Existence of solutions semilinear parabolic equations with singular initial data in the Heisenberg group

This paper is concerned with nonnegative solutions of the fractional semilinear heat equation in the Heisenberg group

with the initial data

where $N\geq 1$, $\alpha\in(0,2]$, $p>1$, and $\mu$ is a nonnegative Radon measure on $\mathbb{H}^{N}$. Here, $\Delta_{\mathbb{H}}$ is the sub-Laplacian on the Heisenberg group $\mathbb{H}^{N}$ and $(-\Delta_{\mathbb{H}})^{\alpha/2}$ denotes the fractional power of $-\Delta_{\mathbb{H}}$ on $\mathbb{H}^{N}$. For the exact definition of $(-\Delta_{\mathbb{H}})^{\alpha/2}$, see (1.6) below.

In this paper we show that every nonnegative solution of (1.1) has a unique nonnegative Radon measure in $\mathbb{H}^{N}$ as the initial trace and study qualitative properties of it. Furthermore, we give sufficient conditions for the solvability of problem (1.1) with (1.2) and obtain sharp estimates of the life span of solutions with small initial data. Throughout of this paper, for $d\geq 1$ and $\alpha\in(0,2]$ denote

The Heisenberg group is the Lie group $\mathbb{H}^{N}=\mathbb{R}^{2N+1}$ equipped with the group law

and with the Haar measure on $\mathbb{H}^{N}$, where $\eta=(x,y,\tau)$, $\eta^{\prime}=(x^{\prime},y^{\prime},\tau^{\prime})\in\mathbb{R}^{2N+1}$, and $\cdot$ is the scalar product in $\mathbb{R}^{N}$. It is well-known that the Haar measure on $\mathbb{H}^{N}$ coincides with the $2N+1$-dimensional Lebesgue measure $\mathcal{L}^{2N+1}$. The identity element for $\mathbb{H}^{N}$ is $0$ and $\eta^{-1}=-\eta$ for all $\eta\in\mathbb{H}^{N}$. The homogeneous Heisenberg norm is defined by

where $|\cdot|$ is the Euclidean norm associated to $\mathbb{R}^{N}$. Then

is a left-invariant distance on $\mathbb{H}^{N}$. The homogeneous dimension of $\mathbb{H}^{N}$ is $Q=2N+2$. The left-invariant vector fields that span the Lie algebra are given by

for $i=1,\cdots,N$. The Heisenberg gradient is given by

and the sub-Laplacian is defined as

where $\Delta_{x}$ and $\Delta_{y}$ stand for the Laplace operators on $\mathbb{R}^{N}$. In particular, the Heisenberg group $\mathbb{H}^{N}$ is the most typical example of metric measure spaces with a structure different from that of $\mathbb{R}^{N}$.

We recall the solvability of (1.1) in $\mathbb{R}^{N}$. For $x\in\mathbb{R}^{N}$ and $r>0$, set $B_{\mathbb{R}^{N}}(x,r):=\{y\in\mathbb{R}^{N}:|x-y|<r\}$. Let us consider nonnegative solutions of the fractional semilinear equation

with the initial data

where $N\geq 1$, $\alpha\in(0,2]$, $q>1$, and $\nu$ is a nonnegative Radon measure on $\mathbb{R}^{N}$. Here, $(-\Delta)^{\alpha/2}$ denotes the fractional power of $-\Delta$ on $\mathbb{R}^{N}$.

Let us consider the case of $\alpha=2$ first. The solvability of problem (1.4) with (1.5) has been studied in many papers since the pioneering work due to Fujita [F] (see, e.g., [QS19], which includes an extensive list of references for (1.4)). It is already known from his work and the subsequent works of Hayakawa [H73], Sugitani [S75], and Kobayashi–Sirao–Tanaka [KST77] that $1<q\leq p_{2,N}$ implies the nonexistence of any positive global-in-time solutions, while if $q>p_{2,N}$, problem (1.4) with (1.5) possesses a positive global-in-time solution for appropriate initial data $\nu$. For this reason, such an exponent $p_{2,N}$ is called the Fujita-exponent. Note that Sugitani [S75] also dealt with the fractional case $\alpha\in(0,2)$.

Among these studies, Baras–Pierre [BP85] obtained necessary conditions for the solvability in the case of $\alpha=2$. Subsequently, the second author of this paper and Ishige [HI18] obtained a generalization to the fractional case $\alpha\in(0,2)$ and sufficient conditions for the solvability. Furthermore, they showed that every nonnegative solution of (1.4) has a unique nonnegatuve Radon measure on $\mathbb{R}^{N}$ as the initial trace (see also [IKO20], which deals with the case where $\alpha$ is a positive even integer as well as the case of $\alpha\in(0,2]$). More precisely, the following results have already been obtained.

• (a)

  Let $N\geq 1$, $\alpha\in(0,2]$, and $q>1$. Let $v$ be a nonnegative solution of (1.4) in $\mathbb{R}^{N}\times(0,T)$, where $T\in(0,\infty)$. Then there exists a unique nonnegative Radon measure $\nu$ on $\mathbb{R}^{N}$ such that

  $$\operatorname*{ess\,lim}_{t\to 0^{+}}\int_{\mathbb{R}^{N}}v(y,t)\phi(y)\,\mathop{}\!\mathrm{d}y=\int_{\mathbb{R}^{N}}\phi(y)\,\mathop{}\!\mathrm{d}\nu(y)$$

  for all $\phi\in C_{c}(\mathbb{R}^{N})$.

• (b)

  Let $\nu$ be as in assertion (a). Then $\nu$ must satisfy the following:

  • –

    If $1<q<p_{\alpha,N}$, then $\displaystyle{\sup_{x\in\mathbb{R}^{N}}\nu(B_{\mathbb{R}^{N}}(x,T^{\frac{1}{\alpha}}))\leq\gamma T^{\frac{N}{\alpha}-\frac{1}{q-1}}}$;

  • –

    If $q=p_{\alpha,N}$, then $\displaystyle{\sup_{x\in\mathbb{R}^{N}}\nu(B_{\mathbb{R}^{N}}(x,\sigma))\leq\gamma\left[\log\left(e+\frac{T^{1/\alpha}}{\sigma}\right)\right]^{-{\frac{N}{\alpha}}}}$ for all $0<\sigma<T^{\frac{1}{\alpha}}$;

  • –

    If $q>p_{\alpha,N}$, then $\displaystyle{\sup_{x\in\mathbb{R}^{N}}\nu(B_{\mathbb{R}^{N}}(x,\sigma))\leq\gamma\sigma^{N-\frac{\alpha}{q-1}}}$ for all $0<\sigma<T^{\frac{1}{\alpha}}$.

  Here $\gamma>0$ is a constant depending only on $N$, $\alpha$, and $q$.

From this necessary condition (b) we can find a large constant $C_{*}>0$ with the following property:

• (c)

  Problem (1.4) with (1.5) possesses no local-in-time solutions if $\nu$ is a nonnegative measurable function in $\mathbb{R}^{N}$ satisfying $\nu(x)\geq C_{*}\Phi(x)$ in a neighborhood of the origin, where

  $$\Phi(x):=\left\{\begin{array}[]{ll}\displaystyle{|x|^{-N}\left[\log\left(e+\frac{1}{|x|}\right)\right]^{-\frac{N}{\alpha}-1}}&\mbox{if}\quad q=p_{\alpha,N},\vspace{3pt}\\ \displaystyle{|x|^{-\frac{\alpha}{q-1}}}&\mbox{if}\quad q>p_{\alpha,N}.\vspace{3pt}\\ \end{array}\right.$$

On the other hand, the following sufficient condition for the solvability has also been obtained:

• (d)

  There exists a small constant $c_{*}>0$ such that if $\nu$ satisfies $0\leq\nu(x)\leq c_{*}\Phi(x)$ in $\mathbb{R}^{N}$, then problem (1.4) with (1.5) possesses a local-in-time solution. In particular, when $p>p_{\alpha,N}$, this solution is a global-in-time one.

Note that these results show that $q=p_{\alpha,N}$ is the Fujita-exponent. For (d), see also the papers written by Kozono–Yamazaki [KY94] , Robinson–Sierż\kega [RS13] (the case where $\alpha=2$ and $q>p_{2,N}$), and Ishige–Kawakami–Okabe [IKO20] (the case where $\alpha\in(0,2]$ and $q\geq p_{\alpha,N}$). The conditions (c) and (d) demonstrate that strength of the singularity at the origin of the function $\Phi$ is the critical threshold for the local-in-time solvability of problem (1.4) with (1.5). We term such a singularity in the initial data an optimal singularity for the solvability of problem (1.4) with (1.5). For studies on optimal singularities, see e.g. [FHIL23, FHIL24, H24, HI18, HIT23, HS24, HT21, IKO20].

Let us go back to (1.1). For problem (1.1) with (1.2) in the case of $\alpha=2$, Zhang [Z98] proved that $1<p<p_{2,Q}$ implies the nonexistence of any positive global-in-time solutions, while if $p>p_{2,Q}$, problem (1.1) with (1.2) possesses a positive global-in-time solution for appropriate initial data $\mu$. Later, Pohozaev–Véron [PV00] considered more general equations in $\mathbb{H}^{N}$ and proved that $p=p_{2,Q}$ is included in the nonexistence case. Namely, we already know that $p=p_{2,Q}$ is the Fujita exponent. Recently, Georgiev–Palmieri [GP21] obtained sharp estimates of the life span of solutions of problem (1.1) with small initial data in the case of $1<p\leq p_{2,Q}$ and also proved the global-in-time solvability in the case of $p>p_{2,Q}$. Many other studies on the global-in-time solvability in $\mathbb{H}^{N}$ have been undertaken, see e.g. [A01, AJS15, FRT24, JKS16, P98, P99, RY22]. We would like to emphasize that, in the case of $\alpha\in(0,2)$, there were no previous studies on problem (1.1) with (1.2) to our knowledge.

In this paper we consider (1.1) and obtain analogous results to [HI18] (i.e. assertions (a)–(d)) in the Heisenberg group $\mathbb{H}^{N}$. Our conditions are optimal and, as an application, we show that our conditions can lead the sharp estimates of the life span of solutions with $\mu$ having a polynomial decay at the space infinity.

In order to state our main results, we prepare some notation and formulate the definition of solutions. For any measurable set $A\subset\mathbb{H}^{N}$, $|A|$ denotes the Haar measure of $A$. For any $\eta\in\mathbb{H}^{N}$ and $r>0$, set $B(\eta,r):=\{\zeta\in\mathbb{H}^{N}:\mathsf{d}_{\mathbb{H}}(\eta,\zeta)<r\}$. Furthermore, for any $L^{1}_{\rm loc}$ function $f$, we set

Following [Folland], for $\alpha\in(0,2)$ we define the fractional sub-Laplacian by

where $\Gamma$ is the Gamma function, $e^{t\Delta_{\mathbb{H}}}$ is the heat flow, and $f\in L^{2}(\mathbb{H}^{N})$ is any function for which the relevant limit exists in $L^{2}$ norm. For the simplicity of notation, for $\alpha\in(0,2]$, denote

Let $G_{\alpha}:=G_{\alpha}(\eta,t)$ be the fundamental solution of

where $\alpha\in(0,2]$. When $\alpha=2$, we briefly write $G(\eta,t)$ instead of $G_{\alpha}(\eta,t)$. For any locally integrable function $f$ in $\mathbb{H}^{N}$, it is well-known that

See e.g., [FS] (for $\alpha=2$) and [MPS] (for $\alpha\in(0,2)$).

The key to our arguments is the following two sided estimate of the fundamental solution $G_{\alpha}$ of (1.7).

For the proof of Proposition 1.1, see Subsection 2.3 below. In the case of $\alpha=2$, (1.8) is well-known (see e.g. [FS]). But in the case of $\alpha\in(0,2)$, this is new. To prove the estimates (1.8) we employ a subordination formula in [G], which allows us to estimate the kernel $G_{\alpha}(\eta,t)$ via the heat kernel of the sub-Laplacian $-\Delta_{\mathbb{H}}$.

We formulate the definition of solutions.

Now we are ready to state the main results of this paper. In the first theorem we show the existence and the uniqueness of the initial trace of solutions of (1.1) and obtain analogous results to assertions (a) and (b).

Since $\mu$ in (1.11) is unique, the following holds:

As a corollary of Theorem A, we have:

Theorem A can be regarded as a generalization of the result in [HI18, Theorem 1.1]. Let $u$ be a solution of problem (1.1) with (1.2) in $\mathbb{H}^{N}\times[0,T)$, where $T\in(0,\infty)$. We first prove the existence and the uniqueness of the initial trace of the solution $u$ and then obtain necessary conditions for the solvability (i.e. assertions (i)–(iii) in Theorem A). The proof of our necessary conditions in the case of $\alpha\in(0,2)$ follows the arguments in the proofs of [FHIL23, H24, HIT23, HS24, LS21], which enable us to more easily obtain the necessary conditions in [HI18, Theorem 1.1]. Let $\zeta\in\mathbb{H}^{N}$. Following these results, we get an integral inequality related to

and then obtain the desired estimates by applying the existence theorem for ordinary differential equations to this estimate. See Lemma 2.10 below.

However, the proof in the case of $\alpha=2$ is completely different from those in [FHIL23, H24, HIT23, HS24, LS21] and [HI18]. First, we extend solutions to the framework of weak ones and then employ a suitable cut-off function as a test function. This method was developed by Mitidieri–Pohozaev [MP01] and can be applied not only to semilinear heat equations but also to a wider class of equations. For the case of semilinear heat equations, see e.g. [GP21, IKO20, IS19]. In this paper we follow the arguments in [IKO20].

The reason why we adopt two methods comes from the fact that in Proposition 1.1, $c_{1}=c_{2}$ does generally not hold in the case of $\alpha=2$. In order to apply the arguments in [FHIL23, H24, HIT23, HS24, LS21], we have to get

for $\eta,\zeta\in\mathbb{H}^{N}$ and $0<s<t$. However, in the case of $\alpha=2$, we can not get such an estimate from Proposition 1.1, since $g_{\alpha}$ is an exponential function. On the other hand, in the case of $\alpha\in(0,2)$, it can be regarded as $c_{1}=c_{2}=1$ and we can avoid this problem, since $g_{\alpha}$ is a polynomial function. In the previous studies dealing with $\mathbb{R}^{N}$ [H24, HI18, HIT23, HS24, LS21, FHIL23], this problem did not arise because they were either dealing with the fractional case or had an explicit formula of $G_{\alpha}$.

By Theorem A we have

We give sufficient conditions for the solvability of problem (1.1) with (1.2). The proofs follow the arguments in [HI18].

As a corollary of Theorems A, C, D, and E, we have

From this corollary it can be seen that $\Phi_{\alpha}$ is an optimal singularity for the local-in-time solvability. In particular, together with Remark 1.3, it can be seen that $p_{\alpha,Q}=1+\alpha/Q$ is the Fujita-exponent.

The rest of this paper is organized as follows. In Section 2 we collect properties of $\mathbb{H}^{N}$ and $G_{\alpha}$ and prepare some preliminary lemmas. In Section 3 we prove (1.11) in Theorem A and Theorem B. In Section 4 we prove assertions (i)–(iii) in Theorem A and complete the proof of Theorem A. In Section 5 we prove Theorems C–E. In Section 6, as an application of our theorems, we obtain estimates of the life span of solutions of problem (1.1) with small initial data.

In this subsection we collect properties of the Heisenberg group $\mathbb{H}^{N}$ and the fundamental solution $G_{\alpha}$. The following lemma is used when we calculate integrals in the Heisenberg group $\mathbb{H}^{N}$ and is well-known (see e.g. [BHQ24, P98]). Therefore, we omit the proof.

For $\lambda>0$ and $\eta=(x,y,\tau)\in\mathbb{H}^{N}$, define

The following properties are taken from [FS, MPS]. Given $\alpha\in(0,2]$, the function $G_{\alpha}$ has the following properties:

for all $\eta\in\mathbb{H}^{N}$, $0<s<t$, and $\lambda>0$.

Since $X^{*}_{i}=-X_{i}$, $Y_{i}^{*}=-Y_{i}$ for $i=1,\ldots,N$ and $\Delta_{\mathbb{H}}=\sum_{i=1}^{N}(X_{i}^{2}+Y_{i}^{2})$, by integration by parts we have:

for all $\phi,\psi\in C_{0}^{\infty}(\mathbb{H}^{N})$.

For $\eta\in\mathbb{H}^{N}$ and a nonempty subset $A\subset\mathbb{H}^{N}$, set

Recall that the Hardy-Littlewood maximal function $\mathcal{M}$ is defined by

where the supremum is taken over all balls $B$ containing $\eta$ and $r_{B}>0$ is the radius of $B$. It is well-known that $\mathcal{M}$ is bounded on $L^{p}(\mathbb{H}^{N})$ for $1<p\leq\infty$.

We have the following result whose proof is quite elementary and will be omitted.

In this subsection, we obtain estimates of the fundamental solution $G_{\alpha}$ and collect the basic properties of $\Lambda_{\alpha}$.

First, we prove Proposition 1.1 in Subsection 1.3.