u_{0}(x)&\text{ in }\Omega;\end{cases}

where $\displaystyle\gamma\in\mathcal{C}(\overline{\Omega}_{T})$ and is positive. Then, $\displaystyle v_{2}\geq v_{1}$ in $\displaystyle\Omega_{T}$ .

Proof.

Define $\displaystyle u=v_{1}-v_{2}$ , then $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(\Omega_{T})$ . We show that $\displaystyle u^{+}=0$ . By hypotheses, we get

u_{t}-\Delta u+(-\Delta)^{s}u\leq f\left(\frac{1}{\left(v_{1}+a\right)^{\gamma(x,t)}}-\frac{1}{\left(v_{2}+a\right)^{\gamma(x,t)}}\right).

Now since $\displaystyle\left(\frac{1}{\left(v_{1}+a\right)^{\gamma(x,t)}}-\frac{1}{\left(v_{2}+a\right)^{\gamma(x,t)}}\right)\leq 0$ in the set $\displaystyle\{u\geq 0\}$ , then using Propositions 2.18 and 2.19, it holds that

(u_{+})_{t}-\Delta(u_{+})+(-\Delta)^{s}(u_{+})\leq 0.

Again we notice that $\displaystyle u_{+}(x,0)\leq 0$ ; therefore by comparison principle as of Theorem 2.13, we have $\displaystyle u_{+}\equiv 0$ , and then we conclude. ∎

Corollary 2.21.

As a consequence of the previous comparison principle we get that for $\displaystyle a>0$ and $\displaystyle\gamma$ fixed, if $\displaystyle\left(f,u_{0}\right)$ are non-negative functions with $\displaystyle\left(f,u_{0}\right)\in L^{\sigma}(\Omega_{T})\times L^{\sigma}(\Omega)$ , $\displaystyle\sigma\geq 2$ , then the problem

{}\begin{array}[]{lcr}\begin{cases}v_{t}-\Delta v+(-\Delta)^{s}v=\frac{f}{(v+a)^{\gamma(x,t)}}&\text{ in }\Omega_{T},\\ v(x,t)=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ v(x,0)=u_{0}(x)&\text{ in }\Omega;\end{cases}\end{array}

(2.7)

has a unique energy solution $\displaystyle v_{a}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ .
We note that for $\displaystyle\gamma$ being a constant, the existence of an energy solution can be shown using a monotonicity argument by observing that $\displaystyle\underline{v}=0$ is a subsolution and $\displaystyle\bar{v}$ , the unique solution to the problem

\displaystyle\displaystyle\begin{cases}\bar{v}_{t}-\Delta\bar{v}+(-\Delta)^{s}\bar{v}=\frac{f}{a^{\gamma}}&\text{ in }\Omega_{T},\\ \bar{v}(x,t)=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ \bar{v}(x,0)=u_{0}(x)&\text{ in }\Omega;\end{cases}

is a supersolution. Then Lemma 2.20 allows us to get the existence of a unique solution $\displaystyle v$ to Eq. 2.7 such that $\displaystyle v\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle 0\leq v\leq\bar{v}$ in $\displaystyle\Omega_{T}$ .
Now if $\displaystyle\gamma$ is not constant and belongs to $\displaystyle\mathcal{C}(\overline{\Omega}_{T})$ , we denote

\gamma^{*}=\underset{(x,t)\in\Omega_{T}}{\operatorname{sup}}\gamma(x,t)\text{ and }\gamma_{*}=\underset{(x,t)\in\Omega_{T}}{\operatorname{inf}}\gamma(x,t),

and observe that the unique nonnegative solutions to the problems

\displaystyle\displaystyle\begin{cases}\bar{v}_{t}-\Delta\bar{v}+(-\Delta)^{s}\bar{v}=\frac{f}{a^{\gamma^{*}}}&\text{ in }\Omega_{T},\\ \bar{v}(x,t)=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ \bar{v}(x,0)=u_{0}(x)&\text{ in }\Omega;\end{cases}

and

\displaystyle\displaystyle\begin{cases}\bar{v}_{t}-\Delta\bar{v}+(-\Delta)^{s}\bar{v}=\frac{f}{a^{\gamma_{*}}}&\text{ in }\Omega_{T},\\ \bar{v}(x,t)=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ \bar{v}(x,0)=u_{0}(x)&\text{ in }\Omega;\end{cases}

are sub and supersolution (respectively super and subsolution) of

{}\begin{array}[]{rcl}\begin{cases}\bar{v}_{t}-\Delta\bar{v}+(-\Delta)^{s}\bar{v}=\frac{f}{a^{\gamma(x,t)}}&\text{ in }\Omega_{T},\\ \bar{v}(x,t)=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ \bar{v}(x,0)=u_{0}(x)&\text{ in }\Omega;\end{cases}\end{array}

(2.8)

for $\displaystyle a\geq 1$ (respectively for $\displaystyle a<1$ ). Then, by monotonicity argument and comparison principle similar to Lemma 2.20, we get the existence of a unique nonnegative solution to Eq. 2.8, which will be a supersolution to Eq. 2.7 with $\displaystyle\underline{v}=0$ being its subsolution. So Eq. 2.7 has a unique nonnegative solution in this case too. We note that these solutions also satisfy the comparison principle, in the sense that if $\displaystyle a_{1}<a_{2}$ , then we have $\displaystyle v_{a_{2}}\leq v_{a_{1}}$ .

We now list approximated problems for $\displaystyle\gamma$ being a constant. Firstly when $\displaystyle(f,u_{0})\in L^{\infty}(\Omega_{T})\times L^{\infty}(\Omega)$ , we consider the following problem for each $\displaystyle k\in\mathbb{N}$ ,

{}\begin{array}[]{lcr}\begin{cases}(u_{k})_{t}-\Delta u_{k}+(-\Delta)^{s}u_{k}=\frac{f(x,t)}{(u_{k}+\frac{1}{k})^{\gamma}}&\mbox{ in }\Omega_{T},\\ u_{k}=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u_{k}(x,0)=u_{0}(x)&\mbox{ in }\Omega.\end{cases}\end{array}

(2.9)

We now take $\displaystyle f$ and $\displaystyle u_{0}$ may not be bounded and consider

{}\begin{array}[]{lcr}\begin{cases}(u_{k})_{t}-\Delta u_{k}+(-\Delta)^{s}u_{k}=\frac{f_{k}(x,t)}{(u_{k}+\frac{1}{k})^{\gamma}}&\mbox{ in }\Omega_{T},\\ u_{k}=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u_{k}(x,0)=u_{0k}(x)&\mbox{ in }\Omega;\end{cases}\end{array}

(2.10)

where $\displaystyle f_{k}(x,t)=T_{k}(f(x,t))$ and $\displaystyle u_{0k}(x)=T_{k}(u_{0}(x))$ .
To treat variable exponent, we take $\displaystyle(f,u_{0})\in L^{1}(\Omega_{T})\times L^{1}(\Omega)$ and consider the following approximating problem.

{}\begin{array}[]{lcr}\begin{cases}(u_{k})_{t}-\Delta u_{k}+(-\Delta)^{s}u_{k}=\frac{f_{k}(x,t)}{(u_{k}+\frac{1}{k})^{\gamma(x,t)}}&\mbox{ in }\Omega_{T},\\ u_{k}=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u_{k}(x,0)=u_{0k}(x)&\mbox{ in }\Omega.\end{cases}\end{array}

(2.11)

We will prescribe conditions on $\displaystyle f$ and $\displaystyle u_{0}$ later on. With these settings, we have the following existence result.

Theorem 2.22.

For each of the above problems Eqs. 2.9, 2.10 and 2.11, there exists a unique nonnegative energy solution $\displaystyle u_{k}$ for each $\displaystyle k\in\mathbb{N}$ satisfying the followings:
(a) each $\displaystyle u_{k}$ is bounded,
(b) the sequence $\displaystyle\{u_{k}\}_{k}$ is increasing in $\displaystyle k$ ,
(c) for each $\displaystyle t_{0}>0$ and $\displaystyle\omega\subset\subset\Omega_{T}$ , $\displaystyle\exists C(\omega,t_{0},n,s)$ such that $\displaystyle u_{k}\geq C(\omega,t_{0},n,s)$ .

Proof.

The existence, uniqueness and nonnegativity of $\displaystyle u_{k}$ ’s follow from the comparison principle in Lemma 2.20 and Corollary 2.21. Since $\displaystyle f$ and $\displaystyle u_{0}$ are nonnegative, therefore $\displaystyle 0\leq f_{k}\leq f_{k+1}$ and $\displaystyle 0\leq u_{0k}\leq u_{0(k+1)}$ for each $\displaystyle k$ and then using the similar technique as that of Lemma 2.20, we obtain that the sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is increasing in $\displaystyle k$ . Therefore $\displaystyle u_{k}\geq u_{1}$ for all $\displaystyle k$ . Now for $\displaystyle f\in L^{\infty}(\Omega_{T})$ we have $\displaystyle\frac{f}{(u_{1}+1)^{\gamma}}\leq f$ and for $\displaystyle f\notin L^{\infty}(\Omega_{T})$ we have $\displaystyle\frac{f_{1}}{(u_{1}+1)^{\gamma(x,t)}}\leq f_{1}$ and hence $\displaystyle\frac{f_{1}}{(u_{1}+1)^{\gamma(x,t)}}\in L^{\infty}(\Omega_{T})$ . Therefore $\displaystyle u_{1}\in L^{\infty}(\Omega_{T})$ and by Proposition 2.17, we obtain that for all $\displaystyle\omega\subset\subset\Omega$ and for all $\displaystyle t_{0}>0$ , $\displaystyle u_{1}(x,t)\geq$ $\displaystyle C\left(\omega,t_{0},n,s\right)$ in $\displaystyle\omega\times\left[t_{0},T\right)$ . Thus for all $\displaystyle k\geq 1$ , we have $\displaystyle u_{k}(x,t)\geq C\left(\omega,t_{0},n,s\right)$ in $\displaystyle\omega\times\left[t_{0},T\right)$ . Also, we observe that for each $\displaystyle k$ , $\displaystyle\frac{f_{k}}{(u_{k}+\frac{1}{k})^{\gamma(x,t)}}\leq\operatorname{max}\{1,k^{\gamma^{*}}\}f_{k},$ where $\displaystyle\gamma^{*}=\underset{(x,t)\in\Omega_{T}}{\operatorname{sup}}\gamma(x,t)$ . Therefore each $\displaystyle u_{k}$ is bounded. ∎

Remark 2.23.

As each $\displaystyle u_{k}\in\mathcal{C}([0,T],L^{2}(\Omega))\subset\mathcal{C}([0,T],L^{1}(\Omega))$ , therefore at each time level $\displaystyle t\in[0,T]$ , we have $\displaystyle u_{k}(\cdot,t)\in L^{1}(\Omega)$ , this along with the monotonicity of $\displaystyle\{u_{k}\}_{k}$ in $\displaystyle k$ allows us to define the pointwise limit (a.e.) $\displaystyle u$ of $\displaystyle\{u_{k}\}_{k}$ in $\displaystyle\Omega$ at each time $\displaystyle t\in[0,T]$ . Also $\displaystyle u$ satisfies $\displaystyle u(\cdot,0)=u_{0}(\cdot)$ in $\displaystyle L^{1}$ sense. Again as each $\displaystyle u_{k}=0$ in $\displaystyle(\mathbb{R}^{n}\backslash\Omega)\times(0,T)$ , therefore $\displaystyle u$ also satisfies the same. Further we have $\displaystyle u\geq u_{k}$ for each $\displaystyle k$ and hence $\displaystyle u(x,t)\geq C\left(\omega,t_{0},n,s\right)$ in $\displaystyle\omega\times\left[t_{0},T\right)$ for each $\displaystyle\omega\subset\subset\Omega$ and $\displaystyle t_{0}>0$ .

To treat the singular case, we define next as:

Definition 2.24.

Let $\displaystyle\left(f,u_{0}\right)\in L^{1}\left(\Omega_{T}\right)\times L^{1}(\Omega)$ be a pair of non-negative functions and $\displaystyle\gamma>0$ is a constant. We say that $\displaystyle u$ is a very weak solution to the problem

{}\begin{array}[]{lcr}\begin{cases}u_{t}-\Delta u+(-\Delta)^{s}u=\frac{f(x,t)}{u^{\gamma}}&\mbox{ in }\Omega_{T},\\ u=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u(x,0)=u_{0}(x)&\mbox{ in }\Omega;\end{cases}\end{array}

(2.12)

if $\displaystyle u\in L^{1}(\Omega_{T})$ satisfying $\displaystyle u\equiv 0$ in $\displaystyle(\mathbb{R}^{n}\backslash\Omega)\times(0,T)$ , $\displaystyle\forall\omega\subset\subset\Omega$ and $\displaystyle\forall t_{0}>0$ , $\displaystyle\exists c\equiv c(\omega,t_{0},n,s)>0$ such that $\displaystyle u(x,t)\geq c>0$ , in $\displaystyle\omega\times\left[t_{0},T\right)$ , $\displaystyle u(\cdot,0)=u_{0}(\cdot)$ , and for all $\displaystyle\varphi\in\mathcal{C}_{0}^{\infty}\left(\Omega_{T}\right)$ , we have

\iint_{\Omega_{T}}u\left(-\varphi_{t}-\Delta\varphi+(-\Delta)^{s}\varphi\right)dxdt=\iint_{\Omega_{T}}\frac{f\varphi}{u^{\gamma}}dxdt.

Definition 2.25.

Let $\displaystyle\left(f,u_{0}\right)$ be a pair of non-negative functions with $\displaystyle u_{0}\in L^{1}(\Omega)$ , $\displaystyle f\in L^{1}(\Omega_{T})$ and $\displaystyle\gamma\in\mathcal{C}(\overline{\Omega}_{T})$ be a positive function. Then we say that $\displaystyle u$ , such that $\displaystyle u\equiv 0$ in $\displaystyle(\mathbb{R}^{n}\backslash\Omega)\times(0,T)$ , is a very weak solution to the problem

{}\begin{array}[]{lcr}\begin{cases}u_{t}-\Delta u+(-\Delta)^{s}u=\frac{f(x,t)}{u^{\gamma(x,t)}}&\mbox{ in }\Omega_{T},\\ u=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u(x,0)=u_{0}(x),&\mbox{ in }\Omega;\end{cases}\end{array}

(2.13)

if $\displaystyle u\in L^{1}(\Omega_{T})$ and $\displaystyle\forall\omega\subset\subset\Omega$ and $\displaystyle\forall t_{0}>0$ , $\displaystyle\exists c\equiv c(\omega,t_{0},n,s)>0$ such that $\displaystyle u(x,t)\geq c>0$ , in $\displaystyle\omega\times\left[t_{0},T\right)$ , $\displaystyle u(\cdot,0)=u_{0}(\cdot)$ , and for all $\displaystyle\varphi\in\mathcal{C}_{0}^{\infty}\left(\Omega_{T}\right)$ , we have

\iint_{\Omega_{T}}u\left(-\varphi_{t}-\Delta\varphi+(-\Delta)^{s}\varphi\right)dxdt=\iint_{\Omega_{T}}\frac{f\varphi}{u^{\gamma(x,t)}}dxdt.

2.4 Main Results

First, we consider $\displaystyle\gamma$ to be a constant and then take it as a positive continuous function. In this section, we state our main results depending on $\displaystyle\gamma$ and the regularity of initial conditions.

2.4.1 Existence for bounded data for $\displaystyle\gamma$ being a constant

In the case of bounded data, we will have the next existence result.

Theorem 2.26.

Let $\displaystyle\left(f,u_{0}\right)\in L^{\infty}\left(\Omega_{T}\right)\times L^{\infty}(\Omega)$ be a pair of non-negative functions and $\displaystyle\gamma>0$ . Then Eq. 2.12 has a bounded very weak positive solution $\displaystyle u$ in the sense of Definition 2.24 such that $\displaystyle u^{\frac{\gamma+1}{2}}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle u\in\mathcal{C}([0,T],L^{2}(\Omega))$ . Moreover if $\displaystyle\gamma\leq 1$ or $\displaystyle\operatorname{Supp}(f)\subset\subset\Omega_{T}$ , then $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))$ .

2.4.2 Existence for general data for $\displaystyle\gamma$ being a constant

In this subsection, we will consider general data. According to the regularity of initial conditions, we will consider two cases as $\displaystyle u_{0}\in L^{\gamma+1}(\Omega)$ and $\displaystyle u_{0}\in L^{1}(\Omega)$ . Let us state the following existence results.

Theorem 2.27.

Let $\displaystyle\left(f,u_{0}\right)\in L^{1}\left(\Omega_{T}\right)\times L^{\gamma+1}(\Omega)$ be a pair of non-negative functions and $\displaystyle\gamma>0$ . Then Eq. 2.12 has a very weak positive solution $\displaystyle u$ in the sense of Definition 2.24 such that $\displaystyle u^{\frac{\gamma+1}{2}}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle u\in L^{2}\left(0,T;L^{\sigma}(\Omega)\right)$ , where $\displaystyle\sigma=\frac{n(1+\gamma)}{n-2}$ and $\displaystyle\sigma\geq 2$ if $\displaystyle\gamma\geq 1$ .

Theorem 2.28.

Let $\displaystyle\left(f,u_{0}\right)\in L^{1}(\Omega_{T})\times L^{1}(\Omega)$ be a pair of non-negative functions and $\displaystyle\gamma>0$ . Then Eq. 2.12 has a very weak positive solution $\displaystyle u$ in the sense of Definition 2.24 such that $\displaystyle T_{k}(u)^{\frac{\gamma+1}{2}}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ for all $\displaystyle k>0$ .

2.4.3 Improved results for $\displaystyle\gamma$ being a constant

We now break $\displaystyle\gamma$ in three parts namely $\displaystyle 0<\gamma<1$ , $\displaystyle\gamma=1$ and $\displaystyle\gamma>1$ . We improve our results to find solutions in better spaces. For this, we take $\displaystyle u_{0}\in L^{\operatorname{max}(\gamma+1,2)}(\Omega).$ The case $\displaystyle\gamma=1$ will be same as that of Theorem 2.27. For $\displaystyle\gamma>1$ , we cannot find solutions in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))$ . In fact if we look for $\displaystyle L^{2}(0,T;H^{1}(\Omega))$ estimates, we will only get them in $\displaystyle L^{2}(t_{0},T;H_{\operatorname{loc}}^{1}(\Omega))$ for each $\displaystyle t_{0}>0$ . We state our next result for $\displaystyle\gamma>1$ as follows:

Theorem 2.29.

Let $\displaystyle\gamma>1$ . Assume that $\displaystyle\left(f,u_{0}\right)\in L^{1}\left(\Omega_{T}\right)\times L^{\gamma+1}(\Omega)$ be a pair of non-negative functions. Then Eq. 2.12 has a very weak positive solution $\displaystyle u$ in the sense of Definition 2.24 such that $\displaystyle u^{\frac{\gamma+1}{2}}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle u\in L^{2}(t_{0},T;H_{\operatorname{loc}}^{1}(\Omega))\cap L^{\infty}(0,T;L^{1+\gamma}(\Omega))$ for each $\displaystyle t_{0}>0$ .

Now we consider the case $\displaystyle 0<\gamma<1$ , and state our results as follows:

Theorem 2.30.

Let $\displaystyle 0<\gamma<1$ . Assume that $\displaystyle u_{0}\in L^{2}(\Omega)$ with $\displaystyle u_{0}\geq 0$ and $\displaystyle f\geq 0$ is such that
$\displaystyle i)$ $\displaystyle f\in L^{\frac{2}{\gamma+1}}\left(0,T;L^{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}(\Omega)\right)$ , or
$\displaystyle ii)$ $\displaystyle f\in L^{\bar{m}}\left(\Omega_{T}\right)$ with $\displaystyle\bar{m}:=\frac{2(n+2)}{2(n+2)-n(1-\gamma)}$ .
Then Eq. 2.12 has a very weak solution $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))$ in the sense of Definition 2.24.

Remark 2.31.

As $\displaystyle\gamma<1$ , so

\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}=\frac{2n}{2n-(1-\gamma)(n-2)}<\bar{m}<\frac{2}{\gamma+1},

and the two spaces $\displaystyle L^{\frac{2}{\gamma+1}}\left(0,T;L^{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}(\Omega)\right)$ and $\displaystyle L^{\bar{m}}\left(\Omega_{T}\right)$ are not comparable. Also the case $\displaystyle\gamma=1$ cannot be considered here since in the proof of Theorem 2.30, we will use Hölder inequality with exponents $\displaystyle\frac{2}{\gamma+1}$ and $\displaystyle\frac{2}{1-\gamma}$ . If $\displaystyle\gamma\to 1$ , then $\displaystyle\bar{m}\to 1$ and $\displaystyle\frac{2}{\gamma+1}$ and $\displaystyle\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}$ both tend to $\displaystyle 1$ , so that $\displaystyle f$ will belong to $\displaystyle L^{1}\left(\Omega_{T}\right)$ .

Now for $\displaystyle m<\bar{m}$ , we no longer find solutions in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))$ but in a larger space depending on $\displaystyle m$ .

Theorem 2.32.

Let $\displaystyle 0<\gamma<1$ . Assume that $\displaystyle 0\leq f\in L^{m}\left(\Omega_{T}\right)$ , with $\displaystyle 1\leq m<\bar{m}$ , and that $\displaystyle u_{0}\in L^{2}(\Omega)$ be nonnegative. Then the problem Eq. 2.12 admits a very weak solution $\displaystyle u\in L^{\bar{q}}(0,T;W_{0}^{1,\bar{q}}(\Omega))\cap L^{\infty}(0,T;L^{1+\gamma}(\Omega))$ , with

\bar{q}=\frac{m(\gamma+1)(n+2)}{n+2-m(1-\gamma)}.

Moreover $\displaystyle u\in L^{\sigma}\left(\Omega_{T}\right)$ , where

\sigma=\frac{m(\gamma+1)(n+2)}{n-2(m-1)}.

Remark 2.33.

Observe that we can get rid of the fact that $\displaystyle u\in L^{\bar{q}}(0,T;W_{0}^{s_{1},\bar{q}}(\Omega))$ with $\displaystyle s_{1}<s$ as is done in [1, Theorem 11]. For our case, due to the presence of the leading Laplacian operator, we will get $\displaystyle u\in L^{\bar{q}}(0,T;W_{0}^{s,\bar{q}}(\Omega))$ .

Remark 2.34.

Clearly $\displaystyle\bar{q}\geq m(\gamma+1)>1$ and $\displaystyle\sigma\geq m(\gamma+1)>1$ . Also $\displaystyle m<\bar{m}$ is equivalent to $\displaystyle\bar{q}<2$ which implies $\displaystyle L^{2}\left(0,T;H_{0}^{1}(\Omega)\right)\subset$ $\displaystyle L^{\bar{q}}(0,T;W_{0}^{1,\bar{q}}(\Omega))$ . In Theorem 2.32 the case $\displaystyle\gamma=1$ is not allowed since it yields $\displaystyle\bar{q}=2m\geq 2$ which contradicts $\displaystyle\bar{q}<2$ .

2.4.4 Further summability for $\displaystyle\gamma$ being a constant

Here we state two results regarding the optimal summability of our solutions in terms of summability of the initial data $\displaystyle f\in L^{r}(0,T;L^{q}(\Omega))$ . For the $\displaystyle L^{\infty}$ -boundedness, we take $\displaystyle u_{0}\in L^{\infty}(\Omega)$ , and $\displaystyle 1/r$ and $\displaystyle 1/q$ are in the Aronson-Serrin domain, see [2, 29]. We now state the corresponding theorem as:

Theorem 2.35.

Assume that $\displaystyle f\in L^{r}(0,T;L^{q}(\Omega))$ with $\displaystyle r,q$ satisfying

\frac{1}{r}+\frac{n}{2q}<1,

and suppose that $\displaystyle u_{0}\in L^{\infty}(\Omega)$ . Then there exists a positive constant $\displaystyle c$ such that the unique finite energy solution of Eq. 2.10 satisfies

\left\|u_{k}(x,t)\right\|_{L^{\infty}\left(\Omega_{T}\right)}\leq c,

and the solution $\displaystyle u$ obtained as the limit of $\displaystyle u_{k}$ satisfies, $\displaystyle u\in L^{\infty}(\Omega_{T})$ . Moreover for $\displaystyle\gamma\leq 1$ , $\displaystyle u\in L^{2}(0,T;H^{1}_{0}(\Omega))$ .

Outside the Aronsom-Serrin zone, the solutions are not expected to be bounded. We divide the region by the straight line $\displaystyle\frac{1}{r}=\frac{n}{n-2}\frac{1}{q}-\frac{2}{n-2}$ in two parts. Further, we take $\displaystyle u_{0}\equiv 0$ . The following summability results we will get for this case.

Theorem 2.36.

Assume $\displaystyle u_{0}(x)\equiv 0$ and $\displaystyle f\in L^{r}\left(0,T;L^{q}(\Omega)\right)$ , with $\displaystyle r>1,q>1$ satisfy

1<\frac{1}{r}+\frac{n}{2q}.

Further, assume that for $\displaystyle\gamma<1$ ,
$\displaystyle i)$ if $\displaystyle\frac{1}{r}<\frac{n}{n-2}\frac{1}{q}-\frac{2}{n-2}$ , then $\displaystyle q>\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}$ , and
$\displaystyle ii)$ if $\displaystyle\frac{1}{r}\geq\frac{n}{n-2}\frac{1}{q}-\frac{2}{n-2}$ , then $\displaystyle r>\frac{2}{1+\gamma}$ .
Then there exists a positive constant $\displaystyle c$ such that the sequence of finite energy solutions of Eq. 2.10 satisfies

\left\|u_{k}\right\|_{L^{\infty}\left(0,T;L^{2\sigma}(\Omega)\right)}+\left\|u_{k}\right\|_{L^{2\sigma}\left(0,T;L^{2^{*}\sigma}\right)}\leq c,

where

\sigma=\left\{\begin{array}[]{lll}\frac{q(n-2)(\gamma+1)}{2(n-2q)}&\text{ if }\frac{1}{r}<\frac{n}{n-2}\frac{1}{q}-\frac{2}{n-2},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \frac{qrn(\gamma+1)}{2(nr+2q-2qr)}&\text{ if }\frac{1}{r}\geq\frac{n}{n-2}\frac{1}{q}-\frac{2}{n-2}.\end{array}\right.

Further, the solution $\displaystyle u$ obtained as the limit of $\displaystyle u_{k}$ satisfies, $\displaystyle u\in L^{\infty}(0,T;L^{2\sigma}(\Omega))\cap L^{2\sigma}(0,T;L^{2^{*}\sigma}(\Omega))$ .

Remark 2.37.

In Theorem 2.36, we observe that the conditions needed for $\displaystyle\gamma<1$ are obvious, as Theorem 2.30 implies that if $\displaystyle r=\frac{2}{1+\gamma}$ and $\displaystyle q=\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}$ , then by Sobolev embedding, $\displaystyle u\in L^{2}(0,T;L^{2^{*}}(\Omega))$ which matches with Theorem 2.36 as we get $\displaystyle\sigma=1$ in this case.

Remark 2.38.

We note that, for $\displaystyle\gamma\geq 1$ , the singularity allows us to get better summability results as we can go up to $\displaystyle r=1$ and $\displaystyle q=1$ , which is not allowed for the nonsingular case (see [10, 29]), whereas for $\displaystyle\gamma<1$ , the singularity gives us better summability for the spatial exponent $\displaystyle q$ only.

2.4.5 Existence results for $\displaystyle\gamma$ being a function

We now consider $\displaystyle\gamma$ to be a positive continuous function on $\displaystyle\overline{\Omega}_{T}$ . We will mainly see the behaviour of $\displaystyle\gamma$ near the parabolic boundary, and accordingly, we will state two existence results. We recall the strip around the parabolic boundary given by $\displaystyle(\Omega_{T})_{\delta}=\{(x,t)\in\Omega_{T}:\operatorname{dist}((x,t),\Gamma_{T})<\delta\}$ for $\displaystyle\delta>0$ , where $\displaystyle\Gamma_{T}=(\Omega\times\{t=0\})\cup(\partial\Omega\times(0,T))$ .

Theorem 2.39.

Let $\displaystyle\exists\delta>0$ such that $\displaystyle\gamma(x,t)\leq 1$ in $\displaystyle(\Omega_{T})_{\delta}$ . Also let $\displaystyle u_{0}\in L^{2}(\Omega)$ with $\displaystyle u_{0}\geq 0$ and $\displaystyle f\geq 0$ satisfies
$\displaystyle i)$ $\displaystyle f\in L^{2}\left(0,T;L^{\left(\frac{2n}{n+2}\right)}(\Omega)\right)$ , or
$\displaystyle ii)$ $\displaystyle f\in L^{\bar{r}}\left(\Omega_{T}\right)$ with $\displaystyle\bar{r}:=\frac{2(n+2)}{n+4}$ .
Then Eq. 2.13 has a very weak solution $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))$ in the sense of Definition 2.25.

Remark 2.40.

Since $\displaystyle\frac{2n}{n+2}<\frac{2(n+2)}{n+4}=\bar{r}<2$ , so the two spaces $\displaystyle L^{2}\left(0,T;L^{\left(\frac{2n}{n+2}\right)}(\Omega)\right)$ and $\displaystyle L^{\bar{r}}\left(\Omega_{T}\right)$ are not comparable. Also for the constant case (Theorem 2.30) we got larger possible spaces $\displaystyle L^{\frac{2}{\gamma+1}}\left(0,T;L^{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}(\Omega)\right)$ and $\displaystyle L^{\bar{m}}\left(\Omega_{T}\right)$ for the belonging of initial data $\displaystyle f$ , which gives us broader results, this is the cause of considering the constant and nonconstant cases separately.

Theorem 2.41.

Assume that for some $\displaystyle\gamma^{*}>1$ and some $\displaystyle\delta>0$ , we have $\displaystyle\|\gamma\|_{L^{\infty}((\Omega_{T})_{\delta})}<\gamma^{*}$ . Assume that $\displaystyle u_{0}\in L^{\gamma^{*}+1}(\Omega)$ with $\displaystyle u_{0}\geq 0$ and $\displaystyle f\geq 0$ is such that
$\displaystyle i)$ $\displaystyle f\in L^{\gamma^{*}+1}\left(0,T;L^{\left(\frac{n(\gamma^{*}+1)}{n+2\gamma^{*}}\right)}(\Omega)\right)$ , or
$\displaystyle ii)$ $\displaystyle f\in L^{\tilde{r}}\left(\Omega_{T}\right)$ with $\displaystyle\tilde{r}:=\frac{(n+2)(\gamma^{*}+1)}{n+2(\gamma^{*}+1)}$ .
Then Eq. 2.13 admits a very weak nonnegative solution $\displaystyle u$ in the sense of Definition 2.25 such that $\displaystyle u^{\frac{\gamma^{*}+1}{2}}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle u\in L^{2}(t_{0},T;H_{\operatorname{loc}}^{1}(\Omega))\cap L^{\infty}(0,T;L^{1+\gamma^{*}}(\Omega))$ for each $\displaystyle t_{0}>0$ .

Remark 2.42.

Since $\displaystyle\frac{n(\gamma^{*}+1)}{n+2\gamma^{*}}<\frac{(n+2)(\gamma^{*}+1)}{n+2(\gamma^{*}+1)}=\tilde{r}<\gamma^{*}+1,$ so the two spaces $\displaystyle L^{\gamma^{*}+1}\left(0,T;L^{\left(\frac{n(\gamma^{*}+1)}{n+2\gamma^{*}}\right)}(\Omega)\right)$ and $\displaystyle L^{\tilde{r}}\left(\Omega_{T}\right)$ are not comparable. Also, for the constant case (Theorem 2.29), we got the largest possible space $\displaystyle L^{1}(\Omega_{T})$ for the belonging of initial data $\displaystyle f$ .

3 Proof of main results

Proof of Theorem 2.26

We consider the approximated problems Eq. 2.9 in this case and show that $\displaystyle\left\{u_{k}\right\}_{k}$ is bounded in $\displaystyle L^{\infty}\left(\Omega_{T}\right)$ . Note that the existence and other properties of $\displaystyle\{u_{k}\}_{k}$ follow by Theorem 2.22. We define $\displaystyle w_{k}=$ $\displaystyle H\left(u_{k}\right)=u_{k}^{\gamma+1}$ , and note that as $\displaystyle u_{k}\in L^{2}(0,T;H^{1}_{0}(\Omega))$ is bounded, so $\displaystyle u_{k}^{\gamma+1}\in L^{2}(0,T;H^{1}_{0}(\Omega))$ . Then, using Kato inequality as in Proposition 2.18, we get

\left(w_{k}\right)_{t}-\Delta w_{k}+(-\Delta)^{s}w_{k}\leq H^{\prime}(u_{k})\left(\left(u_{k}\right)_{t}-\Delta u_{k}+(-\Delta)^{s}u_{k}\right)\leq(\gamma+1)f\quad\text{ in weak sense}.

Now let $\displaystyle\vartheta$ be the unique solution to the problem

\displaystyle\displaystyle\begin{cases}\vartheta_{t}-\Delta\vartheta+(-\Delta)^{s}\vartheta=(\gamma+1)f&\text{ in }\Omega_{T},\\ \vartheta=0&\text{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ \vartheta(x,0)=H(u_{0}(x))&\text{ in }\Omega.\end{cases}

As $\displaystyle f$ and $\displaystyle u_{0}$ are bounded, so by Remark 2.14, $\displaystyle\vartheta\in L^{\infty}(\Omega_{T})$ and by comparison principle $\displaystyle w_{k}\leq\vartheta$ for all $\displaystyle k$ . Hence, $\displaystyle u_{k}^{\gamma+1}\leq\vartheta$ and the claim follows.
Now since $\displaystyle u_{k}\in L^{\infty}(\Omega_{T})\cap L^{2}(0,T;H^{1}_{0}(\Omega))$ and nonnegative, then for any $\displaystyle\varepsilon>0$ and $\displaystyle\theta>0$ , $\displaystyle(\left(u_{k}(x,t)+\varepsilon\right)^{\theta}-\varepsilon^{\theta})\in L^{2}(0,T;H^{1}_{0}(\Omega))$ . So choosing $\displaystyle 0<\varepsilon<1/k$ , for $\displaystyle t\in(0,T]$ , we take $\displaystyle(\left(u_{k}(x,\theta)+\varepsilon\right)^{\gamma}-\varepsilon^{\gamma})\chi_{(0,t)}$ as a test function in Eq. 2.9, and it holds that

\begin{array}[]{l}\int_{0}^{t}\int_{\Omega}\left(u_{k}\right)_{t}\left((u_{k}+\varepsilon))^{\gamma}-\varepsilon^{\gamma}\right)dxd\theta+\int_{0}^{t}\int_{\Omega}\nabla u_{k}\cdot\nabla\left((u_{k}+\varepsilon)^{\gamma}-\varepsilon^{\gamma}\right)dxd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{(u_{k}(x,\theta)-u_{k}(y,\theta))((u_{k}(x,\theta)+\varepsilon)^{\gamma}-\left(u_{k}(y,\theta)+\varepsilon\right)^{\gamma})}{|x-y|^{n+2s}}dxdyd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\iint_{\Omega_{t}}f(x,\theta)dxd\theta.\end{array}

Now letting $\displaystyle\varepsilon\to 0$ , by Fatou’s lemma, we get for all $\displaystyle t\leq T$

{}\begin{array}[]{l}\frac{1}{\gamma+1}\int_{\Omega}u_{k}^{\gamma+1}(x,t)dx+\int_{0}^{t}\int_{\Omega}\nabla u_{k}\cdot\nabla u_{k}^{\gamma}dxd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{\left(u_{k}^{\gamma}(x,\theta)-u_{k}^{\gamma}(y,\theta)\right)\left(u_{k}(x,\theta)-u_{k}(y,\theta)\right)}{|x-y|^{n+2s}}dxdyd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\iint_{\Omega_{t}}fdxd\theta+\frac{1}{\gamma+1}\int_{\Omega}u_{0}^{\gamma+1}(x)dx\leq\|f\|_{L^{\infty}(\Omega_{T})}|\Omega_{T}|+\frac{1}{\gamma+1}\|u_{0}\|_{L^{\infty}(\Omega)}^{\gamma+1}|\Omega|.\end{array}

(3.1)

Since we know $\displaystyle\nabla u_{k}\cdot\nabla u_{k}^{\gamma}=\gamma u_{k}^{\gamma-1}|\nabla u_{k}|^{2},$ and by Lemma 2.11

\left(u_{k}^{\gamma}(x,\theta)-u_{k}^{\gamma}(y,\theta)\right)\left(u_{k}(x,\theta)-u_{k}(y,\theta)\right)\geq C(\gamma)\left(u_{k}^{\frac{\gamma+1}{2}}(x,\theta)-u_{k}^{\frac{\gamma+1}{2}}(y,\theta)\right)^{2},

we get taking supremum over $\displaystyle t\in(0,T]$ in Eq. 3.1, that the sequence $\displaystyle\left\{u_{k}^{\frac{\gamma+1}{2}}\right\}_{k}$ is bounded in $\displaystyle L^{\infty}(0,T,L^{2}(\Omega))\cap L^{2}(0,T;X_{0}^{s}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))\equiv L^{\infty}(0,T,L^{2}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))$ .
Now by Theorem 2.22 and Remark 2.23, as the sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is increasing in $\displaystyle k$ , so the pointwise limit $\displaystyle u$ of $\displaystyle\{u_{k}\}_{k}$ exists; and satisfies $\displaystyle u(\cdot,0)=u_{0}(\cdot)$ and $\displaystyle u=0$ in $\displaystyle(\mathbb{R}^{n}\backslash\Omega)\times(0,T)$ . Also since $\displaystyle\left\{u_{k}^{\frac{\gamma+1}{2}}\right\}_{k}$ is bounded in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))$ , therefore $\displaystyle u_{k}^{\frac{\gamma+1}{2}}\rightharpoonup u^{\frac{\gamma+1}{2}}$ in $\displaystyle L^{2}(0,T;H^{1}_{0}(\Omega))$ . Again by Beppo Levi theorem $\displaystyle u_{k}\rightarrow u$ in $\displaystyle L^{1}(\Omega_{T})$ . Using Fatou’s Lemma, we have $\displaystyle u^{\frac{\gamma+1}{2}}\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{\infty}(\Omega_{T})$ . Also, by Remark 2.23 we get $\displaystyle u(x,t)\geq C\left(\omega,t_{0},n,s\right)$ in $\displaystyle\omega\times\left[t_{0},T\right)$ for any $\displaystyle\omega\subset\subset\Omega$ and $\displaystyle t_{0}>0$ . Using this positivity, it is then easy to show that $\displaystyle u$ is a very weak solution in the sense of Definition 2.24 and is shown in detail in the proof of Theorem 2.27.
In order to show that $\displaystyle u\in\mathcal{C}([0,T],L^{2}(\Omega))$ , we fix $\displaystyle k\geq l$ and hence $\displaystyle u_{k}\geq u_{l}$ and note that the function $\displaystyle\tilde{u}=(u_{k}+\frac{1}{k})-(u_{l}+\frac{1}{l})\in L^{2}(0,T;H^{1}(\Omega))$ satisfies the equation

\begin{array}[]{lcr}\begin{cases}\tilde{u}_{t}-\Delta\tilde{u}+(-\Delta)^{s}\tilde{u}=\frac{f(x,t)}{(u_{k}+\frac{1}{k})^{\gamma}}-\frac{f(x,t)}{(u_{l}+\frac{1}{l})^{\gamma}},&\mbox{ in }\Omega_{T},\\ \tilde{u}=\frac{1}{k}-\frac{1}{l},&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \tilde{u}(x,0)=\frac{1}{k}-\frac{1}{l},&\mbox{ in }\Omega;\end{cases}\end{array}

Now as $\displaystyle\left(\frac{f}{(u_{k}+\frac{1}{k})^{\gamma}}-\frac{f}{(u_{l}+\frac{1}{l})^{\gamma}}\right)\leq 0$ in the set $\displaystyle\{\tilde{u}\geq 0\}$ , using Propositions 2.18 and 2.19, it holds that

(\tilde{u}_{+})_{t}-\Delta(\tilde{u}_{+})+(-\Delta)^{s}(\tilde{u}_{+})\leq 0.

Again we notice that $\displaystyle\tilde{u}_{+}\in L^{2}(0,T;H^{1}_{0}(\Omega))$ , has finite energy and $\displaystyle\tilde{u}_{+}(x,0)=0$ , therefore by comparison principle, it holds that $\displaystyle\tilde{u}_{+}\equiv 0$ , and then we have $\displaystyle(u_{k}+\frac{1}{k})\leq(u_{l}+\frac{1}{l})$ for $\displaystyle k\geq l$ . Therefore we get $\displaystyle 0\leq u_{k}-u_{l}\leq\frac{1}{l}-\frac{1}{k}$ , for $\displaystyle k\geq l$ . This implies that the sequence $\displaystyle\{u_{k}\}_{k}$ is Cauchy in $\displaystyle\mathcal{C}([0,T],L^{2}(\Omega))$ and hence $\displaystyle u\in\mathcal{C}([0,T],L^{2}(\Omega))$ .
We now consider that $\displaystyle\gamma\leq 1$ , then using $\displaystyle u_{k}\chi_{(0,t)}$ as a test function in Eq. 2.9, we get that

\begin{array}[]{l}\frac{1}{2}\int_{\Omega}u_{k}^{2}(x,t)dx+\int_{0}^{t}\int_{\Omega}|\nabla u_{k}|^{2}dxd\theta+\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{\left(u_{k}(x,\theta)-u_{k}(y,\theta)\right)^{2}}{|x-y|^{n+2s}}dxdyd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\iint_{\Omega_{\mathrm{t}}}fu_{k}^{1-\gamma}dxd\theta+\frac{1}{2}\int_{\Omega}u_{0}^{2}(x)dx.\end{array}

Since $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{\infty}\left(\Omega_{T}\right)$ , and $\displaystyle\gamma\leq 1$ , so $\displaystyle\iint_{\Omega_{t}}fu_{k}^{1-\gamma}dxd\theta\leq C\|f\|_{L^{\infty}\left(\Omega_{T}\right)},$ and then we conclude that $\displaystyle\left\{u_{k}\right\}_{k}$ is bounded in the space $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(\Omega_{T})$ . We note that the same conclusion holds if $\displaystyle\operatorname{Supp}(f)\subset\subset\Omega_{T}$ taking into consideration that $\displaystyle u_{k}\geq u_{1}\geq C$ in $\displaystyle\operatorname{Supp}(f)$ for each $\displaystyle k$ .

Proof of Theorem 2.27

We consider here the approximating problem Eq. 2.10 and refer Theorem 2.22 for properties of $\displaystyle u_{k}$ . As $\displaystyle f_{k}$ and $\displaystyle u_{0k}$ are bounded and nonnegative, therefore $\displaystyle u_{k}\in L^{2}(0,T;H^{1}_{0}(\Omega))$ is bounded and nonnegative, so we can choose the same test function $\displaystyle(\left(u_{k}(x,\theta)+\varepsilon\right)^{\gamma}-\varepsilon^{\gamma})\chi_{(0,t)},0<\varepsilon<1/k,$ in Eq. 2.10 also, and let $\displaystyle\varepsilon\to 0$ to get

\begin{array}[]{l}\quad\frac{1}{\gamma+1}\int_{\Omega}u_{k}^{\gamma+1}(x,t)dx+\int_{0}^{t}\int_{\Omega}\nabla u_{k}\cdot\nabla u_{k}^{\gamma}dxd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{\left(u_{k}^{\gamma}(x,\theta)-u_{k}^{\gamma}(y,\theta)\right)\left(u_{k}(x,\theta)-u_{k}(y,\theta)\right)}{|x-y|^{n+2s}}dxdyd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\iint_{\Omega_{t}}f_{k}dxd\theta+\frac{1}{\gamma+1}\int_{\Omega}u_{0k}^{\gamma+1}(x)dx\leq\|f\|_{L^{1}\left(\Omega_{T}\right)}+\frac{1}{\gamma+1}\left\|u_{0}\right\|_{L^{\gamma+1}\left(\Omega\right)}.\end{array}

Similarly like the proof of Theorem 2.26, it holds that $\displaystyle\left\{u_{k}^{\frac{\gamma+1}{2}}\right\}_{k}$ is bounded in $\displaystyle L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))$ .
Now as $\displaystyle\left\{u_{k}\right\}_{k}$ is increasing in $\displaystyle k$ , we get by Fatou’s Lemma and Beppo Levi’s Lemma, that the pointwise limit (as mentioned in Remark 2.23) $\displaystyle u$ satisfies $\displaystyle u^{\frac{\gamma+1}{2}}\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle u_{k}\uparrow u$ strongly in $\displaystyle L^{1}(\Omega_{T})$ . Since $\displaystyle u^{\frac{\gamma+1}{2}}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle\frac{n(1+\gamma)}{n-2}=\frac{2^{*}(1+\gamma)}{2}$ , embedding result on $\displaystyle u^{\frac{\gamma+1}{2}}$ implies $\displaystyle u\in L^{2}\left(0,T;L^{\left(\frac{n(1+\gamma)}{n-2}\right)}(\Omega)\right)$ . Also, we have $\displaystyle u(x,t)\geq$ $\displaystyle C\left(\omega,t_{0},n,s\right)$ in $\displaystyle\omega\times\left[t_{0},T\right)$ for any $\displaystyle\omega\subset\subset\Omega$ and $\displaystyle t_{0}>0$ . We show that $\displaystyle u$ is a very weak solution to Eq. 2.12 in the sense of Definition 2.24. Let us take arbitrary $\displaystyle\phi\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ , we then have

\iint_{\Omega_{T}}(\left(u_{k}\right)_{t}-\Delta u_{k}+(-\Delta)^{s}u_{k})\phi\,dxdt=\iint_{\Omega_{T}}\frac{f_{k}}{(u_{k}+\frac{1}{k})^{\gamma}}\phi dxdt.

Now $\displaystyle\phi\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ implies $\displaystyle\phi_{t},\Delta\phi$ and $\displaystyle(-\Delta)^{s}\phi$ all are bounded. Then as $\displaystyle u_{k}\rightarrow u$ strongly in $\displaystyle L^{1}\left(\Omega_{T}\right)$ , we have

\begin{array}[]{l}\quad\iint_{\Omega_{T}}(\left(u_{k}\right)_{t}-\Delta u_{k}+(-\Delta)^{s}u_{k})\phi\,dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ =\iint_{\Omega_{T}}u_{k}(-(\phi)_{t}-\Delta\phi+(-\Delta)^{s}\phi)dxdt\rightarrow\iint_{\Omega_{T}}u(-(\phi)_{t}-\Delta\phi+(-\Delta)^{s}\phi)dxdt.\end{array}

Now since for all $\displaystyle\omega\subset\subset\Omega$ and for all $\displaystyle t_{0}>0,$ $\displaystyle u_{k}(x,t)\geq C\left(\omega,t_{0},n,s\right)$ in $\displaystyle\omega\times\left[t_{0},T\right)$ , we get the positivity of $\displaystyle\{u_{k}\}_{k}$ as $\displaystyle u_{k}(x,t)\geq C$ in $\displaystyle\operatorname{Supp}\phi$ for all $\displaystyle k\geq 1$ . Hence, we can use the dominated convergence theorem to obtain

\iint_{\Omega_{T}}\frac{f_{k}}{(u_{k}+\frac{1}{k})^{\gamma}}\phi\,dxdt\rightarrow\iint_{\Omega_{T}}\frac{f}{u^{\gamma}}\phi\,dxdt\text{ as }k\rightarrow\infty.

Thus

\iint_{\Omega_{T}}(u_{t}-\Delta u+(-\Delta)^{s}u)\phi\,dxdt=\iint_{\Omega_{T}}\frac{f}{u^{\gamma}}\phi\,dxdt.

Proof of Theorem 2.28

Here we consider Eq. 2.10 but with suffix $\displaystyle m\in\mathbb{N}$ as

{}\begin{array}[]{lcr}\begin{cases}(u_{m})_{t}-\Delta u_{m}+(-\Delta)^{s}u_{m}=\frac{f_{m}(x,t)}{(u_{m}+\frac{1}{m})^{\gamma}}&\mbox{ in }\Omega_{T}:=\Omega\times(0,T),\\ u_{m}=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u_{m}(x,0)=u_{0m}(x)&\mbox{ in }\Omega.\end{cases}\end{array}

(3.2)

The properties of $\displaystyle u_{m}$ ’s follow from Theorem 2.22. As $\displaystyle u_{m}\in L^{2}(0,T;H^{1}_{0}(\Omega))$ is nonnegative and $\displaystyle T_{k}(u_{m})$ is bounded, so using $\displaystyle\left(\left(T_{k}\left(u_{m}\right)+\varepsilon\right)^{\gamma}-\varepsilon^{\gamma}\right)\chi_{(0,t)}$ as a test function in Eq. 3.2, we get

\begin{array}[]{l}\quad\int_{0}^{t}\int_{\Omega}\left(u_{m}\right)_{t}\left((T_{k}(u_{m})+\varepsilon)^{\gamma}-\varepsilon^{\gamma}\right)dxd\theta+\int_{0}^{t}\int_{\Omega}\nabla u_{k}\cdot\nabla\left((T_{k}(u_{m})+\varepsilon)^{\gamma}-\varepsilon^{\gamma}\right)dxd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{(u_{k}(x,\theta)-u_{k}(y,\theta))((T_{k}(u_{m}(x,\theta))+\varepsilon)^{\gamma}-\left(T_{k}(u_{m}(y,\theta))+\varepsilon\right)^{\gamma})}{|x-y|^{n+2s}}dxdyd\theta\leq\iint_{\Omega_{T}}f(x,\theta)dxd\theta.\end{array}

We have chosen $\displaystyle 0<\varepsilon<1/k$ arbitrarily in above. Now letting $\displaystyle\varepsilon\to 0$ , by Fatou’s lemma, we get for all $\displaystyle t\leq T$

\begin{array}[]{l}\quad\int_{\Omega}L_{k}\left(u_{m}(x,t)\right)dx+\int_{0}^{t}\int_{\Omega}\nabla u_{m}\cdot\nabla T_{k}^{\gamma}(u_{m})dxd\theta\\ +\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{\left(T_{k}^{\gamma}\left(u_{m}(x,\theta)\right)-T_{k}^{\gamma}\left(u_{m}(y,\theta)\right)\right)\left(u_{m}(x,\theta)-u_{m}(y,\theta)\right)}{|x-y|^{n+2s}}dxdyd\theta\\ \leq\|f\|_{L^{1}\left(\Omega_{T}\right)}+C_{3}(k)\left\|u_{0}\right\|_{L^{1}(\Omega)}+C_{4}(k)|\Omega|,\end{array}

where $\displaystyle L_{k}(\rho)=\int_{0}^{\rho}\left(T_{k}(\xi)\right)^{\gamma}d\xi$ . Notice that

\begin{array}[]{l}\left(T_{k}^{\gamma}\left(u_{m}(x,\theta)\right)-T_{k}^{\gamma}\left(u_{m}(y,\theta)\right)\right)\left(u_{m}(x,\theta)-u_{m}(y,\theta)\right)\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \geq\left(T_{k}^{\gamma}\left(u_{m}(x,\theta)\right)-T_{k}^{\gamma}\left(u_{m}(y,\theta)\right)\right)\left(T_{k}\left(u_{m}(x,\theta)\right)-T_{k}\left(u_{m}(y,\theta)\right)\right)\end{array}

and so

\nabla u_{m}\cdot\nabla T_{k}^{\gamma}(u_{m})\geq\gamma T_{k}^{\gamma-1}(u_{m})|\nabla T_{k}(u_{m})|^{2}

and for $\displaystyle\rho>0$ , trivial calculation yields that

C_{1}(k)\rho-C_{2}(k)\leq L_{k}(\rho)\leq C_{3}(k)\rho+C_{4}(k)\text{ and }L_{k}(\rho)\geq C\left(T_{k}(\rho)\right)^{\gamma+1},

where $\displaystyle C_{1},C_{2},C_{3},C_{4},C$ are positive constants depending only on $\displaystyle k$ . Therefore $\displaystyle\left\{\left(T_{k}\left(u_{m}\right)\right)^{\frac{\gamma+1}{2}}\right\}_{m}$ is bounded in $\displaystyle L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle\left\{u_{m}\right\}_{m}$ is bounded in $\displaystyle L^{\infty}(0,T;L^{1}(\Omega))$ .
Now as $\displaystyle\left\{u_{m}\right\}_{m}$ is increasing in $\displaystyle m$ , by Fatou’s Lemma and Beppo Levi’s theorem, the pointwise limit $\displaystyle u$ of $\displaystyle\{u_{m}\}_{m}$ (as mentioned in Remark 2.23) satisfies $\displaystyle\left(T_{k}(u)\right)^{\frac{\gamma+1}{2}}\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))$ , $\displaystyle u_{m}\uparrow u$ strongly in $\displaystyle L^{1}(\Omega_{T})$ . Also $\displaystyle u(\cdot,0)=u_{0}(\cdot)$ in $\displaystyle L^{1}$ sense and $\displaystyle u$ is $\displaystyle 0$ outside $\displaystyle\Omega$ . Then $\displaystyle u$ can be shown to be a very weak solution of Eq. 2.12 in the sense of Definition 2.24, and the proof is same as that of Theorem 2.27.

Remark 3.1.

In view of Theorem 2.26, if we consider the approximating problem

\begin{array}[]{lcr}\begin{cases}(u_{k})_{t}-\Delta u_{k}+(-\Delta)^{s}u_{k}=\frac{f_{k}(x,t)}{u_{k}^{\gamma}}&\mbox{ in }\Omega_{T},\\ u_{k}=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega)\times(0,T),\\ u_{k}(x,0)=u_{0k}(x)&\mbox{ in }\Omega;\end{cases}\end{array}

for $\displaystyle\gamma\leq 1$ , then for each $\displaystyle k$ , $\displaystyle u_{k}$ exists and $\displaystyle u_{k}\in L^{2}(0,T;H^{1}_{0}(\Omega))\cap\mathcal{C}([0,T],L^{2}(\Omega))$ . Also, each $\displaystyle u_{k}$ is unique (see Remark 3.3). Further, the sequence $\displaystyle\{u_{k}\}_{k}$ satisfy the followings:
(a) each $\displaystyle u_{k}$ is bounded and nonnegative,
(b) the sequence $\displaystyle\{u_{k}\}_{k}$ is increasing in $\displaystyle k$ (can be shown similarly like Lemma 2.20),
(c) for each $\displaystyle t_{0}>0$ and $\displaystyle\omega\subset\subset\Omega_{T}$ , $\displaystyle\exists C(\omega,t_{0},n,s)$ such that $\displaystyle u_{k}\geq C(\omega,t_{0},n,s)$ .
Now for $\displaystyle\gamma\leq 1$ , we have $\displaystyle(u_{k}+\varepsilon)^{\gamma}-\varepsilon^{\gamma}\leq u_{k}^{\gamma}$ , and hence $\displaystyle\frac{f_{k}((u_{k}+\varepsilon)^{\gamma}-\varepsilon^{\gamma})}{u_{k}^{\gamma}}\leq f_{k}$ . Therefore we can follow the same procedure as that of Theorem 2.27 to get that the pointwise limit $\displaystyle u$ of $\displaystyle\{u_{k}\}_{k}$ is a very weak solution of Eq. 2.12 and satisfies the corresponding properties of Theorem 2.27. Now let $\displaystyle k\geq l$ be fixed, then $\displaystyle u_{k}\geq u_{l}$ and

\left(u_{k}-u_{l}\right)_{t}-\Delta(u_{k}-u_{l})+(-\Delta)^{s}\left(u_{k}-u_{l}\right)=\frac{f_{k}(x,t)}{u_{k}^{\gamma}}-\frac{f_{l}(x,t)}{u_{l}^{\gamma}}\leq\frac{f_{k}-f_{l}}{u_{k}^{\gamma}}\text{ in }\Omega_{T}.

Hence using $\displaystyle\left((u_{k}-u_{l}+\varepsilon)^{\gamma}-\varepsilon^{\gamma}\right)\chi_{(0,t)}$ , $\displaystyle 0<\varepsilon<<1$ as a test function in the above inequality and letting $\displaystyle\varepsilon\to 0$ it holds that

\begin{array}[]{l}\frac{1}{\gamma+1}\int_{\Omega}\left(u_{k}(x,t)-u_{l}(x,t)\right)^{\gamma+1}dx\leq\iint_{\Omega_{t}}\left(f_{k}-f_{l}\right)dxd\theta+\frac{1}{\gamma+1}\int_{\Omega}\left(u_{0k}(x)-u_{0l}(x)\right)^{\gamma+1}(x)dx.\end{array}

Now as by Dominated Convergence Theorem, $\displaystyle f_{k}\uparrow f$ strongly in $\displaystyle L^{1}(\Omega_{T})$ and $\displaystyle u_{0k}\uparrow u_{0}$ strongly in $\displaystyle L^{\gamma+1}(\Omega)$ if $\displaystyle u_{0}\in L^{\gamma+1}(\Omega)$ , we get that the sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is a Cauchy sequence in the space $\displaystyle\mathcal{C}([0,T];L^{\gamma+1}(\Omega))$ and hence in $\displaystyle\mathcal{C}([0,T];L^{1}(\Omega))$ . Therefore this approximation allows us to have $\displaystyle u\in\mathcal{C}([0,T],L^{1}(\Omega))$ .
Note that the same approximation technique will not work for $\displaystyle\gamma>1$ , as in that case we will have $\displaystyle u_{k}^{\frac{\gamma+1}{2}}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ which may not give us $\displaystyle u_{k}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and the monotonicity of $\displaystyle\{u_{k}\}_{k}$ cannot be proven in similar way like Lemma 2.20. However, we will take approximations as Eq. 2.10 for convenience even for $\displaystyle\gamma\leq 1$ .
Further, this approximation technique will not work for $\displaystyle u_{0}\in L^{1}(\Omega)$ , as in that case, we need to take test functions like $\displaystyle\left((T_{m}\left(u_{k}-u_{l})+\varepsilon\right)^{\gamma}-\varepsilon^{\gamma}\right)\chi_{(0,t)}$ and will end up with $\displaystyle\int_{\Omega}L_{m}\left(u_{k}(x,t)-u_{l}(x,t)\right)dx$ in the left which will not give us anything desired.

Proof of Theorem 2.29

We consider the approximated problems Eq. 2.10 here and refer Theorem 2.22. Since each $\displaystyle u_{k}\in L^{2}(0,T;H^{1}_{0}(\Omega))$ is bounded and $\displaystyle\gamma>1$ so we can use $\displaystyle u_{k}^{\gamma}\chi_{(0,t)}$ as a test function in Eq. 2.10, to get that, for all $\displaystyle t\leq T$ ,

{}\begin{array}[]{l}\quad\frac{1}{\gamma+1}\int_{\Omega}u_{k}^{\gamma+1}(x,t)dx+\int_{0}^{t}\int_{\Omega}\nabla u_{k}\cdot\nabla u_{k}^{\gamma}dxd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \quad+\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{\left(u_{k}^{\gamma}(x,\theta)-u_{k}^{\gamma}(y,\theta)\right)\left(u_{k}(x,\theta)-u_{k}(y,\theta)\right)}{|x-y|^{n+2s}}dxdyd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\iint_{\Omega_{t}}fdxd\theta+\frac{1}{\gamma+1}\int_{\Omega}u_{0k}^{\gamma+1}(x)dx\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\|f\|_{L^{1}\left(\Omega_{T}\right)}+\frac{1}{\gamma+1}\left\|u_{0}\right\|_{L^{\gamma+1}\left(\Omega\right)}.\end{array}

(3.3)

Using item $\displaystyle(i)$ of Lemma 2.11 and taking supremum over $\displaystyle 0<t\leq T$ , we get that $\displaystyle\left\{u_{k}^{\frac{\gamma+1}{2}}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{\infty}(0,T;L^{\gamma+1}(\Omega))$ .
We now show that $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(t_{0},T;H_{loc}^{s}(\Omega))\cap L^{2}(t_{0},T;H_{loc}^{1}(\Omega))$ for each $\displaystyle t_{0}>0$ . Since $\displaystyle\gamma>1$ , and $\displaystyle\Omega_{T}$ is bounded, and $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{\infty}(0,T;L^{\gamma+1}(\Omega))$ , we deduce that $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(0,T;L^{2}(\Omega))=L^{2}(\Omega_{T})$ , in particular in $\displaystyle L^{2}(K\times(t_{0},T))$ , for every subset $\displaystyle K$ compactly contained in $\displaystyle\Omega$ and for each $\displaystyle t_{0}>0$ . Further, as $\displaystyle K\times K\subset\Omega\times\Omega\subset Q$ and all the integrals in the left-hand-side of Eq. 3.3 are positive, hence we have,

\begin{array}[]{l}\int_{t_{0}}^{T}\int_{K}\int_{K}\frac{\left(u_{k}(x,t)-u_{k}(y,t)\right)\left(u_{k}^{\gamma}(x,t)-u_{k}^{\gamma}(y,t)\right)}{|x-y|^{n+2s}}dxdydt\leq 2\|f\|_{L^{1}\left(\Omega_{T}\right)}+\frac{2}{\gamma+1}\left\|u_{0}\right\|_{L^{\gamma+1}\left(\Omega\right)},\end{array}

and

\int_{t_{0}}^{T}\int_{K}u_{k}^{\gamma-1}|\nabla u_{k}|^{2}dxdt\leq\frac{1}{\gamma}\left(\|f\|_{L^{1}\left(\Omega_{T}\right)}+\frac{1}{\gamma+1}\left\|u_{0}\right\|_{L^{\gamma+1}\left(\Omega\right)}\right),

for every $\displaystyle K\subset\subset\Omega$ and for each $\displaystyle t_{0}>0$ .
We now apply the item $\displaystyle(iii)$ of Lemma 2.11, to get

\int_{t_{0}}^{T}\int_{K}\int_{K}\frac{\left|u_{k}(x,t)-u_{k}(y,t)\right|^{2}\left|u_{k}(x,t)+u_{k}(y,t)\right|^{\gamma-1}}{|x-y|^{n+2s}}dxdydt\leq 2C_{\gamma}\left(\|f\|_{L^{1}\left(\Omega_{T}\right)}+\left\|u_{0}\right\|_{L^{\gamma+1}(\Omega)}\right).

Using the positivity of $\displaystyle u_{k}$ in $\displaystyle\omega\times[t_{0},T)$ for all $\displaystyle k$ , we get

{}\int_{t_{0}}^{T}\int_{K}\int_{K}\frac{\left|u_{k}(x,t)-u_{k}(y,t)\right|^{2}}{|x-y|^{n+2s}}dxdydt\leq\frac{2^{2-\gamma}C_{\gamma}}{c_{(K,t_{0})}^{\gamma-1}}\left(\|f\|_{L^{1}\left(\Omega_{T}\right)}+\left\|u_{0}\right\|_{L^{\gamma+1}(\Omega)}\right),

(3.4)

and

{}\int_{t_{0}}^{T}\int_{K}|\nabla u_{k}|^{2}dxdydt\leq\frac{1}{\gamma c_{(K,t_{0})}^{\gamma-1}}\left(\|f\|_{L^{1}\left(\Omega_{T}\right)}+\frac{1}{\gamma+1}\left\|u_{0}\right\|_{L^{\gamma+1}(\Omega)}\right).

(3.5)

Hence $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(t_{0},T;H_{loc}^{s}(\Omega))\cap L^{2}(t_{0},T;H_{loc}^{1}(\Omega))\equiv L^{2}(t_{0},T;H_{loc}^{1}(\Omega))$ for each $\displaystyle t_{0}>0$ .
Since we have $\displaystyle\left\{u_{k}^{\frac{\gamma+1}{2}}\right\}$ is uniformly bounded in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))\subset$ $\displaystyle L^{2}\left(\Omega_{T}\right)$ , this implies that the increasing sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{1}\left(\Omega_{T}\right)$ . Then, there exists a measurable function $\displaystyle u$ such that $\displaystyle u_{k}\rightarrow u$ a.e. in $\displaystyle\Omega_{T}$ and by Beppo Levi’s theorem $\displaystyle u_{k}\rightarrow u$ in $\displaystyle L^{1}(\Omega_{T})$ . Since $\displaystyle u_{k}=0$ in $\displaystyle\left(\mathbb{R}^{n}\backslash\Omega\right)\times(0,T)$ , extending $\displaystyle u$ by zero outside of $\displaystyle\Omega$ we conclude that $\displaystyle u_{k}\rightarrow u$ a.e. in $\displaystyle\mathbb{R}^{n}\times(0,T)$ with $\displaystyle u=$ 0 in $\displaystyle\left(\mathbb{R}^{n}\backslash\Omega\right)\times(0,T)$ . Now we use Fatou’s lemma in Eqs. 3.3, 3.4 and 3.5, to obtain for each $\displaystyle t_{0}>0$ , $\displaystyle u\in L^{2}(t_{0},T;H_{loc}^{1}(\Omega))\cap L^{\infty}(0,T;L^{\gamma+1}(\Omega))$ and $\displaystyle u^{\frac{\gamma+1}{2}}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ . As $\displaystyle u_{k}(x,0)=u_{0k}(x)$ for each $\displaystyle k$ , so $\displaystyle u(x,0)=u_{0}(x)$ in $\displaystyle L^{1}$ sense (see Remark 2.23). The rest of the proof will follow similarly as that of Theorem 2.27.

Proof of Theorem 2.30

Here also, we consider the approximating problems Eq. 2.10. Since $\displaystyle u_{k}\in L^{2}(0,T;H^{1}_{0}(\Omega))$ , we take $\displaystyle u_{k}(x,t)\chi_{(0,\tau)}(t)$ as a test function in Eq. 2.10, to have

{}\begin{array}[]{l}\sup_{0\leq\tau\leq T}\int_{\Omega}u_{k}^{2}(x,\tau)dx+2\int_{0}^{T}\int_{\Omega}|\nabla u_{k}|^{2}dxdt+\int_{0}^{T}\int_{Q}\frac{\left(u_{k}(x,t)-u_{k}(y,t)\right)^{2}}{|x-y|^{n+2s}}dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq 2\iint_{\Omega_{T}}fu_{k}^{1-\gamma}dxdt+\left\|u_{0}\right\|_{L^{2}(\Omega)}.\end{array}

(3.6)

Case 1: $\displaystyle f\in L^{\frac{2}{\gamma+1}}\left(0,T;L^{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}(\Omega)\right)$
For this case, since $\displaystyle f\in L^{\frac{2}{\gamma+1}}\left(0,T;L^{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}(\Omega)\right)$ , we apply the Hölder inequality two times, first for the space integral and then for the time integral, to obtain

\begin{array}[]{rcl}\iint_{\Omega_{T}}fu_{k}^{1-\gamma}dxdt&\leq&\int_{0}^{T}\left(\int_{\Omega}|f(x,t)|^{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}dx\right)^{\frac{1}{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}}\left(\int_{\Omega}\left|u_{k}(x,t)\right|^{2^{*}}dx\right)^{\frac{1-\gamma}{2^{*}}}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &=&\int_{0}^{T}\|f\|_{L^{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}{(\Omega)}}\left\|u_{k}\right\|_{L^{2^{*}}(\Omega)}^{1-\gamma}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&\left(\int_{0}^{T}\|f\|_{L^{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}(\Omega)}^{\frac{2}{1+\gamma}}dt\right)^{\frac{1+\gamma}{2}}\left(\int_{0}^{T}\left\|u_{k}\right\|_{L^{2^{*}}(\Omega)}^{2}dt\right)^{\frac{1-\gamma}{2}}.\end{array}

We now apply the Sobolev embedding as of Theorem 2.8 in the last term on the right-hand-side to get

\iint_{\Omega_{T}}fu_{k}^{1-\gamma}dxdt\quad\leq(C(n))^{\frac{1-\gamma}{2}}\|f\|_{L^{\frac{2}{\gamma+1}}\left(0,T;L^{\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}}(\Omega)\right)}\left[\int_{0}^{T}\int_{\Omega}|\nabla u_{k}|^{2}dxdt\right]^{\frac{1-\gamma}{2}}.

Since $\displaystyle\gamma<1$ , so we use Young’s inequality to deduce from Eq. 3.6 that

\begin{array}[]{l}\sup_{0\leq\tau\leq T}\int_{\Omega}u_{k}^{2}(x,\tau)dx+\int_{0}^{T}\int_{\Omega}|\nabla u_{k}|^{2}dxdt+\int_{0}^{T}\int_{Q}\frac{\left(u_{k}(x,t)-u_{k}(y,t)\right)^{2}}{|x-y|^{n+2s}}dxdydt\leq C,\end{array}

where $\displaystyle C$ is a positive constant independent of $\displaystyle k$ .
Case 2: $\displaystyle f\in L^{\bar{m}}(\Omega_{T})$
For this case, we notice that as $\displaystyle f\in L^{\bar{m}}\left(\Omega_{T}\right)$ , we can apply the Hölder inequality in the first term on the right-hand-side in Eq. 3.6 with exponents $\displaystyle\bar{m}$ and $\displaystyle\bar{m}^{\prime}$ , to get

{}\begin{array}[]{l}\sup_{0\leq\tau\leq T}\int_{\Omega}u_{k}^{2}(x,\tau)dx+\int_{0}^{T}\int_{\Omega}|\nabla u_{k}|^{2}dxdt+\int_{0}^{T}\int_{Q}\frac{\left|u_{k}(x,t)-u_{k}(y,t)\right|^{2}}{|x-y|^{n+2s}}dxdydt\\ \leq 2\|f\|_{L^{\bar{m}}\left(\Omega_{T}\right)}\left[\iint_{\Omega_{T}}|u_{k}|^{(1-\gamma)\bar{m}^{\prime}}dxdt\right]^{\frac{1}{\bar{m}^{\prime}}}+\left\|u_{0}\right\|_{L^{2}(\Omega)}.\end{array}

(3.7)

We observe that $\displaystyle(1-\gamma)\bar{m}^{\prime}=\frac{2(n+2)}{n}$ and hence using the Hölder inequality with the exponents $\displaystyle\frac{n}{n-2}$ and $\displaystyle\frac{n}{2}$ and by Sobolev embedding (Theorem 2.8), we reach that

\begin{array}[]{rcl}\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{2(n+2)}{n}}dxdt&=&\iint_{\Omega_{T}}|u_{k}|^{2}\left|u_{k}\right|^{\frac{4}{n}}dxdt\\ &\leq&\int_{0}^{T}\left[\int_{\Omega}\left|u_{k}(x,t)\right|^{2}dx\right]^{\frac{2}{n}}\left\|u_{k}\right\|_{L^{2^{*}}(\Omega)}^{2}dt\\ &\leq&C(n)\left[\sup_{0\leq t\leq T}\int_{\Omega}\left|u_{k}(x,t)\right|^{2}dx\right]^{\frac{2}{n}}\int_{0}^{T}\int_{\Omega}|\nabla u_{k}|^{2}dxdt.\end{array}

So using Eq. 3.7 and convexity argument, we get

\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{2(n+2)}{n}}dxdt\leq C(n)2^{\frac{2}{n}}\left(\left(2\|f\|_{L^{\bar{m}}\left(\Omega_{T}\right)}\right)^{\frac{n+2}{n}}\left(\iint_{\Omega_{T}}|u_{k}|^{(1-\gamma){\bar{m}^{\prime}}}\right)^{\frac{n+2}{n\bar{m}^{\prime}}}+\left(\left\|u_{0}\right\|_{L^{2}(\Omega)}\right)^{\frac{n+2}{n}}\right).

Since $\displaystyle\frac{n+2}{n\bar{m}^{\prime}}=\frac{1-\gamma}{2}<1$ , we use the Young inequality to obtain

\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{2(n+2)}{n}}dxdt\leq C,

where $\displaystyle C$ is a positive constant independent of $\displaystyle k$ . Therefore, by Eq. 3.7 we deduce that the sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in the space $\displaystyle L^{2}(0,T;X_{0}^{s}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))\equiv L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))$ .
Now the rest of the proof follows similarly as that of Theorem 2.27. However, for the sake of completeness, we include it here in a bit different way.
Since the sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in the reflexive Banach space $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))$ , there exist a subsequence of $\displaystyle\left\{u_{k}\right\}_{k}$ , still indexed by $\displaystyle k$ , and a measurable function $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))$ such that $\displaystyle u_{k}\rightharpoonup u$ weakly in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle u_{k}\rightarrow u$ strongly in $\displaystyle L^{2}\left(\Omega_{T}\right)$ and a.e. in $\displaystyle\Omega\times(0,T)$ . In addition, since $\displaystyle u_{k}=u=0$ on $\displaystyle\mathcal{C}\Omega\times(0,T)$ , extending $\displaystyle u\equiv 0$ outside $\displaystyle\Omega$ , we obtain $\displaystyle u_{k}\rightarrow u$ for a.e. $\displaystyle(x,t)\in\mathbb{R}^{n}\times(0,T)$ . Again since $\displaystyle\left\{u_{k}\right\}$ is increasing in $\displaystyle k$ , and $\displaystyle u_{k}(x,0)=u_{0k}(x)$ , so we have $\displaystyle u(x,0)=u_{0}(x)$ in $\displaystyle L^{1}$ sense (Remark 2.23). Also, by Fatou’s Lemma, we get that $\displaystyle u\in L^{\infty}(0,T;L^{2}(\Omega))$ . Hence it follows that

\frac{u_{k}(x,t)-u_{k}(y,t)}{|x-y|^{\frac{n+2s}{2}}}\rightarrow\frac{u(x,t)-u(y,t)}{|x-y|^{\frac{n+2s}{2}}}\text{ a.e. in }Q\times(0,T).

We take an arbitrary test function $\displaystyle\varphi\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ in Eq. 2.10 to get

\begin{array}[]{l}\quad-\iint_{\Omega_{T}}u_{k}\varphi_{t}dxdt+\int_{0}^{T}\int_{\Omega}u_{k}(-\Delta\phi)dxdt\\ \quad+\frac{1}{2}\iint_{Q_{T}}\frac{\left(u_{k}(x,t)-u_{k}(y,t)\right)(\varphi(x,t)-\varphi(y,t))}{|x-y|^{n+2s}}dxdydt=\iint_{\Omega_{T}}\frac{f_{k}\varphi}{\left(u_{k}+\frac{1}{k}\right)^{\gamma}}dxdt.\end{array}

Since $\displaystyle\varphi\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ , therefore $\displaystyle\varphi_{t}$ and $\displaystyle\nabla\varphi$ both are in $\displaystyle L^{2}(\Omega_{T})$ and since strong convergence implies weak convergence too, so it is clear that

\lim_{k\rightarrow\infty}\int_{0}^{T}\int_{\Omega}u_{k}\phi_{t}dxdt=\int_{0}^{T}\int_{\Omega}u\phi_{t}dxdt

and by weak convergence in $\displaystyle L^{2}(0,T;H^{1}_{0}(\Omega))$ , we get

\begin{array}[]{rcl}\int_{0}^{T}\int_{\Omega}u_{k}(-\Delta\phi)dxdt&=&\int_{0}^{T}\int_{\Omega}\nabla u_{k}\cdot\nabla\phi dxdt\\ &\to&\int_{0}^{T}\int_{\Omega}\nabla u\cdot\nabla\phi dxdt=\int_{0}^{T}\int_{\Omega}u(-\Delta\phi)dxdt,\text{ as }k\to\infty.\end{array}

We now define

F_{k}(x,y,t)=\frac{u_{k}(x,t)-u_{k}(y,t)}{|x-y|^{\frac{n+2s}{2}}}\text{ and }F(x,y,t)=\frac{u(x,t)-u(y,t)}{|x-y|^{\frac{n+2s}{2}}}\text{. }

Then as $\displaystyle F_{k}\rightarrow F$ a.e. in $\displaystyle Q\times(0,T)$ and $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}\left(0,T;X_{0}^{s}(\Omega)\right)$ , by weak convergence we reach that

\begin{array}[]{l}\lim_{k\rightarrow\infty}\iint_{Q_{T}}\frac{\left(u_{k}(x,t)-u_{k}(y,t)\right)(\varphi(x,t)-\varphi(y,t))}{|x-y|^{n+2s}}dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ =\iint_{Q_{T}}\frac{(u(x,t)-u(y,t))(\varphi(x,t)-\varphi(y,t))}{|x-y|^{n+2s}}dxdydt,\end{array}

for all $\displaystyle\varphi\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ . Now since for all $\displaystyle\omega\subset\subset\Omega$ and for all $\displaystyle t_{0}>0,$ we have $\displaystyle u_{k}(x,t)\geq C\left(\omega,t_{0},n,s\right)$ in $\displaystyle\omega\times\left[t_{0},T\right)$ , so we reach that $\displaystyle u_{k}(x,t)\geq C$ in $\displaystyle\operatorname{Supp}\phi$ (as support of $\displaystyle\phi$ is compact in $\displaystyle\Omega_{T}$ ) for all $\displaystyle k\geq 1$ . Using this fact, we then have

0\leq\left|\frac{f_{k}\varphi}{\left(u_{k}+\frac{1}{k}\right)^{\gamma}}\right|\leq\frac{\|\varphi\|_{L^{\infty}\left(\Omega_{T}\right)}|f|}{C^{\gamma}}\in L^{1}\left(\Omega_{T}\right).

So, by the dominated convergence theorem, we get

\lim_{k\rightarrow\infty}\iint_{\Omega_{T}}\frac{f_{k}\varphi}{\left(u_{k}+\frac{1}{k}\right)^{\gamma}}dxdt=\iint_{\Omega_{T}}\frac{f\varphi}{u^{\gamma}}dxdt.

Finally, we pass to the limit as $\displaystyle k\rightarrow\infty$ to get

\begin{array}[]{l}\quad-\iint_{\Omega_{T}}u(x,t)\varphi_{t}(x,t)dxdt-\int_{0}^{T}\int_{\Omega}u(x,t)\Delta\phi(x,t)dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \quad+\frac{1}{2}\iint_{Q_{T}}\frac{(u(x,t)-u(y,t))(\varphi(x,t)-\varphi(y,t))}{|x-y|^{n+2s}}dxdydt=\iint_{\Omega_{T}}\frac{f(x,t)\varphi(x,t)}{u^{\gamma}(x,t)}dxdt,\end{array}

for all $\displaystyle\phi\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ . Therefore, $\displaystyle u$ is a weak solution to Eq. 2.12.

Proof of Theorem 2.32

As $\displaystyle u_{k}(\geq 0)\in L^{\infty}(\Omega_{T})\cap L^{2}(0,T;H^{1}_{0}(\Omega))$ , so $\displaystyle(\left(u_{k}(x,t)+\varepsilon\right)^{\theta}-\varepsilon^{\theta})\in L^{2}(0,T;H^{1}_{0}(\Omega))$ , for any $\displaystyle\varepsilon,\theta>0$ . So choosing $\displaystyle\gamma\leq\theta<1,0<\varepsilon<1/k$ , we take the test function $\displaystyle(\left(u_{k}(x,t)+\varepsilon\right)^{\theta}-\varepsilon^{\theta})\chi_{(0,\tau)}(t)$ in Eq. 2.10, to get for each $\displaystyle\tau\leq T$

\begin{array}[]{l}\quad\int_{0}^{\tau}\int_{\Omega}\left(u_{k}\right)_{t}\left((u_{k}+\varepsilon))^{\theta}-\varepsilon^{\theta}\right)dxdt+\int_{0}^{\tau}\int_{\Omega}\nabla u_{k}\cdot\nabla\left((u_{k}+\varepsilon)^{\theta}-\varepsilon^{\theta}\right)dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\frac{1}{2}\int_{0}^{\tau}\int_{Q}\frac{(u_{k}(x,t)-u_{k}(y,t))((u_{k}(x,t)+\varepsilon)^{\theta}-\left(u_{k}(y,t)+\varepsilon\right)^{\theta})}{|x-y|^{n+2s}}dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\iint_{\Omega_{T}}f(x,t)\left(u_{k}(x,t)+\varepsilon\right)^{\theta-\gamma}dxdt.\end{array}

Letting $\displaystyle\varepsilon\rightarrow 0$ , integrating the first term, and taking the supremum over $\displaystyle\tau\in[0,T]$ , we get

{}\begin{array}[]{c}\quad\frac{2}{\theta+1}\sup_{0\leq\tau\leq T}\int_{\Omega}\left|u_{k}(x,\tau)\right|^{\theta+1}dx+2\int_{0}^{T}\int_{\Omega}\nabla u_{k}\cdot\nabla u_{k}^{\theta}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\int_{0}^{T}\int_{Q}\frac{\left(u_{k}(x,t)-u_{k}(y,t)\right)\left(u_{k}^{\theta}(x,t)-u_{k}^{\theta}(y,t)\right)}{|x-y|^{n+2s}}dxdydt\leq 2\iint_{\Omega_{T}}fu_{k}^{\theta-\gamma}dxdt+\frac{2}{\theta+1}\left\|u_{0}\right\|_{L^{2}(\Omega)}.\end{array}

(3.8)

Since all terms on the left are positive, taking supremum was allowed. Then by item $\displaystyle i)$ of Lemma 2.11, we get

{}\begin{array}[]{c}\quad\frac{\theta+1}{2\theta}\sup_{0\leq\tau\leq T}\int_{\Omega}\left|u_{k}(x,\tau)\right|^{\theta+1}dx+\frac{(\theta+1)^{2}}{2}\int_{0}^{T}\int_{\Omega}u_{k}^{\theta-1}|\nabla u_{k}|^{2}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\int_{0}^{T}\int_{Q}\frac{|u_{k}^{\frac{\theta+1}{2}}(x,t)-u_{k}^{\frac{\theta+1}{2}}(y,t)|^{2}}{|x-y|^{n+2s}}dxdydt\leq\frac{2}{\theta}\left(\iint_{\Omega_{T}}fu_{k}^{\theta-\gamma}dxdt+\left\|u_{0}\right\|_{L^{2}(\Omega)}\right).\end{array}

In the previous inequality, we have used the fact that $\displaystyle\theta+1<2$ , now we observe that the term $\displaystyle\frac{\theta+1}{2\theta}>1$ in the left-hand-side can be dropped. So we get

{}\begin{array}[]{c}\sup_{0\leq\tau\leq T}\int_{\Omega}\left|u_{k}(x,\tau)\right|^{\theta+1}dx+\frac{(\theta+1)^{2}}{4}\int_{0}^{\tau}\int_{\Omega}u_{k}^{\theta-1}|\nabla u_{k}|^{2}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\int_{0}^{T}\int_{Q}\frac{|u_{k}^{\frac{\theta+1}{2}}(x,t)-u_{k}^{\frac{\theta+1}{2}}(y,t)|^{2}}{|x-y|^{n+2s}}dxdydt\leq\frac{2}{\theta}\left(\iint_{\Omega_{T}}fu_{k}^{\theta-\gamma}dxdt+\left\|u_{0}\right\|_{L^{2}(\Omega)}\right).\end{array}

(3.9)

Again, using the same tool like Theorem 2.30 and the Hölder inequality with exponent $\displaystyle\frac{n}{n-2}$ and $\displaystyle\frac{n}{2}$ we get

\begin{array}[]{rcl}\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{(\theta+1)(n+2)}{n}}dxdt&=&\iint_{\Omega_{T}}\left|u_{k}\right|^{2\frac{\theta+1}{2}}\left|u_{k}\right|^{\frac{4}{n}\frac{\theta+1}{2}}dxdt\\ &\leq&\int_{0}^{T}\left(\int_{\Omega}\left|u_{k}(x,t)\right|^{2^{*}\frac{\theta+1}{2}}dx\right)^{\frac{n-2}{n}}\left(\int_{\Omega}\left|u_{k}(x,t)\right|^{\theta+1}dx\right)^{\frac{2}{n}}dt\\ &=&\int_{0}^{T}\left\|u_{k}^{\frac{\theta+1}{2}}\right\|_{L^{2^{*}}(\Omega)}^{2}\left(\int_{\Omega}\left|u_{k}(x,t)\right|^{\theta+1}dx\right)^{\frac{2}{n}}dt.\end{array}

We now take the supremum over $\displaystyle t\in[0,T]$ , and apply the Sobolev embedding as that of Theorem 2.8, to get that

\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{(\theta+1)(n+2)}{n}}dxdt\leq C(n)\left(\sup_{0\leq t\leq T}\int_{\Omega}\left|u_{k}(x,t)\right|^{\theta+1}dx\right)^{\frac{2}{n}}\iint_{\Omega_{T}}|\nabla u_{k}^{\frac{\theta+1}{2}}|^{2}dxdt.

Using Eq. 3.9 and convexity argument, we now get

{}\begin{array}[]{rcl}\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{(\theta+1)(n+2)}{n}}dxdt&\leq&C(n)\left(\frac{2}{\theta}\right)^{\frac{n+2}{n}}\left(\iint_{\Omega_{T}}fu_{k}^{\theta-\gamma}dxdt+\left\|u_{0}\right\|_{L^{2}(\Omega)}\right)^{\frac{n+2}{n}}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&C(n)2^{\frac{2}{n}}\left(\frac{2}{\theta}\right)^{\frac{n+2}{n}}\left(\left(\iint_{\Omega_{T}}fu_{k}^{\theta-\gamma}dxdt\right)^{\frac{n+2}{n}}+\left\|u_{0}\right\|_{L^{2}(\Omega)}^{\frac{(n+2)}{n}}\right).\end{array}

(3.10)

Case 1: $\displaystyle m=1$
Now for the case $\displaystyle m=1$ , we take $\displaystyle\theta=\gamma$ in the Eq. 3.10. So we obtain

{}\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{(\gamma+1)(n+2)}{n}}dxdt\leq C(n)2^{\frac{2}{n}}\left(\frac{2}{\gamma}\right)^{\frac{n+2}{n}}\left(\|f\|_{L^{1}(\Omega_{T})}^{\frac{n+2}{n}}+\left\|u_{0}\right\|_{L^{2}(\Omega)}^{\frac{(n+2)}{n}}\right).

(3.11)

Also, by Eq. 3.9 we easily have

{}\left\|u_{k}\right\|_{L^{\infty}(0,T;L^{\gamma+1}(\Omega))}\leq\frac{2}{\gamma}\left[\|f\|_{L^{1}\left(\Omega_{T}\right)}+\left\|u_{0}\right\|_{L^{2}(\Omega)}\right].

(3.12)

Case 2: $\displaystyle m>1$
For $\displaystyle m>1$ , we take $\displaystyle\gamma<\theta<1$ . Using Eq. 3.10 and applying Hölder inequality with exponent $\displaystyle m$ and $\displaystyle m^{\prime}$ , we have

\iint_{\Omega_{T}}|u_{k}|^{\frac{(\theta+1)(n+2)}{n}}dxdt\leq C_{1}\|f\|_{L^{m}\left(\Omega_{T}\right)}^{\frac{n+2}{n}}\left[\iint_{\Omega_{T}}|u_{k}|^{m^{\prime}(\theta-\gamma)}dxdt\right]^{\frac{n+2}{nm^{\prime}}}+C_{1}\left\|u_{0}\right\|_{L^{2}(\Omega)}^{\frac{(n+2)}{n}},

where $\displaystyle C_{1}=C(n)2^{\frac{2}{n}}\left(\frac{2}{\theta}\right)^{\frac{n+2}{n}}$ . We now choose $\displaystyle\gamma<\theta<1$ to be such that

\frac{(\theta+1)(n+2)}{n}=m^{\prime}(\theta-\gamma),\text{ i.e. }\theta=\frac{(n+2)(m-1)+nm\gamma}{n-2(m-1)}.

We note that the condition $\displaystyle\theta<1$ is equivalent to $\displaystyle m<\bar{m}$ ; while $\displaystyle\gamma<\theta$ is always fulfilled. Since $\displaystyle\frac{n+2}{nm^{\prime}}<1$ , applying Young’s inequality with $\displaystyle\varepsilon>0$ , we get

\vskip 3.0pt plus 1.0pt minus 1.0pt\iint_{\Omega_{T}}|u_{k}|^{\frac{(\theta+1)(n+2)}{n}}dxdt\leq C_{1}\|f\|_{L^{m}\left(\Omega_{T}\right)}^{\frac{n+2}{n}}\left(\varepsilon\iint_{\Omega_{T}}|u_{k}|^{\frac{(\theta+1)(n+2)}{n}}dxdt+C(\varepsilon)\right)+C_{1}\left\|u_{0}\right\|_{L^{2}(\Omega)}^{\frac{(n+2)}{n}}.

We choose $\displaystyle\varepsilon$ small enough such that $\displaystyle\varepsilon C_{1}\|f\|_{L^{m}\left(\Omega_{T}\right)}^{\frac{n+2}{n}}=\frac{1}{2}$ and using the fact that

\sigma:=\frac{m(\gamma+1)(n+2)}{n-2(m-1)}=\frac{(\theta+1)(n+2)}{n}=m^{\prime}(\theta-\gamma),

we get

{}\iint_{\Omega_{T}}\left|u_{k}\right|^{\sigma}dxdt\leq C,

(3.13)

where $\displaystyle C$ is a positive constant independent of $\displaystyle k$ . Now in Eq. 3.9 we use Hölder inequality, to get

\sup_{0\leq\tau\leq T}\int_{\Omega}u_{k}^{\theta+1}(x,\tau)dx\leq\frac{2}{\theta}\left(\|f\|_{L^{m}\left(\Omega_{T}\right)}\left\|u_{k}\right\|_{L^{\sigma}\left(\Omega_{T}\right)}^{\frac{\sigma}{m^{\prime}}}+\left\|u_{0}\right\|_{L^{2}(\Omega)}\right).

Since $\displaystyle\gamma<\theta$ and by Eq. 3.13 we conclude that the sequence $\displaystyle\left\{u_{k}\right\}$ is uniformly bounded in $\displaystyle L^{\infty}(0,T;L^{\gamma+1}(\Omega))$ , the same thing holds for the case $\displaystyle m=1$ by Eq. 3.12. Finally, by Eq. 3.11 and Eq. 3.13 we conclude that in both cases, that is $\displaystyle 1\leq m<\bar{m}$ , the sequence $\displaystyle\left\{u_{k}\right\}$ is uniformly bounded in $\displaystyle L^{\sigma}(\Omega_{T})$ , $\displaystyle\sigma:=\frac{m(\gamma+1)(n+2)}{n-2(m-1)}$ .
Now from Eq. 3.9, again using Hölder inequality, we estimate as

{}\iint_{\Omega_{T}}|\nabla u_{k}^{\frac{\theta+1}{2}}|^{2}dxdt\leq\frac{2}{\theta}\left(\|f\|_{L^{m}\left(\Omega_{T}\right)}\left\|u_{k}\right\|_{L^{\sigma}\left(\Omega_{T}\right)}^{\frac{\sigma}{m^{\prime}}}+\left\|u_{0}\right\|_{L^{2}(\Omega)}\right)\leq C,

(3.14)

where $\displaystyle C>0$ is a constant independent of $\displaystyle k$ . Let $\displaystyle 1<\bar{q}<2$ will be specified later. By Hölder inequality, we have

{}\begin{array}[]{rcl}\iint_{\Omega_{T}}|\nabla u_{k}|^{\bar{q}}dxdt=\iint_{\Omega_{T}}\frac{|\nabla u_{k}|^{\bar{q}}u_{k}^{\theta-1}}{u_{k}^{\theta-1}}dxdt&\leq&\left(\iint_{\Omega_{T}}|\nabla u_{k}|^{2}u_{k}^{\theta-1}dxdt\right)^{\frac{\bar{q}}{2}}\times\left(\iint_{\Omega_{T}}\frac{u_{k}^{\theta-1}}{u_{k}^{\frac{2(\theta-1)}{2-\bar{q}}}}dxdt\right)^{\frac{2-\bar{q}}{2}}\\ &\stackrel{{\scriptstyle\lx@cref{creftype~refnum}{eqn25}}}{{\leq}}&C\left(\iint_{\Omega_{T}}u_{k}^{\frac{\bar{q}(1-\theta)}{2-\bar{q}}}dxdt\right)^{\frac{2-\bar{q}}{2}}.\end{array}

(3.15)

We now choose $\displaystyle\bar{q}$ to be such that

\frac{\bar{q}(1-\theta)}{2-\bar{q}}=\sigma=\frac{m(\gamma+1)(n+2)}{n-2(m-1)}\text{ i.e. }\bar{q}=\frac{m(\gamma+1)(n+2)}{n+2-m(1-\gamma)}.

We note that $\displaystyle\bar{q}<2$ is equivalent to $\displaystyle m<\bar{m}$ ; while $\displaystyle\bar{q}>1$ is always fulfilled. Then we get by embedding results and using Eqs. 3.13 and 3.15

\begin{array}[]{rcl}\int_{0}^{T}\int_{\Omega}\int_{\Omega}\frac{\left|u_{k}(x,t)-u_{k}(y,t)\right|^{\bar{q}}}{|x-y|^{n+\bar{q}s}}dydxdt\leq C(n,s)\iint_{\Omega_{T}}|\nabla u_{k}|^{\bar{q}}dxdt\leq C_{2}\left(\iint_{\Omega_{T}}u_{k}^{\sigma}(x,t)dxdt\right)^{\frac{2-\bar{q}}{2}}\leq C_{3},\end{array}

where $\displaystyle C_{3}$ is a positive constant independent of $\displaystyle k$ . Thus, $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{\bar{q}}(0,T;W_{0}^{s,\bar{q}}(\Omega))\cap L^{\bar{q}}(0,T;W_{0}^{1,\bar{q}}(\Omega))\equiv L^{\bar{q}}(0,T;W_{0}^{1,\bar{q}}(\Omega))$ .
Since the sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in the reflexive Banach space $\displaystyle L^{\bar{q}}(0,T;W_{0}^{1,\bar{q}}(\Omega))$ , there exist a subsequence of $\displaystyle\left\{u_{k}\right\}_{k}$ still indexed by $\displaystyle k$ and a measurable function $\displaystyle u\in L^{\bar{q}}(0,T;W_{0}^{1,\bar{q}}(\Omega))$ such that $\displaystyle u_{k}\rightharpoonup u$ weakly in $\displaystyle L^{\bar{q}}(0,T;W_{0}^{1,\bar{q}}(\Omega))$ . Also by Fatou’s lemma, we will get $\displaystyle u\in L^{\infty}(0,T;L^{\gamma+1}(\Omega))$ and $\displaystyle u\in L^{\sigma}(\Omega_{T})$ , with $\displaystyle\sigma:=\frac{m(\gamma+1)(n+2)}{n-2(m-1)}$ . As before, using the monotonicity of the sequence $\displaystyle\left\{u_{k}\right\}$ , we get using Beppo Levi’s theorem that $\displaystyle u_{k}\rightarrow u$ strongly in $\displaystyle L^{1}(\Omega_{T})$ and $\displaystyle u_{k}\rightarrow u$ a.e in $\displaystyle\mathbb{R}^{n}\times(0,T)$ . By Remark 2.23, this pointwise limit will satisfy $\displaystyle u(\cdot,0)=u_{0}(\cdot)$ in $\displaystyle L^{1}$ sense. Then, the rest of the proof will follow from the proof of Theorem 2.27.

Remark 3.2.

We observe that the sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{r}(\Omega)$ for every $\displaystyle 1\leq r\leq\sigma$ , then following the same lines of the previous proof, we can show that the sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{q}(0,T;W_{0}^{s,q}(\Omega))\cap L^{q}(0,T;W_{0}^{1,q}(\Omega))\equiv L^{q}(0,T;W_{0}^{1,q}(\Omega))$ for all $\displaystyle 1<q\leq\bar{q}$ where $\displaystyle 1\leq m<\bar{m}$ .

Proof of Theorem 2.35

Let us introduce the following notations: for any measurable function $\displaystyle v$ we define

\begin{array}[]{c}A_{m}(v)=\left\{(x,t)\in\Omega_{T}:v(x,t)>m\right\},\text{ and for a.e. }t\in(0,T),\quad A_{m}^{t}(v)=\{x\in\Omega:v(x,t)>m\}.\end{array}

We use the idea of the classical proof by D.G. Aronson and J. Serrin, which is to prove a uniform $\displaystyle L^{\infty}$ bound for $\displaystyle u_{k}$ in $\displaystyle\Omega\times(0,\tau)$ , for a positive (small) $\displaystyle\tau$ (to be specified later), and then to iterate such an estimate. We consider the approximated problem Eq. 2.10 and take $\displaystyle(u_{k}-m)_{+}\chi_{(0,t)}\in L^{2}(0,T;H^{1}_{0}(\Omega))$ , for $\displaystyle m>0,t\in(0,\tau),\tau<T$ , as a test function to obtain

{}\begin{array}[]{c}\int_{\Omega}\varphi_{m}\left(u_{k}(x,t)\right)dx+\int_{0}^{t}\int_{\Omega}\nabla u_{k}\cdot\nabla(u_{k}-m)_{+}dxd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{\left(u_{k}(x,\theta)-u_{k}(y,\theta)\right)\left((u_{k}-m)_{+}(x,\theta)-(u_{k}-m)_{+}(y,\theta)\right)}{|x-y|^{n+2s}}dxdyd\theta\\ \leq\int_{0}^{\tau}\int_{\Omega}\frac{f_{k}(u_{k}-m)_{+}}{(u_{k}+\frac{1}{k})^{\gamma}}dxd\theta+\int_{\Omega}\varphi_{m}\left(u_{k}(x,0)\right)dx,\end{array}

(3.16)

where $\displaystyle\varphi_{m}(\rho)=\int_{0}^{\rho}(\sigma-m)_{+}d\sigma=\frac{(\rho-m)_{+}^{2}}{2}$ . We choose a $\displaystyle m>0$ large enough such that $\displaystyle m>\operatorname{max}\{\left\|u_{0}\right\|_{L^{\infty}(\Omega)},1\}$ , in order to neglect the last term above. Noting that $\displaystyle\nabla u_{k}\cdot\nabla(u_{k}-m)_{+}=|\nabla(u_{k}-m)_{+}|^{2}$ , and

\left(u_{k}(x,\theta)-u_{k}(y,\theta)\right)\left((u_{k}-m)_{+}(x,\theta)-(u_{k}-m)_{+}(y,\theta)\right)\geq\left((u_{k}-m)_{+}(x,\theta)-(u_{k}-m)_{+}(y,\theta)\right)^{2},

we take supremum over $\displaystyle t\in(0,\tau]$ in Eq. 3.16, to get

{}\begin{array}[]{c}\|(u_{k}-m)_{+}\|^{2}_{L^{\infty}(0,\tau;L^{2}(\Omega))}+\|(u_{k}-m)_{+}\|^{2}_{L^{2}(0,\tau;X^{s}_{0}(\Omega))}+\|(u_{k}-m)_{+}\|^{2}_{L^{2}(0,\tau;H^{1}_{0}(\Omega))}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\int_{0}^{\tau}\int_{A^{t}_{m,k}}f(u_{k}-m)_{+}dxdt.\end{array}

(3.17)

Note that, in order to deal with the singularity, we have used the fact that $\displaystyle m\geq 1$ and hence $\displaystyle u_{k}>1$ in $\displaystyle A^{t}_{m,k}$ , here the subscript $\displaystyle k$ in $\displaystyle A^{t}_{m,k}$ denotes that we are considering the function $\displaystyle u_{k}$ . Now the term of the right-hand side above can be estimated as follows,

{}\int_{0}^{\tau}\int_{A_{m,k}^{t}}f(u_{k}-m)_{+}dxdt\leq\int_{0}^{\tau}\int_{A_{m,k}^{t}}f(u_{k}-m)_{+}^{2}dxdt+\int_{0}^{\tau}\int_{A_{m,k}^{t}}fdxdt.

(3.18)

We now study each member present in the right-hand side of Eq. 3.18. We first define the followings,

\bar{r}=2r^{\prime},\bar{q}=2q^{\prime},\eta=\frac{2\eta_{1}}{n},\hat{r}=\bar{r}(1+\eta),\hat{q}=\bar{q}(1+\eta),

where $\displaystyle\eta_{1}=1-\frac{1}{r}-\frac{n}{2q}$ . Note that $\displaystyle\eta_{1}\in(0,1)$ as $\displaystyle 0<\frac{1}{r}+\frac{n}{2q}<1$ . Further, simple calculation yields that $\displaystyle\frac{1}{\hat{r}}+\frac{n}{2\hat{q}}=\frac{n}{4}$ . Now, applying Hölder inequality repeatedly, we estimate the first term as

\begin{array}[]{rcl}\int_{0}^{\tau}\int_{A_{m,k}^{t}}f(u_{k}-m)_{+}^{2}dxdt&\leq&\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}|f(x,t)|^{q}dx\right)^{\frac{1}{q}}\left(\int_{A_{m,k}^{t}}(u_{k}-m)_{+}^{2q^{\prime}}dx\right)^{\frac{1}{q^{\prime}}}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&\left(\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}|f(x,t)|^{q}dx\right)^{\frac{r}{q}}dt\right)^{\frac{1}{r}}\left(\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}(u_{k}-m)_{+}^{2q^{\prime}}dx\right)^{\frac{r^{\prime}}{q^{\prime}}}dt\right)^{\frac{1}{r^{\prime}}}.\end{array}

As again by Hölder inequality with exponent $\displaystyle(1+\eta)$ , we have

\begin{array}[]{rcl}\left(\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}(u_{k}-m)_{+}^{2q^{\prime}}dx\right)^{\frac{r^{\prime}}{q^{\prime}}}dt\right)^{\frac{1}{r^{\prime}}}&\leq&\left(\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}(u_{k}-m)_{+}^{2q^{\prime}(1+\eta)}dx\right)^{\frac{r^{\prime}}{q^{\prime}(1+\eta)}}|A^{t}_{m,k}|^{\frac{r^{\prime}\eta}{q^{\prime}(1+\eta)}}dt\right)^{\frac{1}{r^{\prime}}}\\ &=&\left(\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}(u_{k}-m)_{+}^{\hat{q}}dx\right)^{\frac{2r^{\prime}}{\hat{q}}}|A^{t}_{m,k}|^{\frac{2r^{\prime}\eta}{\hat{q}}}dt\right)^{\frac{1}{r^{\prime}}}\\ &\leq&\left(\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}(u_{k}-m)_{+}^{\hat{q}}dx\right)^{\frac{\hat{r}}{\hat{q}}}dt\right)^{\frac{1}{r^{\prime}(1+\eta)}}\left(\int_{0}^{\tau}|A^{t}_{m,k}|^{\frac{\hat{r}}{\hat{q}}}\right)^{\frac{\eta}{r^{\prime}(1+\eta)}},\end{array}

therefore

\int_{0}^{\tau}\int_{A_{m,k}^{t}}f(u_{k}-m)_{+}^{2}dxdt\leq\|f\|_{L^{r}(0,T;L^{q}(\Omega))}\|(u_{k}-m)_{+}\|^{2}_{L^{\hat{r}}(0,\tau;L^{\hat{q}}(\Omega))}\mu_{k}(m)^{\frac{2\eta}{\hat{r}}},

where $\displaystyle\mu_{k}(m)=\int_{0}^{\tau}\left|A_{m,k}^{t}\right|^{\frac{\hat{r}}{\hat{q}}}dt$ . We now use Gagliardo-Nirenberg inequality (Theorem 2.10) for $\displaystyle(u_{k}-m)_{+}$ to get

{}\begin{array}[]{rcl}\int_{0}^{\tau}\int_{A_{m,k}^{t}}f(u_{k}-m)_{+}^{2}dxdt&\leq&c\|f\|_{L^{r}(0,T;L^{q}(\Omega))}\mu_{k}(m)^{\frac{2\eta}{\hat{r}}}\left(\int_{0}^{\tau}\left\|(u_{k}-m)_{+}\right\|_{L^{2}(\Omega)}^{(1-\theta)\hat{r}}\left\|\nabla(u_{k}-m)_{+}\right\|_{L^{2}\left(\Omega\right)}^{\hat{r}\theta}dt\right)^{\frac{2}{\hat{r}}}\\ &\leq&c\|f\|\mu_{k}(m)^{\frac{2\eta}{\hat{r}}}\bigg{[}\left\|(u_{k}-m)_{+}\right\|_{L^{\infty}\left(0,\tau;L^{2}(\Omega)\right)}^{2}+\left\|(u_{k}-m)_{+}\right\|_{L^{2}(0,\tau;H^{1}_{0}(\Omega))}^{2}\bigg{]}.\end{array}

(3.19)

where $\displaystyle\hat{r}\theta=2$ , and $\displaystyle c$ is a constant independent of the choice of $\displaystyle k$ and $\displaystyle m$ . Note that we have used the fact that any function in $\displaystyle H^{1}_{0}(\Omega)$ can be considered as a function in $\displaystyle H^{1}(\mathbb{R}^{n})$ . Also, we applied Young’s inequality with exponents $\displaystyle\frac{\hat{r}}{2}$ and its conjugate.
On the other hand, the second term on the right-hand side in Eq. 3.18 can be estimated by Hölder inequality as

{}\begin{array}[]{rcl}\int_{0}^{\tau}\int_{A_{m,k}^{t}}fdxdt&\leq&\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}|f(x,t)|^{q}dx\right)^{\frac{1}{q}}|A^{t}_{m,k}|^{\frac{1}{q^{\prime}}}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&\left(\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}|f(x,t)|^{q}dx\right)^{\frac{r}{q}}dt\right)^{\frac{1}{r}}\left(\int_{0}^{\tau}|A^{t}_{m,k}|^{\frac{r^{\prime}}{q^{\prime}}}dt\right)^{\frac{1}{r^{\prime}}}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&\|f\|_{L^{r}(0,T;L^{q}(\Omega))}\mu_{k}(m)^{\frac{2(1+\eta)}{\hat{r}}}.\end{array}

(3.20)

Denoting

|||(u_{k}-m)_{+}|||^{2}=\|(u_{k}-m)_{+}\|_{L^{\infty}(0,\tau;L^{2}(\Omega))}^{2}+\|(u_{k}-m)_{+}\|_{L^{2}(0,\tau;H^{1}_{0}(\Omega))}^{2},

and using Eq. 3.19, Eq. 3.20, we get from Eq. 3.17,

|||(u_{k}-m)_{+}|||^{2}\leq c\|f\|_{L^{r}(0,T;L^{q}(\Omega))}\left[\mu_{k}(m)^{\frac{2\eta}{\hat{r}}}|||(u_{k}-m)_{+}|||^{2}+\mu_{k}(m)^{\frac{2(1+\eta)}{\hat{r}}}\right],

where $\displaystyle c$ is a constant which does not depend on $\displaystyle k$ and $\displaystyle m$ . Note that $\displaystyle\mu_{k}(m)\leq\tau|\Omega|^{\frac{\hat{r}}{\hat{q}}}$ , for all $\displaystyle k\in\mathbb{N}$ and $\displaystyle m$ , so that we can fix $\displaystyle\tau$ , independent of $\displaystyle u_{k}$ and $\displaystyle m$ , suitable small in such a way that $\displaystyle c\mu_{k}(m)^{\frac{2\eta}{\hat{r}}}\|f\|_{L^{r}(0,T;L^{q}(\Omega))}=\frac{1}{2}$ and use again the Gagliardo-Nirenberg inequality (see Eq. 3.19), to deduce that

{}\left\|(u_{k}-m)_{+}\right\|_{L^{\hat{r}}\left(0,\tau;L^{\hat{q}}(\Omega)\right)}^{2}\leq c|||(u_{k}-m)_{+}|||^{2}\leq c\|f\|_{L^{r}(0,T;L^{q}(\Omega))}\mu_{k}(m)^{\frac{2(1+\eta)}{\hat{r}}}.

(3.21)

Consider $\displaystyle l>m>\operatorname{max}\{\left\|u_{0}\right\|_{L^{\infty}(\Omega)},1\}:=m_{0}$ . Then $\displaystyle A^{t}_{l,k}\subset A^{t}_{m,k}$ , and using Eq. 3.21, we get

\begin{array}[]{rcl}(l-m)\mu_{k}(l)^{\frac{1}{\hat{r}}}=\left(\int_{0}^{\tau}\left((l-m)^{\hat{q}}\left|A_{l,k}^{t}\right|\right)^{\frac{\hat{r}}{\hat{q}}}dt\right)^{\frac{1}{\hat{r}}}&\leq&\left(\int_{0}^{\tau}\left(\int_{A_{l,k}^{t}}(u_{k}-m)_{+}^{\hat{q}}dx\right)^{\frac{\hat{r}}{\hat{q}}}dt\right)^{\frac{1}{\hat{r}}}\\ &\leq&\left(\int_{0}^{\tau}\left(\int_{A_{m,k}^{t}}(u_{k}-m)_{+}^{\hat{q}}dx\right)^{\frac{\hat{r}}{\hat{q}}}dt\right)^{\frac{1}{\hat{r}}}\leq c\mu_{k}(m)^{\frac{1+\eta}{\hat{r}}}.\end{array}

Therefore for all $\displaystyle l>m>m_{0}$ , we have

\mu_{k}(l)\leq\frac{c}{(l-m)^{\hat{r}}}\mu_{k}(m)^{1+\eta},

where c is a constant independent of $\displaystyle k$ . Now applying [27, Lemma B. $\displaystyle 1$ ], we conclude that $\displaystyle\mu_{k}(m_{0}+d_{k})=0,$ where $\displaystyle d_{k}=c[\mu_{k}(m_{0})]^{\eta}2^{\hat{r}(1+\eta)/\eta}$ . As for each $\displaystyle k$ , $\displaystyle d_{k}\leq c\left(T|\Omega|^{\frac{\hat{r}}{\hat{q}}}\right)^{\eta}$ , we get

\left\|u_{k}\right\|_{L^{\infty}(\Omega\times[0,\tau])}\leq d.

We can iterate this procedure in the sets $\displaystyle\Omega\times[\tau,2\tau],\cdots,\Omega\times[j\tau,T]$ , where $\displaystyle T-j\tau\leqslant\tau$ to conclude that

\left\|u_{k}\right\|_{L^{\infty}\left(\Omega_{T}\right)}\leq C\quad\text{ uniformly in }k\in\mathbb{N}.

Now, since the sequence $\displaystyle\{u_{k}\}_{k}$ is increasing in $\displaystyle k$ , we have $\displaystyle\|u\|_{L^{\infty}(\Omega)}\leq C$ . Further, if $\displaystyle\gamma\leq 1$ , testing Eq. 2.10 with the function $\displaystyle u_{k}$ we can deduce $\displaystyle u\in L^{2}(0,T;H^{1}_{0}(\Omega))$ (see Theorem 2.26).

Proof of Theorem 2.36

For $\displaystyle\gamma\geq 1$ , we choose $\displaystyle\delta>\frac{1+\gamma}{2}$ and for $\displaystyle\gamma<1$ , we choose $\displaystyle\delta>1$ , and take $\displaystyle u_{k}^{2\delta-1}\chi_{(0,t)},0<t\leq T$ as test function in Eq. 2.10, with $\displaystyle u_{0}\equiv 0$ . We note that, such a test function is admissible as $\displaystyle u_{k}\in L^{\infty}(\Omega_{T})$ . We get $\displaystyle\forall t\leq T,$

\begin{array}[]{c}\quad\frac{1}{2\delta}\int_{\Omega}u_{k}^{2\delta}(x,t)dx+\int_{0}^{t}\int_{\Omega}\nabla u_{k}\cdot\nabla u_{k}^{2\delta-1}dxd\theta\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \quad+\frac{1}{2}\int_{0}^{t}\int_{Q}\frac{\left(u_{k}^{2\delta-1}(x,\theta)-u_{k}^{2\delta-1}(y,\theta)\right)\left(u_{k}(x,\theta)-u_{k}(y,\theta)\right)}{|x-y|^{n+2s}}dxdyd\theta\leq\iint_{\Omega_{t}}fu_{k}^{2\delta-1-\gamma}dxd\theta.\end{array}

Taking supremum over $\displaystyle t\in(0,T]$ and using item $\displaystyle(i)$ of Lemma 2.11, by Hölder and Sobolev inequalities, we have

{}\begin{array}[]{c}\sup_{t\in[0,T]}\int_{\Omega}u_{k}^{2\delta}(x,t)dx+\frac{2(2\delta-1)}{\delta}\lambda\int_{0}^{T}\left\|u_{k}^{\delta}\right\|_{L^{2^{*}}(\Omega)}^{2}dt+\frac{(2\delta-1)}{\delta}\lambda_{s}\int_{0}^{T}\left\|u_{k}^{\delta}\right\|_{L^{2_{s}^{*}}(\Omega)}^{2}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq 2\delta\left\|f\right\|_{L^{r}\left(0,T;L^{q}(\Omega)\right)}\left(\int_{0}^{T}\left[\int_{\Omega}u_{k}^{(2\delta-1-\gamma)q^{\prime}}dx\right]^{\frac{r^{\prime}}{q^{\prime}}}dt\right)^{\frac{1}{r^{\prime}}}.\end{array}

(3.22)

Note that $\displaystyle\frac{2\delta-1}{\delta}>1$ , so we can ignore the constants in left. Denoting by $\displaystyle A=\left(\int_{0}^{T}\left\|u_{k}\right\|_{L^{(2\delta-1-\gamma)q^{\prime}}(\Omega)}^{(2\delta-1-\gamma)r^{\prime}}dt\right)^{\frac{1}{r^{\prime}}},$ we get from Eq. 3.22,

{}\sup_{t\in[0,T]}\int_{\Omega}u_{k}^{2\delta}dx\leq 2\delta\left\|f\right\|_{L^{r}\left(0,T;L^{q}(\Omega)\right)}A,

(3.23)

and

{}\int_{0}^{T}\left[\int_{\Omega}u_{k}^{2_{s}^{*}\delta}dx\right]^{\frac{2}{2_{s}^{*}}}dt\leq c\int_{0}^{T}\left[\int_{\Omega}u_{k}^{2^{*}\delta}dx\right]^{\frac{2}{2^{*}}}dt\leq\frac{c}{\lambda}2\delta\left\|f\right\|_{L^{r}\left(0,T;L^{q}(\Omega)\right)}A.

(3.24)

Case 1: $\displaystyle 1<q<\frac{nr}{n+2(r-1)}$ i.e. $\displaystyle\frac{1}{r}<\frac{n}{n-2}\frac{1}{q}-\frac{2}{n-2}$
Since, $\displaystyle 1<q<\frac{nr}{n+2(r-1)}$ implies $\displaystyle q<r$ and $\displaystyle\frac{r^{\prime}}{q^{\prime}}<\frac{2}{2^{*}}$ , we apply Hölder inequality with exponent $\displaystyle\frac{2q^{\prime}}{2^{*}r^{\prime}}$ to get

{}A\leq T^{\frac{1}{r^{\prime}}-\frac{2^{*}}{2q^{\prime}}}\left[\int_{0}^{T}\left(\int_{\Omega}u_{k}^{(2\delta-1-\gamma)q^{\prime}}dx\right)^{\frac{2}{2^{*}}}dt\right]^{\frac{2^{*}}{2q^{\prime}}}.

(3.25)

Thus from Eq. 3.24, it follows

{}\int_{0}^{T}\left[\int_{\Omega}u_{k}^{2^{*}\delta}dx\right]^{\frac{2}{2^{*}}}dt\leq 2\delta cA\leq 2\delta c\left[\int_{0}^{T}\left(\int_{\Omega}u_{k}^{(2\delta-1-\gamma)q^{\prime}}dx\right)^{\frac{2}{2^{*}}}dt\right]^{\frac{2^{*}}{2q^{\prime}}}.\vskip 3.0pt plus 1.0pt minus 1.0pt

We now choose $\displaystyle 2^{*}\delta=(2\delta-1-\gamma)q^{\prime}$ , that is, $\displaystyle\delta=\frac{1}{2}\cdot\frac{q(n-2)(\gamma+1)}{(n-2q)}$ . Note that if $\displaystyle\gamma\geq 1$ , then $\displaystyle\delta>\frac{\gamma+1}{2}$ if and only if $\displaystyle q>1$ , and for $\displaystyle\gamma<1$ , $\displaystyle\delta>1$ if and only if $\displaystyle q>\left(\frac{2^{*}}{1-\gamma}\right)^{\prime}$ . Moreover, since $\displaystyle q<\frac{n}{2}$ it follows that $\displaystyle\frac{2^{*}}{2q^{\prime}}<1$ . Thus we get

{}\int_{0}^{T}\left(\int_{\Omega}u_{k}^{2^{*}\delta}dx\right)^{\frac{2}{2^{*}}}dt=\int_{0}^{T}\left(\int_{\Omega}u_{k}^{\left(2\delta-1-\gamma\right)q^{\prime}}dx\right)^{\frac{2}{2^{*}}}dt\leq c,

(3.26)

and therefore from Eq. 3.23, Eq. 3.25 and Eq. 3.26, it follows

\left\|u_{k}\right\|_{L^{\infty}\left(0,T;L^{2\sigma}(\Omega)\right)}+\left\|u_{k}\right\|_{L^{2\sigma}\left(0,T;L^{2^{*}\sigma}(\Omega)\right)}\leq c,\text{ with }\sigma=\frac{q(n-2)(\gamma+1)}{2(n-2q)}>\frac{q}{2}.

Case 2: $\displaystyle\frac{nr}{n+2(r-1)}\leq q<\frac{n}{2}r^{\prime}$ i.e. $\displaystyle\frac{1}{r}\geq\frac{n}{n-2}\frac{1}{q}-\frac{2}{n-2}$ and $\displaystyle\frac{1}{r}+\frac{n}{2q}>1$
In this case we choose $\displaystyle\delta=\frac{1}{2}\frac{qrn(\gamma+1)}{nr-2q(r-1)}$ . Note that for $\displaystyle\gamma\geq 1$ , $\displaystyle\delta>\frac{1+\gamma}{2},$ if and only if $\displaystyle\frac{1}{r}+\frac{n}{2q}<1+\frac{n}{2}$ which is always satisfied and for $\displaystyle\gamma<1$ , $\displaystyle\delta>1$ if and only if $\displaystyle q>\frac{2nr}{nr(\gamma+1)+4(r-1)}$ which is satisfied if $\displaystyle r>\frac{2}{\gamma+1}$ . Now by interpolation, we get that $\displaystyle\frac{1}{(2\delta-1-\gamma)q^{\prime}}=\frac{1-\theta}{2\delta}+\frac{\theta}{2^{*}\delta}$ , where $\displaystyle\theta=\frac{nq(r-1)}{nr(q-1)+2q(r-1)}\in(0,1]$ . Therefore

\int_{0}^{T}\left\|u_{k}\right\|_{L^{(2\delta-1-\gamma)q^{\prime}}(\Omega)}^{r^{\prime}(2\delta-1-\gamma)}dt\leq c\left\|u_{k}\right\|_{L^{\infty}\left(0,T;L^{2\delta}(\Omega)\right)}^{2\delta\mu_{1}}\int_{0}^{T}\left(\int_{\Omega}u_{k}^{2^{*}\delta}dx\right)^{\frac{2}{2^{*}}\mu_{2}}dt,

where $\displaystyle\mu_{1}=\frac{(1-\theta)r^{\prime}(2\delta-1-\gamma)}{2\delta}$ and $\displaystyle\mu_{2}=\frac{\theta r^{\prime}(2\delta-1-\gamma)}{2\delta}$ . Since $\displaystyle\mu_{2}\leq 1$ , we have that

\int_{0}^{T}\left\|u_{k}\right\|_{L^{(2\delta-1-\gamma)q^{\prime}}(\Omega)}^{r^{\prime}(2\delta-1-\gamma)}dt\leq c\left\|u_{k}\right\|_{L^{\infty}\left(0,T;L^{2\delta}(\Omega)\right)}^{2\delta\mu_{1}}\left(\int_{0}^{T}\left[\int_{\Omega}u_{k}^{2^{*}\delta}dx\right]^{\frac{2}{2^{*}}}dt\right)^{\mu_{2}},

and since $\displaystyle\mu_{1}+\mu_{2}<r^{\prime}$ , using Eqs. 3.23 and 3.24, we can conclude the following which will give our final result

\int_{0}^{T}\left(\int_{\Omega}u_{k}^{(2\delta-1-\delta)q^{\prime}}dx\right)^{\frac{r^{\prime}}{q^{\prime}}}dt\leq c.

Proof of Theorem 2.39

Consider the approximated problem Eq. 2.11 and corresponding properties of $\displaystyle\{u_{k}\}_{k}$ , Theorem 2.22. We denote $\displaystyle(\omega_{\tilde{T}})_{\delta}=\Omega_{T}\backslash(\Omega_{T})_{\delta}$ , then $\displaystyle u_{k}(x)\geq C((\omega_{\tilde{T}})_{\delta},n,s)>0$ in $\displaystyle(\omega_{\tilde{T}})_{\delta}$ for all $\displaystyle k$ . Choosing $\displaystyle u_{k}\chi_{(0,\tau)}(t)$ as test function in Eq. 2.11, we obtain

\begin{array}[]{l}\quad\frac{1}{2}\int_{\Omega}u_{k}^{2}(x,\tau)dx+\int_{0}^{\tau}\int_{\Omega}|\nabla u_{k}|^{2}dxdt+\frac{1}{2}\int_{0}^{\tau}\int_{Q}\frac{\left(u_{k}(x,t)-u_{k}(y,t)\right)^{2}}{|x-y|^{n+2s}}dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\iint_{\Omega_{T}}\frac{f_{k}u_{k}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt+\frac{1}{2}\int_{\Omega}u_{0}^{2}(x)dx.\end{array}

Now taking supremum over $\displaystyle\tau\in(0,T]$ , we get

{}\begin{array}[]{l}\quad\sup_{0\leq\tau\leq T}\int_{\Omega}u_{k}^{2}(x,\tau)dx+2\int_{0}^{T}\int_{\Omega}|\nabla u_{k}|^{2}dxdt+\int_{0}^{T}\int_{Q}\frac{\left(u_{k}(x,t)-u_{k}(y,t)\right)^{2}}{|x-y|^{n+2s}}dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq 2\iint_{\Omega_{T}}\frac{f_{k}u_{k}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt+\left\|u_{0}\right\|_{L^{2}(\Omega)}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ =2\iint_{(\Omega_{T})_{\delta}}\frac{f_{k}u_{k}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt+2\iint_{(\omega_{\tilde{T}})_{\delta}}\frac{f_{k}u_{k}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt+\left\|u_{0}\right\|_{L^{2}(\Omega)}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq 2\iint_{(\Omega_{T})_{\delta}}fu_{k}^{1-\gamma(x,t)}dxdt+2\iint_{(\omega_{\tilde{T}})_{\delta}}\frac{fu_{k}}{C_{(\omega_{\tilde{T}})_{\delta}}^{\gamma(x,t)}}dxdt+\left\|u_{0}\right\|_{L^{2}(\Omega)}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq 2\iint_{(\Omega_{T})_{\delta}\cap\left\{u_{k}\leq 1\right\}}fu_{k}^{1-\gamma(x,t)}dxdt+2\iint_{(\Omega_{T})_{\delta}\cap\left\{u_{k}\geq 1\right\}}fu_{k}^{1-\gamma(x,t)}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \quad+2\iint_{(\omega_{\tilde{T}})_{\delta}}\frac{fu_{k}}{C_{(\omega_{\tilde{T}})_{\delta}}^{\gamma(x,t)}}dxdt+\left\|u_{0}\right\|_{L^{2}(\Omega)}\\ \leq\left\|u_{0}\right\|_{L^{2}(\Omega)}+2\|f\|_{L^{1}(\Omega_{T})}+2\left(1+\left\|C_{(\omega_{\tilde{T}})_{\delta}}^{-\gamma(\cdot)}\right\|_{L^{\infty}(\Omega_{T})}\right)\iint_{\Omega_{T}}fu_{k}dxdt.\end{array}

(3.27)

Case 1: $\displaystyle f\in L^{2}\left(0,T;L^{\left(\frac{2n}{n+2}\right)}(\Omega)\right)$
We note that $\displaystyle(2^{*})^{\prime}=\frac{2n}{n+2}$ , then by using Hölder’s and Sobolev’s inequalities, we have

\begin{array}[]{rcl}\int_{0}^{T}\int_{\Omega}fu_{k}dxdt&\leq&C\int_{0}^{T}\left(\int_{\Omega}|f(x,t)|^{(2^{*})^{\prime}}dx\right)^{\frac{1}{(2^{*})^{\prime}}}\left(\int_{\Omega}\left|u_{k}(x,t)\right|^{2^{*}}dx\right)^{\frac{1}{2^{*}}}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&C\int_{0}^{T}\|f\|_{L^{\left({2^{*}}\right)^{\prime}}{(\Omega)}}\left(\int_{\Omega}\left|\nabla u_{k}\right|^{2}dx\right)^{\frac{1}{2}}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&C\left(\int_{0}^{T}\|f\|_{L^{\left({2^{*}}\right)^{\prime}}{(\Omega)}}^{2}dt\right)^{\frac{1}{2}}\left(\int_{0}^{T}\int_{\Omega}\left|\nabla u_{k}\right|^{2}dxdt\right)^{\frac{1}{2}}.\end{array}

In the above, we use Young’s inequality, and then from Eq. 3.27 we get that

\begin{array}[]{l}\quad\sup_{0\leq\tau\leq T}\int_{\Omega}u_{k}^{2}(x,\tau)dx+\left\|u_{k}\right\|_{L^{2}(0,T;H_{0}^{1}(\Omega))}^{2}\leq\left\|u_{0}\right\|_{L^{2}(\Omega)}+2\|f\|_{L^{1}(\Omega_{T})}+C\left(1+\left\|C_{(\omega_{\tilde{T}})_{\delta}}^{-\gamma(\cdot)}\right\|_{L^{\infty}(\Omega_{T})}\right).\end{array}

Therefore the sequence $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))$ .
Case 2: $\displaystyle f\in L^{\bar{r}}\left(\Omega_{T}\right)$
For this case, we apply Hölder inequality with exponents $\displaystyle\bar{r}$ and $\displaystyle\bar{r}^{\prime}$ to get

{}\begin{array}[]{rcl}\int_{0}^{T}\int_{\Omega}fu_{k}dxdt\leq C\|f\|_{L^{\bar{r}}\left(\Omega_{T}\right)}\left[\iint_{\Omega_{T}}|u_{k}|^{\bar{r}^{\prime}}dxdt\right]^{\frac{1}{\bar{r}^{\prime}}}.\end{array}

(3.28)

We observe that $\displaystyle\bar{r}^{\prime}=\frac{2(n+2)}{n}$ and hence using the Hölder inequality with exponents $\displaystyle\frac{n}{n-2}$ and $\displaystyle\frac{n}{2}$ and by Sobolev embedding (Theorem 2.8), we can write

{}\begin{array}[]{rcl}\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{2(n+2)}{n}}dxdt=\iint_{\Omega_{T}}|u_{k}|^{2}\left|u_{k}\right|^{\frac{4}{n}}dxdt&\leq&\int_{0}^{T}\left[\int_{\Omega}\left|u_{k}(x,t)\right|^{2}dx\right]^{\frac{2}{n}}\left\|u_{k}\right\|_{L^{2^{*}}(\Omega)}^{2}dt\\ &\leq&C(n)\left[\sup_{0\leq\tau\leq T}\int_{\Omega}\left|u_{k}(x,\tau)\right|^{2}dx\right]^{\frac{2}{n}}\int_{0}^{T}\int_{\Omega}|\nabla u_{k}|^{2}dxdt.\end{array}

(3.29)

So using Eq. 3.28 in Eq. 3.27, we get from Eq. 3.29 by convexity argument

\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{2(n+2)}{n}}dxdt\leq C(n)2^{\frac{2}{n}}\left(\left(C_{1}C\|f\|_{L^{\bar{r}}\left(\Omega_{T}\right)}\right)^{\frac{n+2}{n}}\left(\iint_{\Omega_{T}}|u_{k}|^{{\bar{r}^{\prime}}}\right)^{\frac{n+2}{n\bar{r}^{\prime}}}+C_{2}^{\frac{n+2}{n}}\right),

where $\displaystyle C_{1}=2\left(1+\left\|C_{(\omega_{\tilde{T}})_{\delta}}^{-\gamma(\cdot)}\right\|_{L^{\infty}(\Omega_{T})}\right)$ and $\displaystyle C_{2}=\left\|u_{0}\right\|_{L^{2}(\Omega)}+2\|f\|_{L^{1}(\Omega_{T})}$ . Now since $\displaystyle\frac{n+2}{n\bar{r}^{\prime}}=\frac{1}{2}$ , we use Young’s inequality to obtain

\iint_{\Omega_{T}}\left|u_{k}\right|^{\bar{r}^{\prime}}dxdt=\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{2(n+2)}{n}}dxdt\leq C,

where $\displaystyle C$ is a positive constant independent of $\displaystyle k$ . Therefore, by Eqs. 3.27 and 3.28 we deduce that $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))$ .
Since $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in the reflexive Banach space $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))$ , there exist a subsequence of $\displaystyle\left\{u_{k}\right\}_{k}$ , still indexed by $\displaystyle k$ , and a measurable function $\displaystyle u\in L^{2}(0,T;H_{0}^{1}(\Omega))$ such that $\displaystyle u_{k}\rightharpoonup u$ weakly in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))$ . Also, since $\displaystyle\left\{u_{k}\right\}$ is increasing in $\displaystyle k$ , it holds by Beppo Levi’s theorem that $\displaystyle u_{k}\rightarrow u$ strongly in $\displaystyle L^{1}\left(\Omega_{T}\right)$ and hence a.e. in $\displaystyle\Omega_{T}$ . Applying Fatou’s Lemma, we get $\displaystyle u\in L^{\infty}(0,T;L^{2}(\Omega))$ . This pointwise limit is actually defined for each $\displaystyle t\in[0,T)$ and satisfies $\displaystyle u(\cdot,0)=u_{0}(\cdot)$ in $\displaystyle L^{1}$ sense (see Remark 2.23). We show that this $\displaystyle u$ is a very weak solution to Eq. 2.13 in the sense of Definition 2.25. Choosing arbitrary $\displaystyle\phi\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ , we have

\iint_{\Omega_{T}}(\left(u_{k}\right)_{t}-\Delta u_{k}+(-\Delta)^{s}u_{k})\phi\,dxdt=\iint_{\Omega_{T}}\frac{f_{k}\phi}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt.

Since $\displaystyle\phi\in\mathcal{C}_{0}^{\infty}(\Omega_{T})$ , therefore $\displaystyle\phi_{t},\Delta\phi$ and $\displaystyle(-\Delta)^{s}\phi$ all are bounded. As $\displaystyle u_{k}\rightarrow u$ strongly in $\displaystyle L^{1}\left(\Omega_{T}\right)$ , we have

\begin{array}[]{rcl}\iint_{\Omega_{T}}(\left(u_{k}\right)_{t}-\Delta u_{k}+(-\Delta)^{s}u_{k})\phi\,dxdt&=&\iint_{\Omega_{T}}u_{k}(-(\phi)_{t}-\Delta\phi+(-\Delta)^{s}\phi)dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\rightarrow&\iint_{\Omega_{T}}u(-(\phi)_{t}-\Delta\phi+(-\Delta)^{s}\phi)dxdt.\end{array}

Now since for all $\displaystyle\omega\subset\subset\Omega$ and for all $\displaystyle t_{0}>0,$ $\displaystyle u_{k}(x,t)\geq C\left(\omega,t_{0},n,s\right)$ in $\displaystyle\omega\times\left[t_{0},T\right)$ , we reach that $\displaystyle u_{k}(x,t)\geq C$ in $\displaystyle\operatorname{Supp}\phi$ (say $\displaystyle\omega\times[t_{1},t_{2}]$ ) for all $\displaystyle k\geq 1$ . Therefore

0\leq\left|\frac{f_{k}\phi}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}\right|\leq\|\phi C_{\omega,t_{1},t_{2}}^{-\gamma(x,t)}\|_{L^{\infty}(\Omega_{T})}f.

Hence by the dominated convergence theorem

\iint_{\Omega_{T}}\frac{f_{k}\phi}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt\rightarrow\iint_{\Omega_{T}}\frac{f\phi}{u^{\gamma(x,t)}}dxdt\text{ as }k\rightarrow\infty.

Thus, we get the following and conclude

\iint_{\Omega_{T}}(u_{t}-\Delta u+(-\Delta)^{s}u)\phi\,dxdt=\iint_{\Omega_{T}}\frac{f\phi}{u^{\gamma(x,t)}}dxdt.

Proof of Theorem 2.41

Consider $\displaystyle u_{k}$ to be the unique nonnegative solution to Eq. 2.11. Since $\displaystyle\gamma^{*}>1$ , so we can use $\displaystyle u_{k}^{\gamma^{*}}\chi_{(0,\tau)}(t)$ as a test function in Eq. 2.11 to get for all $\displaystyle\tau\leq T$ ,

\begin{array}[]{c}\frac{1}{\gamma^{*}+1}\int_{\Omega}u_{k}^{\gamma^{*}+1}(x,\tau)dx+\int_{0}^{\tau}\int_{\Omega}\nabla u_{k}\cdot\nabla u_{k}^{\gamma^{*}}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ +\frac{1}{2}\int_{0}^{\tau}\int_{Q}\frac{(u_{k}^{\gamma^{*}}(x,t)-u_{k}^{\gamma^{*}}(y,t))\left(u_{k}(x,t)-u_{k}(y,t)\right)}{|x-y|^{n+2s}}dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\iint_{\Omega_{T}}\frac{f_{k}u_{k}^{\gamma^{*}}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt+\frac{1}{\gamma^{*}+1}\int_{\Omega}u_{0}^{\gamma^{*}+1}(x)dx.\end{array}

Now taking supremum over $\displaystyle\tau\in(0,T]$ and using item $\displaystyle i)$ of Lemma 2.11, we get

\begin{array}[]{c}\quad\frac{({\gamma^{*}}+1)}{2\gamma^{*}}\sup_{0\leq\tau\leq T}\int_{\Omega}\left|u_{k}(x,\tau)\right|^{{\gamma^{*}}+1}dx+\frac{({\gamma^{*}}+1)^{2}}{2}\int_{0}^{T}\int_{\Omega}u_{k}^{{\gamma^{*}}-1}|\nabla u_{k}|^{2}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \quad+\int_{0}^{T}\int_{Q}\frac{|u_{k}^{\frac{{\gamma^{*}}+1}{2}}(x,t)-u_{k}^{\frac{{\gamma^{*}}+1}{2}}(y,t)|^{2}}{|x-y|^{n+2s}}dxdydt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq\frac{({\gamma^{*}}+1)^{2}}{2\gamma^{*}}\left(\iint_{\Omega_{T}}\frac{f_{k}u_{k}^{\gamma^{*}}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt+\left\|u_{0}\right\|_{L^{\gamma^{*}+1}(\Omega)}\right).\end{array}

We note that $\displaystyle 1<\frac{\gamma^{*}+1}{\gamma^{*}}<2$ and hence it follows

{}\begin{array}[]{c}\sup_{0\leq\tau\leq T}\int_{\Omega}\left|u_{k}(x,\tau)\right|^{{\gamma^{*}}+1}dx+4\int_{0}^{T}\int_{\Omega}|\nabla u_{k}^{\frac{{\gamma^{*}}+1}{2}}|^{2}dxdt\\ +2\int_{0}^{T}\int_{Q}\frac{|u_{k}^{\frac{{\gamma^{*}}+1}{2}}(x,t)-u_{k}^{\frac{{\gamma^{*}}+1}{2}}(y,t)|^{2}}{|x-y|^{n+2s}}dxdydt\leq 4\gamma^{*}\left(\iint_{\Omega_{T}}\frac{f_{k}u_{k}^{\gamma^{*}}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt+\left\|u_{0}\right\|_{L^{\gamma^{*}+1}(\Omega)}\right).\end{array}

(3.30)

Denoting $\displaystyle(\omega_{\tilde{T}})_{\delta}=\Omega_{T}\backslash(\Omega_{T})_{\delta}$ , we have $\displaystyle u_{k}(x,t)\geq C((\omega_{\tilde{T}})_{\delta},n,s)>0$ in $\displaystyle(\omega_{\tilde{T}})_{\delta}$ for all $\displaystyle k$ . We estimate like Theorem 2.39 as

\begin{array}[]{rcl}\iint_{\Omega_{T}}\frac{f_{k}u_{k}^{\gamma^{*}}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt&=&\iint_{(\Omega_{T})_{\delta}}\frac{f_{k}u_{k}^{\gamma^{*}}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt+\iint_{(\omega_{\tilde{T}})_{\delta}}\frac{f_{k}u_{k}^{\gamma^{*}}}{\left(u_{k}+\frac{1}{k}\right)^{\gamma(x,t)}}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&\iint_{(\Omega_{T})_{\delta}}fu_{k}^{{\gamma^{*}}-\gamma(x,t)}dxdt+\iint_{(\omega_{\tilde{T}})_{\delta}}\frac{fu_{k}^{\gamma^{*}}}{C_{(\omega_{\tilde{T}})_{\delta}}^{\gamma(x,t)}}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&\iint_{(\Omega_{T})_{\delta}\cap\left\{u_{k}\leq 1\right\}}fu_{k}^{{\gamma^{*}}-\gamma(x,t)}dxdt+\iint_{(\Omega_{T})_{\delta}\cap\left\{u_{k}\geq 1\right\}}fu_{k}^{\gamma^{*}-\gamma(x,t)}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &&+\iint_{(\omega_{\tilde{T}})_{\delta}}\frac{fu_{k}^{\gamma^{*}}}{C_{(\omega_{\tilde{T}})_{\delta}}^{\gamma(x,t)}}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&\|f\|_{L^{1}(\Omega_{T})}+\left(1+\left\|C_{(\omega_{\tilde{T}})_{\delta}}^{-\gamma(\cdot)}\right\|_{L^{\infty}(\Omega_{T})}\right)\iint_{\Omega_{T}}fu_{k}^{\gamma^{*}}dxdt.\end{array}

Using the above in Eq. 3.30 we get

{}\begin{array}[]{l}\quad\sup_{0\leq\tau\leq T}\int_{\Omega}\left|u_{k}(x,\tau)\right|^{{\gamma^{*}}+1}dx+2\int_{0}^{T}\int_{\Omega}|\nabla u_{k}^{\frac{{\gamma^{*}}+1}{2}}|^{2}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq 4\gamma^{*}\left(1+\left\|C_{(\omega_{\tilde{T}})_{\delta}}^{-\gamma(\cdot)}\right\|_{L^{\infty}(\Omega_{T})}\right)\iint_{\Omega_{T}}fu_{k}^{\gamma^{*}}dxdt+4\gamma^{*}\left(\|f\|_{L^{1}(\Omega_{T})}+\left\|u_{0}\right\|_{L^{\gamma^{*}+1}(\Omega)}\right).\end{array}

(3.31)

For convenience we take $\displaystyle C_{1}=4\gamma^{*}\left(1+\left\|C_{(\omega_{\tilde{T}})_{\delta}}^{-\gamma(\cdot)}\right\|_{L^{\infty}(\Omega_{T})}\right)$ and $\displaystyle C_{2}=4\gamma^{*}\left(\|f\|_{L^{1}(\Omega_{T})}+\left\|u_{0}\right\|_{L^{\gamma^{*}+1}(\Omega)}\right)$ .
Case 1: $\displaystyle f\in L^{\gamma^{*}+1}\left(0,T;L^{\left(\frac{n(\gamma^{*}+1)}{n+2\gamma^{*}}\right)}(\Omega)\right)$
In this case using Hölder inequality first in the space variable and then by Sobolev inequality and another application of Hölder inequality (in time variable) we get

\begin{array}[]{rcl}\int_{0}^{T}\int_{\Omega}fu_{k}^{\gamma^{*}}dxdt&=&\int_{0}^{T}\int_{\Omega}f\left(u_{k}^{\frac{\gamma^{*}+1}{2}}\right)^{\frac{2\gamma^{*}}{\gamma^{*}+1}}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&C\int_{0}^{T}\left(\int_{\Omega}|f(x,t)|^{\frac{n(\gamma^{*}+1)}{n+2\gamma^{*}}}dx\right)^{\frac{n+2\gamma^{*}}{n(\gamma^{*}+1)}}\left(\int_{\Omega}|u_{k}^{\frac{\gamma^{*}+1}{2}}(x,t)|^{2^{*}}dx\right)^{\frac{(n-2)\gamma^{*}}{n(\gamma^{*}+1)}}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&C\int_{0}^{T}\|f(\cdot,t)\|_{L^{\frac{n(\gamma^{*}+1)}{n+2\gamma^{*}}}{(\Omega)}}\left(\int_{\Omega}|\nabla u_{k}^{\frac{\gamma^{*}+1}{2}}|^{2}dx\right)^{\frac{\gamma^{*}}{\gamma^{*}+1}}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&C\left(\int_{0}^{T}\|f(\cdot,t)\|_{L^{\frac{n(\gamma^{*}+1)}{n+2\gamma^{*}}}{(\Omega)}}^{\gamma^{*}+1}dt\right)^{\frac{1}{\gamma^{*}+1}}\times\left(\int_{0}^{T}\int_{\Omega}|\nabla u_{k}^{\frac{\gamma^{*}+1}{2}}|^{2}dxdt\right)^{\frac{\gamma^{*}}{\gamma^{*}+1}}.\end{array}

Now since $\displaystyle\frac{\gamma^{*}}{\gamma^{*}+1}<1$ , so using Young’s inequality in the above estimate, we get from Eq. 3.31 that

\begin{array}[]{l}\quad\sup_{0\leq\tau\leq T}\int_{\Omega}\left|u_{k}(x,\tau)\right|^{{\gamma^{*}}+1}dx+2\int_{0}^{T}\int_{\Omega}|\nabla u_{k}^{\frac{{\gamma^{*}}+1}{2}}|^{2}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \leq C\left\|f\right\|_{L^{\gamma^{*}+1}\left(0,T;L^{\frac{n(\gamma^{*}+1)}{n+2\gamma^{*}}}(\Omega)\right)}\times\left(\varepsilon\int_{0}^{T}\int_{\Omega}|\nabla u_{k}^{\frac{{\gamma^{*}}+1}{2}}|^{2}dxdt+C(\varepsilon)\right)+C_{2}.\end{array}

Choosing $\displaystyle\varepsilon$ small enough such that $\displaystyle\varepsilon C\left\|f\right\|_{L^{\gamma^{*}+1}\left(0,T;L^{\frac{n(\gamma^{*}+1)}{n+2\gamma^{*}}}(\Omega)\right)}=1,$ we get

\sup_{0\leq\tau\leq T}\int_{\Omega}\left|u_{k}(x,\tau)\right|^{{\gamma^{*}}+1}dx+\int_{0}^{T}\int_{\Omega}|\nabla u_{k}^{\frac{{\gamma^{*}}+1}{2}}|^{2}dxdt\leq C,

which implies that the sequence $\displaystyle\left\{u_{k}^{\frac{\gamma^{*}+1}{2}}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))$ and $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{\infty}(0,T;L^{\gamma^{*}+1}(\Omega))$ .
Case 2: $\displaystyle f\in L^{\tilde{r}}\left(\Omega_{T}\right)$
For this case, we apply Hölder inequality with exponents $\displaystyle\tilde{r}$ and $\displaystyle\tilde{r}^{\prime}$ to get

{}\begin{array}[]{rcl}\int_{0}^{T}\int_{\Omega}fu_{k}^{\gamma^{*}}dxdt\leq C\|f\|_{L^{\tilde{r}}\left(\Omega_{T}\right)}\left[\iint_{\Omega_{T}}|u_{k}|^{\gamma^{*}\tilde{r}^{\prime}}dxdt\right]^{\frac{1}{\tilde{r}^{\prime}}}.\end{array}

(3.32)

We note that $\displaystyle\gamma^{*}\tilde{r}^{\prime}=\frac{(n+2)(\gamma^{*}+1)}{n}$ . Again, we use the same technique as that of Theorem 2.30, Theorem 2.32 and Theorem 2.39. Using the Hölder inequality with exponent $\displaystyle\frac{n}{n-2}$ and $\displaystyle\frac{n}{2}$ we get

\begin{array}[]{rcl}\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{({\gamma^{*}}+1)(n+2)}{n}}dxdt&=&\iint_{\Omega_{T}}\left|u_{k}\right|^{2\frac{{\gamma^{*}}+1}{2}}\left|u_{k}\right|^{\frac{4}{n}\frac{{\gamma^{*}}+1}{2}}dxdt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &\leq&\int_{0}^{T}\left(\int_{\Omega}\left|u_{k}(x,t)\right|^{2^{*}\frac{{\gamma^{*}}+1}{2}}dx\right)^{\frac{n-2}{n}}\left(\int_{\Omega}\left|u_{k}(x,t)\right|^{{\gamma^{*}}+1}dx\right)^{\frac{2}{n}}dt\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &=&\int_{0}^{T}\|u_{k}^{\frac{{\gamma^{*}}+1}{2}}\|_{L^{2^{*}}(\Omega)}^{2}\left(\int_{\Omega}\left|u_{k}(x,t)\right|^{{\gamma^{*}}+1}dx\right)^{\frac{2}{n}}dt.\end{array}

Now taking supremum over $\displaystyle t\in[0,T]$ and applying the Sobolev embedding as that of Theorem 2.8, we can write

\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{({\gamma^{*}}+1)(n+2)}{n}}dxdt\leq C(n)\left(\sup_{0\leq t\leq T}\int_{\Omega}\left|u_{k}(x,t)\right|^{{\gamma^{*}}+1}dx\right)^{\frac{2}{n}}\left(\iint_{\Omega_{T}}|\nabla u_{k}^{\frac{{\gamma^{*}}+1}{2}}|^{2}dxdt\right).

Using Eq. 3.31 and Eq. 3.32, from the above inequality we get by convexity argument

\begin{array}[]{rcl}\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{({\gamma^{*}}+1)(n+2)}{n}}dxdt&\leq&C(n)\left(C_{1}\iint_{\Omega_{T}}fu_{k}^{{\gamma^{*}}}dxdt+C_{2}\right)^{\frac{n+2}{n}}\\ &\leq&C(n)2^{\frac{2}{n}}\left(\left(C_{1}\iint_{\Omega_{T}}fu_{k}^{{\gamma^{*}}}dxdt\right)^{\frac{n+2}{n}}+C_{2}^{\frac{n+2}{n}}\right)\\ &\leq&C(n)2^{\frac{2}{n}}\left(\left(C_{1}C\|f\|_{L^{\tilde{r}}(\Omega_{T})}\right)^{\frac{n+2}{n}}\left(\iint_{\Omega_{T}}|u_{k}|^{\gamma^{*}\tilde{r}^{\prime}}dxdt\right)^{\frac{n+2}{n\tilde{r}^{\prime}}}+C_{2}^{\frac{n+2}{n}}\right).\end{array}

Now as $\displaystyle\frac{n+2}{n\tilde{r}^{\prime}}=\frac{\gamma^{*}}{\gamma^{*}+1}<1$ , therefore using Young’s inequality we get

\iint_{\Omega_{T}}\left|u_{k}\right|^{\gamma^{*}\tilde{r}^{\prime}}dxdt=\iint_{\Omega_{T}}\left|u_{k}\right|^{\frac{({\gamma^{*}}+1)(n+2)}{n}}dxdt\leq C,

where $\displaystyle C$ is a positive constant independent of $\displaystyle k$ . The above boundedness along with Eqs. 3.32 and 3.31 gives that the sequence $\displaystyle\left\{u_{k}^{\frac{\gamma^{*}+1}{2}}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))$ and $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{\infty}(0,T;L^{\gamma^{*}+1}(\Omega))$ . We now show $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(t_{0},T;H_{loc}^{1}(\Omega))$ for each $\displaystyle t_{0}>0$ . Since $\displaystyle\gamma^{*}>1$ , and $\displaystyle\Omega_{T}$ is bounded and $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{\infty}(0,T;L^{\gamma^{*}+1}(\Omega))$ , we thus have $\displaystyle\left\{u_{k}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(0,T;L^{2}(\Omega))=L^{2}(\Omega_{T})$ , in particular in $\displaystyle L^{2}(K\times(t_{0},T))$ , for every $\displaystyle K\subset\subset\Omega$ and for each $\displaystyle t_{0}>0$ . Again as $\displaystyle\left\{u_{k}^{\frac{\gamma^{*}+1}{2}}\right\}_{k}$ is uniformly bounded in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega))$ , so we have by nonnegativity of $\displaystyle u_{k}$

\begin{array}[]{c}\int_{t_{0}}^{T}\int_{K}u_{k}^{\gamma^{*}-1}|\nabla u_{k}|^{2}dxdt\leq\int_{0}^{T}\int_{\Omega}u_{k}^{\gamma^{*}-1}|\nabla u_{k}|^{2}dxdt\leq\frac{4C}{(\gamma^{*}+1)^{2}},\end{array}

for every $\displaystyle K\subset\subset\Omega$ and for each $\displaystyle t_{0}>0$ . Using the positivity of $\displaystyle u_{k}$ in $\displaystyle\omega\times[t_{0},T)$ for all $\displaystyle k$ , we conclude

\int_{t_{0}}^{T}\int_{K}|\nabla u_{k}|^{2}dxdydt\leq\frac{C}{\gamma^{*}c_{(K,t_{0})}^{\gamma^{*}-1}}.

Since $\displaystyle\left\{u_{k}\right\}_{k}$ is increasing in $\displaystyle k$ and is uniformly bounded in $\displaystyle L^{\infty}(0,T;L^{\gamma^{*}+1}(\Omega))$ , we get by Beppo Levi’s Lemma $\displaystyle u_{k}\uparrow u$ strongly in $\displaystyle L^{\gamma^{*}+1}\left(\Omega_{T}\right)$ and hence in $\displaystyle L^{1}(\Omega_{T})$ , where $\displaystyle u$ is the pointwise limit of $\displaystyle u_{k}$ (possibly infinite). Then employing Fatou’s lemma we will get $\displaystyle u^{\frac{\gamma^{*}+1}{2}}\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\displaystyle u\in L^{2}(t_{0},T;H_{loc}^{1}(\Omega))$ for each $\displaystyle t_{0}>0$ . Remark 2.23 implies that $\displaystyle u$ satisfies $\displaystyle u(\cdot,0)=u_{0}(\cdot)$ in $\displaystyle L^{1}$ sense. Also, we have $\displaystyle u(x,t)\geq$ $\displaystyle C\left(\omega,t_{0},n,s\right)$ in $\displaystyle\omega\times\left[t_{0},T\right)$ for all $\displaystyle\omega\subset\subset\Omega$ and for each $\displaystyle t_{0}>0$ . This $\displaystyle u$ will be a very weak solution to Eq. 2.13 in the sense of Definition 2.25, and the proof follows exactly similar to Theorem 2.39.

Remark 3.3.

In the above existence results, the cases where $\displaystyle u\in L^{2}(0,T;H^{1}_{0}(\Omega))$ , we can show that the weak solution $\displaystyle u$ is unique, if it has finite energy. For this we assume that $\displaystyle v$ is another finite energy solution to Eq. 2.12 or Eq. 2.13. Then, by the construction of $\displaystyle u$ in each existence result, we get $\displaystyle v$ is a supersolution to all the approximating problems. Hence by the comparison principle in Lemma 2.20, we deduce that $\displaystyle v\geq u_{k}$ for all $\displaystyle k$ . Then $\displaystyle v\geq u$ . To prove the inverse inequality, let $\displaystyle\phi=v-u$ , then $\displaystyle\phi\geq 0$ and $\displaystyle\phi\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and hence we can use both $\displaystyle\phi_{+}$ or $\displaystyle\phi_{-}$ as test functions. Also, we note that $\displaystyle\phi$ solves the problem

\displaystyle\displaystyle\begin{cases}\phi_{t}-\Delta\phi+(-\Delta)^{s}\phi=f\left(\frac{1}{v^{\gamma(x,t)}}-\frac{1}{u^{\gamma(x,t)}}\right)&\text{ in }\Omega_{T},\\ \phi(x,t)=0&\text{ in }\left(\mathbb{R}^{n}\backslash\Omega\right)\times(0,T),\\ \phi(x,0)=0&\text{ in }\Omega.\end{cases}

Since $\displaystyle f\left(\frac{1}{v^{\gamma(x,t)}}-\frac{1}{u^{\gamma(x,t)}}\right)\leq 0$ in $\displaystyle\Omega_{T}$ , by comparison principle, we get $\displaystyle\phi\leq 0$ in $\displaystyle\Omega_{T}$ . Thus $\displaystyle\phi=0$ i.e. $\displaystyle u=v$ .

Remark 3.4.

As we can notice, in each of the existence results, we have $\displaystyle u\in L^{2}(t_{0},T;W^{1,1}_{\operatorname{loc}}(\Omega))$ for each $\displaystyle t_{0}>0$ and this gives that our definition of a weak solution is well motivated in the sense that we have derived them integrating by parts twice upon multiplying by a test function.

4 Asymptotic behaviour

This section is devoted to the study of the asymptotic behaviour of finite energy solution to the problem Eq. 2.12, as $\displaystyle t\rightarrow\infty$ , for the particular case where $\displaystyle f$ depends only on $\displaystyle x$ and that $\displaystyle f>0$ with $\displaystyle f\in L^{q}(\Omega)$ and $\displaystyle q$ will be specified later. We first state the following existence and uniqueness result to the corresponding mixed local-nonlocal elliptic problem, which can be done as a direct application of Theorem 2.27. Consider the problem

{}\begin{array}[]{lcr}\begin{cases}-\Delta w+(-\Delta)^{s}w=\frac{f}{w^{\gamma}}&\mbox{ in }\Omega,\\ w=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega).\end{cases}\end{array}

(4.1)

We define the weak solution to the above problem as:

Definition 4.1.

Suppose that $\displaystyle f\in L^{1}(\Omega)$ is a nonnegative function and $\displaystyle\gamma>0$ . We say that $\displaystyle w$ is a very weak solution to problem Eq. 4.1 if $\displaystyle w\in L^{1}(\Omega)$ satisfies the boundary conditions and $\displaystyle\forall\omega\subset\subset\Omega,$ $\displaystyle\exists c_{\omega}>0$ such that $\displaystyle w(x)\geq c_{\omega}>0,\text{ a.e. in }\omega,$ and for all $\displaystyle\varphi\in\mathcal{C}_{0}^{\infty}(\Omega)$ , we have

\int_{\Omega}w(-\Delta\varphi+(-\Delta)^{s}\varphi)dx=\int_{\Omega}\frac{f\varphi}{w^{\gamma}}dx.

Theorem 4.2.

Let $\displaystyle f\in L^{1}(\Omega)$ be a non-negative function. Then for all $\displaystyle\gamma>0$ , the problem Eq. 4.1 has a nonnegative very weak solution $\displaystyle w$ such that $\displaystyle w^{\frac{\gamma+1}{2}}\in H_{0}^{1}(\Omega)$ in the sense of Definition 4.1.

Proof.

Let $\displaystyle w_{k}$ be the unique positive bounded solution to the approximating problem

{}\begin{array}[]{lcr}\begin{cases}-\Delta w_{k}+(-\Delta)^{s}w_{k}=\frac{f_{k}}{(w_{k}+\frac{1}{k})^{\gamma}}&\mbox{ in }\Omega,\\ w_{k}>0&\mbox{ in }\Omega,\\ w_{k}=0&\mbox{ in }(\mathbb{R}^{n}\backslash\Omega);\end{cases}\end{array}

(4.2)

with $\displaystyle f_{k}:=\min(k,f)$ . We note that the existence of $\displaystyle w_{k}$ follows using a simple monotonicity argument. More precisely, one can prove a similar version of existence results and comparison principle as of Theorem 2.13 for the elliptic case also, and using that, one can find the elliptic version of Lemma 2.20 and Corollary 2.21. Again, by the comparison principle, we deduce that the sequence $\displaystyle\left\{w_{k}\right\}_{k}$ is increasing in $\displaystyle k$ . By the Harnack inequality, see [22, Theorem 8.3], it holds that for any set $\displaystyle\omega\subset\subset\Omega$ , for all $\displaystyle k$ we have, $\displaystyle w_{k}(x)\geq w_{1}(x)\geq c(\omega,n,s)\text{ in }\omega.$ Also, the details of existence, uniqueness, positivity and monotonicity of $\displaystyle w_{k}$ ’s can be found in [26, Lemma 3.2].
Now taking $\displaystyle((w_{k}+\varepsilon)^{\gamma}-\varepsilon^{\gamma}),0<\varepsilon<1/k,$ as a test function in Eq. 4.2 and letting $\displaystyle\varepsilon\to 0$ , it follows that

\int_{\Omega}\nabla w_{k}\cdot\nabla w_{k}^{\gamma}+\frac{1}{2}\iint_{\mathbb{R}^{2n}}\frac{\left(w_{k}(x)-w_{k}(y)\right)\left(w_{k}^{\gamma}(x)-w_{k}^{\gamma}(y)\right)}{|x-y|^{n+2s}}dxdy\leq\int_{\Omega}f(x)dx.

By Lemma 2.11, we have

\left(w_{k}(x)-w_{k}(y)\right)\left(w_{k}^{\gamma}(x)-w_{k}^{\gamma}(y)\right)\geq C\left(w_{k}^{\frac{\gamma+1}{2}}(x)-w_{k}^{\frac{\gamma+1}{2}}(y)\right)^{2},

and on the other hand

\nabla w_{k}\cdot\nabla w_{k}^{\gamma}=\gamma w_{k}^{\gamma-1}|\nabla w_{k}|^{2},

we deduce that the sequence $\displaystyle\left\{w_{k}^{\frac{\gamma+1}{2}}\right\}_{k}$ is bounded in the reflexive Banach space $\displaystyle X_{0}^{s}(\Omega)\cap H_{0}^{1}(\Omega)$ . Now by the monotonicity of $\displaystyle\left\{w_{k}\right\}_{k}$ , using Beppo-Levi’s theorem, we get the existence of a measurable function $\displaystyle w$ such that $\displaystyle w_{k}\uparrow w$ a.e. in $\displaystyle\mathbb{R}^{n},$ $\displaystyle w_{k}^{\frac{\gamma+1}{2}}\rightharpoonup w^{\frac{\gamma+1}{2}}$ weakly in $\displaystyle X_{0}^{s}(\Omega)\cap H_{0}^{1}(\Omega)$ . Clearly $\displaystyle w\geq c(\omega)$ for any set $\displaystyle\omega\subset\subset\Omega$ . Then, letting $\displaystyle k\rightarrow\infty$ in the approximating problems and using the dominated convergence theorem, we can show that $\displaystyle w$ is a very weak solution to problem Eq. 4.1 with $\displaystyle w^{\frac{\gamma+1}{2}}\in H_{0}^{1}(\Omega)$ . We refer [26, Theorem $\displaystyle 2.16$ ] for the boundedness of $\displaystyle w$ . One can find conditions when $\displaystyle w\in H^{1}_{0}(\Omega)$ in [26]. Further, [26, Corollary $\displaystyle 5.2$ ] gives such solution is unique. ∎

We now write the asymptotic behaviour of solutions to problem Eq. 2.12 under suitable conditions on $\displaystyle f$ and $\displaystyle u_{0}$ .

Theorem 4.3.

Suppose that $\displaystyle f\in L^{q}(\Omega)$ is a non-negative function depending only on $\displaystyle x$ , and $\displaystyle q$ is such that solution to the problem Eq. 4.1 is in $\displaystyle H^{1}_{0}(\Omega)$ and is unique. Let $\displaystyle u$ be a finite energy solution to problem Eq. 2.12 with $\displaystyle 0\leq u_{0}\leq w$ , where $\displaystyle w$ is the solution to Eq. 4.1. Then $\displaystyle u(\cdot,t)\rightarrow w$ as $\displaystyle t\rightarrow\infty$ , in $\displaystyle L^{\sigma}(\Omega)$ , where $\displaystyle\sigma<2^{*}$ .

Proof.

We divide the proof into two cases according to the value of the initial condition $\displaystyle u_{0}$ .
The first case $\displaystyle u_{0}=0$ : In this case using the comparison principle as in Lemma 2.20, we get that $\displaystyle u\leq w$ in $\displaystyle\Omega\times(0,T)$ for all $\displaystyle T>0$ . Hence, $\displaystyle u$ is globally defined. We now show that $\displaystyle u(x,\cdot)$ is increasing in $\displaystyle t$ . Since $\displaystyle u$ has finite energy, we fix $\displaystyle t_{1}>0$ and define $\displaystyle v(x,t)=u(x,t_{1}+t),$ then $\displaystyle v$ satisfies the problem

\displaystyle\displaystyle\begin{cases}v_{t}-\Delta v+(-\Delta)^{s}v=\frac{f(x)}{v^{\gamma}(x,t)}&\text{ in }\Omega_{T},\\ v(x,t)=0&\text{ in }\left(\mathbb{R}^{n}\backslash\Omega\right)\times(0,T),\\ v(x,0)=u\left(x,t_{1}\right)&\text{ in }\Omega.\end{cases}

Now as $\displaystyle u\left(x,t_{1}\right)\geq u_{0}\equiv 0$ in $\displaystyle\Omega$ , using the comparison principle as in Lemma 2.20 again, we have $\displaystyle u\leq v$ in $\displaystyle\Omega_{T}$ . Therefore for all $\displaystyle t_{1}>0$ , we have $\displaystyle u(x,t)\leq u\left(x,t_{1}+t\right)$ for all $\displaystyle x\in\mathbb{R}^{n}$ . Therefore $\displaystyle u(\cdot,t)$ is increasing in $\displaystyle t$ . Since $\displaystyle u\leq w$ , so there exists a measurable function $\displaystyle\hat{u}(\cdot)=\underset{t\rightarrow\infty}{\operatorname{lim}\operatorname{sup}}~{}u(\cdot,t)$ . By dominated convergence theorem, $\displaystyle u(\cdot,t)\rightarrow\hat{u}(\cdot)$ as $\displaystyle t\rightarrow\infty$ in $\displaystyle L^{\sigma}(\Omega)$ for $\displaystyle\sigma<2^{*}$ . Also, we have $\displaystyle\hat{u}\leq w$ . It is now sufficient to show that $\displaystyle\hat{u}$ is a very weak solution to problem Eq. 4.1.
Let $\displaystyle\varphi\in\mathcal{C}_{0}^{\infty}(\Omega)$ , then using $\displaystyle\varphi$ as a test function in Eq. 2.12 and integrating in $\displaystyle\Omega\times(k,k+1)$ , we get

\int_{\Omega}(u(x,k+1)-u(x,k))\varphi(x)dx+\int_{k}^{k+1}\int_{\Omega}u(x,t)(-\Delta\varphi+(-\Delta)^{s}\varphi)dxdt=\int_{k}^{k+1}\int_{\Omega}\frac{f(x)}{u^{\gamma}(x,t)}\varphi(x)dxdt.

Now clearly we have

\int_{\Omega}(u(x,k+1)-u(x,k))|\varphi(x)|dx\rightarrow 0\text{ as }k\rightarrow\infty.

On the other hand, in the set $\displaystyle\Omega\times(k,k+1)$ using the monotonicity of $\displaystyle u$ in the variable $\displaystyle t$ , we have

\frac{f(x)}{u^{\gamma}(x,k+1)}|\varphi(x)|\leq\frac{f(x)}{u^{\gamma}(x,t)}|\varphi(x)|\leq\frac{f(x)}{u^{\gamma}(x,k)}|\varphi(x)|,\quad\forall t\in(k,k+1),

u(x,k)\left|-\Delta\varphi\right|\leq u(x,t)\left|-\Delta\varphi\right|\leq u(x,k+1)\left|-\Delta\varphi\right|,\quad\forall t\in(k,k+1),

and

u(x,k)\left|(-\Delta)^{s}\varphi\right|\leq u(x,t)\left|(-\Delta)^{s}\varphi\right|\leq u(x,k+1)\left|(-\Delta)^{s}\varphi\right|,\quad\forall t\in(k,k+1).

Then, since $\displaystyle\phi,\Delta\phi,(-\Delta)^{s}\phi$ are bounded, so by the dominated convergence theorem, we will get that, as $\displaystyle k\rightarrow\infty$ ,

\int_{k}^{k+1}\int_{\Omega}u(x,t)(-\Delta\varphi+(-\Delta)^{s}\varphi)dxdt\rightarrow\int_{\Omega}\hat{u}(x)(-\Delta\varphi+(-\Delta)^{s}\varphi)dx,

and

\int_{k}^{k+1}\int_{\Omega}\frac{f(x)}{u^{\gamma}(x,t)}\varphi(x)dxdt\rightarrow\int_{\Omega}\frac{f(x)}{\hat{u}^{\gamma}(x)}\varphi(x)dx.

Hence, $\displaystyle\hat{u}$ is a very weak solution to problem Eq. 4.1. By the uniqueness we then get $\displaystyle\hat{u}=w$ .
The second case $\displaystyle 0<u_{0}\leq w$ : Let us denote by $\displaystyle\hat{u}$ a finite energy solution to problem Eq. 2.12 in this case. Then by the comparison principle in Lemma 2.20, we deduce that $\displaystyle u\leq\hat{u}\leq w$ , where $\displaystyle u$ is the weak solution to problem Eq. 2.12 with $\displaystyle u_{0}=0$ . Hence by the convergence result of the first case, we get that $\displaystyle\hat{u}(\cdot,t)\rightarrow w$ , as $\displaystyle t\rightarrow\infty$ , strongly in $\displaystyle L^{\sigma}(\Omega)$ for all $\displaystyle\sigma<2^{*}$ . ∎

References

[1] Boumediene Abdellaoui, María Medina, Ireneo Peral, and Ana Primo. The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian. J. Differential Equations, 260(11):8160–8206, 2016.
[2] D. G. Aronson and James Serrin. Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal., 25:81–122, 1967.
[3] Rakesh Arora and Vicenţiu D. Rădulescu. Combined effects in mixed local–nonlocal stationary problems. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, page 1–47, 2023.
[4] Mehdi Badra, Kaushik Bal, and Jacques Giacomoni. Existence results to a quasilinear and singular parabolic equation. Discrete Contin. Dyn. Syst., pages 117–125, 2011.
[5] Mehdi Badra, Kaushik Bal, and Jacques Giacomoni. Some results about a quasilinear singular parabolic equation. Differ. Equ. Appl., 3(4):609–627, 2011.
[6] Mehdi Badra, Kaushik Bal, and Jacques Giacomoni. A singular parabolic equation: existence, stabilization. J. Differential Equations, 252(9):5042–5075, 2012.
[7] Begoña Barrios, Ida De Bonis, María Medina, and Ireneo Peral. Semilinear problems for the fractional Laplacian with a singular nonlinearity. Open Math., 13(1):390–407, 2015.
[8] Kheireddine Biroud. Mixed local and nonlocal equation with singular nonlinearity having variable exponent. J. Pseudo-Differ. Oper. Appl., 14(1):Paper No. 13, 24, 2023.
[9] Lucio Boccardo and Luigi Orsina. Semilinear elliptic equations with singular nonlinearities. Calc. Var. Partial Differential Equations, 37(3-4):363–380, 2010.
[10] Lucio Boccardo, Maria Michaela Porzio, and Ana Primo. Summability and existence results for nonlinear parabolic equations. Nonlinear Anal., 71(3-4):978–990, 2009.
[11] Brahim Bougherara and Jacques Giacomoni. Existence of mild solutions for a singular parabolic equation and stabilization. Adv. Nonlinear Anal., 4(2):123–134, 2015.
[12] Stefano Buccheri, João Vítor da Silva, and Luís Henrique de Miranda. A system of local-nonlocal $\displaystyle p$ -Laplacians: the eigenvalue problem and its asymptotic limit as $\displaystyle p\to\infty$ . Asymptotic Analysis, 128(2):149–181, 2022.
[13] Annamaria Canino, Luigi Montoro, Berardino Sciunzi, and Marco Squassina. Nonlocal problems with singular nonlinearity. Bull. Sci. Math., 141(3):223–250, 2017.
[14] José Carmona and Pedro J. Martínez-Aparicio. A singular semilinear elliptic equation with a variable exponent. Adv. Nonlinear Stud., 16(3):491–498, 2016.
[15] M. G. Crandall, P. H. Rabinowitz, and L. Tartar. On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differential Equations, 2(2):193–222, 1977.
[16] Stuti Das. Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem. Journal of Mathematical Analysis and Applications, page 128140, 2024.
[17] Ida de Bonis and Linda Maria De Cave. Degenerate parabolic equations with singular lower order terms. Differential Integral Equations, 27(9-10):949–976, 2014.
[18] Cristiana De Filippis and Giuseppe Mingione. Gradient regularity in mixed local and nonlocal problems. Mathematische Annalen, pages 1–68, 2022.
[19] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci. Hitchhiker’s guide to the fractional sobolev spaces. Bulletin des sciences mathématiques, 136(5):521–573, 2012.
[20] Lawrence C Evans. Partial Differential Equations, volume 19. American Mathematical Soc., 2010.
[21] Yuzhou Fang, Bin Shang, and Chao Zhang. Regularity theory for mixed local and nonlocal parabolic $\displaystyle p$ -Laplace equations. J. Geom. Anal., 32(1):Paper No. 22, 33, 2022.
[22] Prashanta Garain and Juha Kinnunen. On the regularity theory for mixed local and nonlocal quasilinear elliptic equations. Trans. Amer. Math. Soc., 375(8):5393–5423, 2022.
[23] Prashanta Garain and Juha Kinnunen. Weak Harnack inequality for a mixed local and nonlocal parabolic equation. Journal of Differential Equations, 360:373–406, 2023.
[24] Prashanta Garain and Erik Lindgren. Higher hölder regularity for mixed local and nonlocal degenerate elliptic equations. Calculus of Variations and Partial Differential Equations, 62(2):67, 2023.
[25] Prashanta Garain and Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Commun. Pure Appl. Anal., 19(11):5059–5075, 2020.
[26] Prashanta Garain and Alexander Ukhlov. Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems. Nonlinear Anal., 223:Paper No. 113022, 35, 2022.
[27] David Kinderlehrer and Guido Stampacchia. An introduction to variational inequalities and their applications. SIAM, 2000.
[28] A. C. Lazer and P. J. McKenna. On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc., 111(3):721–730, 1991.
[29] Tommaso Leonori, Ireneo Peral, Ana Primo, and Fernando Soria. Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete Contin. Dyn. Syst., 35(12):6031–6068, 2015.
[30] Bin Shang and Chao Zhang. Hölder regularity for mixed local and nonlocal $\displaystyle p$ -Laplace parabolic equations. Discrete Contin. Dyn. Syst., 42(12):5817–5837, 2022.
[31] Bin Shang and Chao Zhang. Harnack inequality for mixed local and nonlocal parabolic $\displaystyle p$ -Laplace equations. J. Geom. Anal., 33(4):Paper No. 124, 24, 2023.
[32] Zhuoqun Wu, Jingxue Yin, and Chunpeng Wang. Elliptic & parabolic equations. World Scientific, 2006.

On a mixed local-nonlocal evolution equation with singular nonlinearity

Abstract

1 Introduction

Organization of the article

2 Preliminaries

2.1 Notations

2.2 Function Spaces

Definition 2.1.

Definition 2.2.

Lemma 2.3.

Lemma 2.4.

Lemma 2.5.

Lemma 2.6.

Definition 2.7.

Theorem 2.8.

Theorem 2.9.

Theorem 2.10.

Lemma 2.11.

2.3 Weak Solutions

Definition 2.12.

Theorem 2.13.

Proof.

Remark 2.14.

Definition 2.15.

Theorem 2.16.

Proof.

Proposition 2.17.

Proof.

Proposition 2.18.

Proof.

Remark 2.19.

Lemma 2.20.

Proof.

Corollary 2.21.

Theorem 2.22.

Proof.

Remark 2.23.

Definition 2.24.

Definition 2.25.

2.4 Main Results

2.4.1 Existence for bounded data for γ\displaystyle\gamma being a constant

Theorem 2.26.

2.4.2 Existence for general data for γ\displaystyle\gamma being a constant

Theorem 2.27.

Theorem 2.28.

2.4.3 Improved results for γ\displaystyle\gamma being a constant

Theorem 2.29.

Theorem 2.30.

Remark 2.31.

Theorem 2.32.

Remark 2.33.

Remark 2.34.

2.4.4 Further summability for γ\displaystyle\gamma being a constant

Theorem 2.35.

Theorem 2.36.

Remark 2.37.

Remark 2.38.

2.4.5 Existence results for γ\displaystyle\gamma being a function

Theorem 2.39.

Remark 2.40.

Theorem 2.41.

Remark 2.42.

3 Proof of main results

Proof of Theorem 2.26

Proof of Theorem 2.27

Proof of Theorem 2.28

Remark 3.1.

Proof of Theorem 2.29

Proof of Theorem 2.30

Proof of Theorem 2.32

Remark 3.2.

Proof of Theorem 2.35

Proof of Theorem 2.36

Proof of Theorem 2.39

Proof of Theorem 2.41

Remark 3.3.

Remark 3.4.

4 Asymptotic behaviour

Definition 4.1.

Theorem 4.2.

2.4.1 Existence for bounded data for $\displaystyle\gamma$ being a constant

2.4.2 Existence for general data for $\displaystyle\gamma$ being a constant

2.4.3 Improved results for $\displaystyle\gamma$ being a constant

2.4.4 Further summability for $\displaystyle\gamma$ being a constant

2.4.5 Existence results for $\displaystyle\gamma$ being a function