Global and local existence of solutions for nonlinear systems of time-fractional diffusion equations

Let $\partial_{t}^{\alpha}$ be the fractional derivative of order $\alpha\in(0,1)$ defined on the fractional Sobolev spaces, which is defined as the closure of the classical Caputo derivative

(see Section 2 for the details). Let $\Omega\subset\mathbb{R}^{d},d=1,2,3$ be a bounded domain with smooth boundary $\partial\Omega$. Moreover by $\nu$ we denote the unit outward normal vector to $\partial\Omega$ at $x\in\partial\Omega$.

We consider an initial-boundary value problem for the following nonlinear system of time-fractional equations with the homogeneous Neumann boundary condition

where we assume some conditions on the nonlinear terms $f$ and $g$, as Assumption 1 in Section 2 describes.

When $\alpha=1$, the system is reduced to a classical reaction-diffusion system, and can be regarded as a special case of the Klausmeier-Gray-Scott model Kla ; WS . This system is considered to be a model equation for the vegetation pattern formation, which describes the self-organization of vegetation spatial patterns resulting from the interaction between water source distribution and plant growth. In Pi and Pi2 , it was proved that a weak solution in $L^{1}$ exists in the case where the coefficients in front of the spatial diffusion terms are not necessarily equal. However for the our fractional derivative case $0<\alpha<1$, we can not apply the same technique as for $\alpha=1$, so that we have to assume that the coefficients of spatial diffusion terms are equal, that is, $1$. It should be a future work to discuss more general elliptic operators in (1.1).

For $0<\alpha<1$, in contrast to the traditional reaction-diffusion process, we introduce a fractional order in the time derivative, indicating that the interaction between vegetation growth and water is influenced by soil medium heterogeneity. The fractional derivatives enable the representation of time memory effects, characterized by their nonlocal properties. Although there has been some research on various nonlinear fractional-order equation or coupled linear fractional-order equation systems, as seen in BP ; LHL ; FLY , to the best of the authors’ knowledge, there has been no investigation concerning nonlinear fractional-order systems.

Our approach involves the truncation of the nonlinear terms, which is inspired by Pierre Pi ; Pi2 . Through a comparison principle and a compact mapping property, we establish a convergent sequence of non-negative functions for the linear case. Moreover, $u+v$ has the energy estimation in the sense of $L^{1}$-norm, implying the convergence of the sequence in the $L^{1}$ sense. Ultimately, we obtain weak solutions in the $L^{1}$ norm.

The rest of this paper is organized as follows. In Section 2, we present the result regarding the global existence of weak solutions to the initial-boundary value problems for the nonlinear system of time-fractional diffusion equations. Section 3 is devoted to proving two key lemmas, which serve as the basis for the proofs of the main results. These lemmas establish the non-negativity of solution to the linear case, provided that the initial value and source term are non-negative, and demonstrate that the mapping from initial value and non-homogeneous term to solution of the corresponding linear system, is compact with suitable norms. Section 4 completes the proof of the first main result. Section 5 discusses the blow-up of solutions under other conditions on the nonlinear terms, which rejects the global existence in time, in general. Section 6 provides some conclusions and remarks.

In this section, we deal with the following initial-boundary value problem for the nonlinear system of time-fractional equations (1.1) with the time-fractional derivative of order $\alpha\in(0,1)$

along with the initial condition (2.2) formulated below.

For $\alpha\in(0,1)$, let $\mathrm{d}^{\alpha}_{t}$ denote the classical Caputo derivative:

Here $\Gamma(\cdot)$ denotes the gamma function. For a consistent treatment of nonlinear time-fractional diffusion equations, we extend the classical Caputo derivative $\mathrm{d}^{\alpha}_{t}$ as follows. First of all, we define the Sobolev-Slobodeckij space $H^{\alpha}(0,T)$ with the norm $\|\cdot\|_{H^{\alpha}(0,T)}$ for $0<\alpha<1$:

(e.g., AdamsAd ). Furthermore, we set $H^{0}(0,T):=L^{2}(0,T)$ and

with the norms defined by

Moreover, for $\beta>0$, we set

It was proved, for instance, in Gorenflo, Luchko, and YamamotoGLY , that the operator $J^{\alpha}:L^{2}(0,T)\to H_{\alpha}(0,T)$ is an isomorphism for $\alpha\in(0,1)$.

Now we are ready to give the definition of the extended Caputo derivative

Henceforth $\mathcal{D}(\cdot)$ represents the domain of an operator under consideration. It has been demonstrated that $\partial_{t}^{\alpha}$ represents the minimal closed extension of $\mathrm{d}^{\alpha}_{t}$, where $\mathcal{D}(\mathrm{d}^{\alpha}_{t}):=\{v\in C^{1}[0,T];v(0)=0\}$. The relation $\partial_{t}^{\alpha}w=\mathrm{d}^{\alpha}_{t}w$ is maintained for $w\in C^{1}[0,T]$ with $w(0)=0$. For further details, we can refer to Gorenflo et al. GLY and YamamotoY4 .

Now we will define initial condition for problem (2.1) as follows:

and write down a complete formulation of an initial-boundary value problem for the nonlinear system of time-fractional equations (1.1):

It is worth mentioning that the terms $\partial_{t}^{\alpha}(u-a)$ and $\partial_{t}^{\alpha}(v-b)$ in the first two lines of (2.3) are well-defined for almost all $x\in\Omega$ and $0<t<T$, due to the inclusion formulated in the last line of (2.3). Especially for $\frac{1}{2}<\alpha<1,$ noting that $w\in H_{\alpha}(0,T)$ implies $w(0)=0$ by the trace theorem, we can understand that the left-hand side means that $u(x,0)=a(x)$ and $v(x,0)=b(x)$ in the trace sense with respect to $t$. While for $\alpha<\frac{1}{2}$, we do not need any initial conditions.

Now we are well prepared to investigate the initial-boundary value problem (2.3). We first provide the definition of weak solutions for equations (2.3). Henceforth we set

and

for $\psi\in C^{1}([0,T];L^{1}(\Omega))$. Then:

If a weak solution $(u,v)\in L^{1}(0,T;L^{1}(\Omega))^{2}$ is sufficiently smooth, then we can verify by the definition that $(u,v)$ satisfies (1.1) pointwise in the classical sense.

Based on the definition, we can give a lemma immediately.

Here, compared to the conditions required in Pi , we impose stronger requirements on $f$ and $g$, which are crucial for the convergence of the truncated non-linear terms in Sections 4. Unlike Pi , we no longer restrict $\lambda$ to be less than or equal to 1.

Now we state the first main result in this article, which validates the well-posedness for given $T>0$ of the initial-boundary value problem (2.3).

We do not know the uniqueness of weak solution under our assumption, which is the same as for the case $\alpha=1$.

Before starting with the proof of Theorem 2.1, we introduce some notations and derive several results necessary for the proof.

For an arbitrary constant $p_{0}>0$, define an elliptic operator $A$ as follows:

Henceforth, by $\|\cdot\|$ and $(\cdot,\cdot)$ we denote the standard norm and the scalar product in $L^{2}(\Omega)$, respectively. It is well-konwn that the operator $A$ is self-adjoint in $L^{2}(\Omega)$. Moreover, for a sufficiently large constant $p_{0}>0$, we can verify that $A$ is positive definiteLY1 . Therefore, by choosing a constant $p_{0}$ large enough, the spectrum of the operator $A$ is comprised of discrete positive eigenvalues, herein represented as $0<\lambda_{1}\leq\lambda_{2}\leq\cdots$, each uniquely designated by its multiplicity. Additionally, $\lambda_{n}\to\infty$ as $n\to\infty$. Set $\varphi_{n}$ be the eigenfunction corresponding to the eigenvalue $\lambda_{n}$ such that $A\varphi_{n}=\lambda_{n}\varphi_{n}$, and $(\varphi_{i},\varphi_{j})=\delta_{ij}$. Then the sequence $\{\varphi_{n}\}_{n\in\mathbb{N}}$ is orthonormal basis in $L^{2}(\Omega)$. For any $\gamma>0$, we can define the fractional power $A^{\gamma}$ of the operator $A$ by the following relation (see, e.g.,Pa ):

where

and

We have $\mathcal{D}(A^{\gamma})\subset H^{2\gamma}(\Omega)$ for $\gamma>0$. Since $\mathcal{D}(A^{\gamma})\subset L^{2}(\Omega)$, identifying the dual $(L^{2}(\Omega))^{\prime}$ with itself, we have $\mathcal{D}(A^{\gamma})\subset L^{2}(\Omega)\subset\mathcal{D}(A^{\gamma})^{\prime}$. Henceforth we set $\mathcal{D}(A^{-\gamma})=\mathcal{D}(A^{\gamma})^{\prime}$, which consists of bounded linear functionals on $\mathcal{D}(A^{\gamma})$. For $w\in\mathcal{D}(A^{-\gamma})$ and $v\in\mathcal{D}(A^{\gamma})$, by ${}_{-\gamma}\langle w,v\rangle_{\gamma}$, we denote the value which is obtained by operating $w$ to $v$. We note that $\mathcal{D}(A^{-\gamma})$ is a Hilbert space with norm:

We further note that

Moreover we define two other operators $S(t)$ and $K(t)$ as:

and

where $E_{\alpha,\beta}(z)$ denotes the Mittag-Leffler function defined by a convergent series as follows:

It follows directly from the definitions given above that $A^{\gamma}K(t)a=K(t)A^{\gamma}a$ and $A^{\gamma}S(t)a=S(t)A^{\gamma}a$ for $a\in\mathcal{D}(A^{\gamma})$. Moreover, the following norm estimates were proved inSY :

For the proof of Theorem 2.1, we show several lemmas.

The proof is simialr to Luchko and Yamamoto LY ; LY1 where the coefficient $c$ is assumed to be more smooth than $L^{\infty}(Q)$. For the case $c\in L^{\infty}(Q)$, approximating $c$ by functions in $C^{\infty}_{0}(Q)$ in the space $L^{\kappa}(Q)$ with arbitrarily fixed $\kappa>1$. For completeness, we provide the proof in Appendix.

Here and henceforth we set $A=-\Delta+p_{0}$. For $w_{0}\in H^{1}(\Omega)$ and $F\in L^{2}(Q)$, we consider

Then we know (e.g., KRY ) that there exists a unique solution $w\in L^{2}(0,T;H^{2}(\Omega))$ to (3.2) satisfying $w-w_{0}\in H_{\alpha}(0,T;L^{2}(\Omega))$. By $w(w_{0},F)$ we denote the solution.

We can prove a compactness property: