A Liouville-type theorem of a weighted semilinear parabolic equation on weighted manifolds with boundary

1.   Introduction

Let $(\hat{M}^{n}, g, e^{-h}dv)$ be a weighted manifold, which is in fact an $n$-dimensional complete Riemannian manifold endowed with a weighted measure $e^{-h}dv$, where $h$ is a smooth function on $\hat{M}^{n}$ and $dv$ is the volume element of the metric $g$. The associated weighted Laplacian is defined as $\triangle_{h}: = \triangle-\nabla h\cdot\nabla$, where $\triangle$ and $\nabla$, respectively, denote the rough Laplacian and the Levi-Civita connection. The Bakry-$\rm\acute{E}mery$ Ricci curvature is given by ${\rm Ric}_{h}: = {\rm Ric}+{\rm Hess}h$, where ${\rm Ric}$ and ${\rm Hess}$, respectively, are the Ricci curvature of $\hat{M}^{n}$ and the Hessian operator with respect to $g$ (see ^[1]).

Recently, many authors have devoted themselves to studying Liouville-type theorems of parabolic equations on weighted manifolds with or without a boundary, and there have been plenty of results obtained (see ^[2,3,4] and the references therein). For example, Wu ^[5] proved elliptic gradient estimates for positive solutions to the linear heat equation

on $(\hat{M}^{n}, g, e^{-h}dv)$ without a boundary and obtained Liouville theorems for positive ancient solutions (i.e., solutions defined in all space and negative time) satisfying some growth restriction near infinity.

Abolarinwa ^[6] showed Souplet-Zhang gradient estimates for positive solutions to the weighted semilinear parabolic equation

on weighted manifolds without a boundary, where $\alpha\in\mathbb R$ and the function $q(x, t)$ is $C^{1}$ in $x$ and $C^{0}$ in $t$. In particular, he obtained the following Liouville-type theorem.

Theorem A. Let $(\hat{M}^{n}, g, e^{-h}dv)$ be a complete noncompact weighted manifold (without a boundary) with ${\rm Ric}_{h}\geq0$. Assume that $q(x, t) = q(x)\neq 0$, that is, it is time-independent and satisfies $\|q^{+}\|_{L^{\infty}(B_{\rho}(x_{0}))} = o(\rho^{-(\alpha-1)})$ and $\|\nabla q\|_{L^{\infty}(B_{\rho}(x_{0}))} = o(\rho^{-(\alpha-1)})$ as $\rho\rightarrow \infty$. If $u$ is a positive ancient solution to Eq (1.2) satisfying $u(x, t) = o(r^{\frac{1}{2}}(x)+|t|^{\frac{1}{4}})$ as $r(x)\rightarrow \infty$ and $t\rightarrow -\infty$, then $u$ is a constant, where $q^{+} = \max\{q(x), 0\}$, $B_{\rho}(x_{0})$ is a geodesic ball with the center at $x_{0}$ and a radius $\rho$.

For further generalization of the result above, see also ^[7] and a recent paper ^[8]. Inspired by the works of Kunikawa and Sakurai ^[9], Dung et al. ^[10] gave elliptic gradient estimates for positive solutions to Eq (1.1) on weighted manifolds with a compact boundary. They also showed the following result.

Theorem B. Let $(\hat{M}^{n}, g, e^{-h}dv)$ be a complete noncompact weighted manifold with the compact boundary $\partial\hat{M}$, ${\rm Ric}_{h}\geq 0$ and ${\rm H}_{h}\geq 0$. Assume that $u$ is a positive ancient solution to Eq (1.1) with a Dirichlet boundary condition (i.e., the solution is constant on the boundary). If $u_{\eta}\geq 0$, $u_{t}\leq 0$ over $\partial\hat{M}\times(-\infty, 0]$, and $u(x, t) = e^{o(r_{\partial\hat{M}}(x)+|t|)}$ (as $r_{\partial \hat{M}}(x)\rightarrow \infty$ and $t\rightarrow -\infty$, $o(r_{\partial\hat{M}}(x)+|t|)$ is an infinitely small quantity), then $u$ must be constant.

Here and below, ${\rm H}_{h}$ stands for the weighted mean curvature on $\partial\hat{M}$ and is defined as ${\rm H}_{h}: = {\rm H}-\nabla h\cdot\eta$, where ${\rm H}$ denotes the mean curvature of the boundary $\partial\hat{M}$, $\eta$ is the outer unit normal vector to $\partial\hat{M}$, $u_{\eta}$ stands for the derivative of $u$ along the direction $\eta$, and $r_{\partial\hat{M}}(x)$ is the distance function from the boundary.

In this paper, on weighted manifolds with a compact boundary, we study the Liouville property of positive ancient solutions to Eq (1.2). On the basis of Souplet-Zhang gradient estimates for positive solutions to Eq (1.2) with a Dirichlet boundary condition, we obtain the following Liouville-type theorem, in the spirit of Theorem 3.3 of Souplet and Zhang in ^[11].

Main theorem. Let $(\hat{M}^{n}, g, e^{-h}dv)$ be a complete noncompact weighted manifold with the compact boundary $\partial\hat{M}$, ${\rm Ric}_{h}\geq0$ and ${\rm H}_{h}\geq0$. Assume that $q(x, t) = q(x)\neq 0$ satisfies $\|q^{+}\|_{L^{\infty}(B_{\rho}(\partial\hat{M}))} = o(\rho^{-(\alpha-1)})$ and $\|\nabla q\|_{L^{\infty}(B_{\rho}(\partial\hat{M}))} = o(\rho^{-(\alpha-1)})$ as $\rho\rightarrow \infty$. Let $u$ be a positive ancient solution to Eq (1.2) with a Dirichlet boundary condition. If $u_{\eta}\geq 0$, $u_{t}\le qu^{\alpha}$ over $\partial\hat{M}\times(-\infty, 0]$ and $u(x, t) = e^{o(r_{\partial\hat{M}}(x)+|t|)}$ (as $r_{\partial\hat{M}}(x)\rightarrow \infty$ and $t\rightarrow -\infty$, $o(r_{\partial\hat{M}}(x)+|t|)$ is an infinitely small quantity), then $u$ must be constant, where $B_{\rho}(\partial\hat{M}): = \{x\in\hat{M}^{n}|d(x, \partial\hat{M}) < \rho\}$.

Remark. If $q(x, t)\equiv 0$, Eq (1.2) reduces to Eq (1.1), and hence our result generalizes the corresponding result of Dung et al. in ^[10]. When $q(x, t)\equiv-1$, $\alpha = 1$, and $\hat{M} = (-\infty, 0]$, $h = x$. It can be checked that $u = e^{x-t}$ is a positive ancient solution to Eq (1.2), where $u_{\eta}\geq 0$, $u_{t}\leq qu^{\alpha}$ over $\partial\hat{M}\times(-\infty, 0]$ and its growth rate is $e^{|x|+|t|}$. The example shows that our growth condition is necessary and sharp in both the spatial and time directions. Hence, it is better than the condition $u(x, t) = o(r^{\frac{1}{2}}(x)+|t|^{\frac{1}{4}})$ (as $r(x)\rightarrow \infty$ and $t\rightarrow -\infty$) used in ^[6].

2.   Basic lemmas

In this section, we present some definitions and results. On a weighted manifold $(\hat{M}^{n}, g, e^{-h}dv)$ with the compact boundary $\partial\hat{M}$, the distance function from the boundary is given by

This is a smooth function outside of the cut locus for the boundary (see ^[12]). We introduce the weighted Laplacian comparison theorem for the distance function on weighted manifolds with a boundary.

Lemma 2.1. ^[13] Let $(\hat{M}^{n}, g, e^{-h}dv)$ be an $n$-dimensional weighted manifold with the compact boundary $\partial\hat{M}$. If ${\rm Ric}_{h}\geq-(n-1)K$ and ${\rm H}_{h}\geq-L$ for some non-negative constants $K$ and $L$, then

for all $x\in B_{\rho}(\partial\hat{M})$.

We give the following useful derivative equality, which is called the Reilly's formula.

Lemma 2.2. ^[14] Let $\varphi$ be a smooth function on a weighted manifold $(\hat{M}^{n}, g, e^{-h}dv)$ with the compact boundary $\partial\hat{M}$. Then

where $\Pi$ is the second fundamental form of $\partial\hat{M}$.

Next, we introduce a smooth cut-off function originally developed by Li-Yau. It is very useful in the proof of elliptic gradient estimates.

Lemma 2.3. ^[15] Let $(\hat{M}^{n}, g, e^{-h}dv)$ be an $n$-dimensional weighted manifold with the compact boundary $\partial\hat{M}$. A smooth cut-off function $\psi = \psi(x, t)$ supported in $\mathcal{Q}_{\rho, T}(\partial\hat{M}): = B_{\rho}(\partial\hat{M})\times[-T, 0]$ exists such that

(ⅰ) $\psi = \psi(r_{\partial\hat{M}}(x), t)\equiv\psi(r, t)$; $\psi(r, t) = 1$ in $\mathcal{Q}_{\rho/2, T/2}(\partial\hat{M})$, $0\leq\psi\leq 1$;

(ⅱ) $\psi$ is decreasing as a radial function in the spatial variables, and $\psi_{r} = 0$ in $\mathcal{Q}_{\rho/2, T}(\partial\hat{M})$;

(ⅲ) $|\psi_{t}|\leq\frac{C\psi^{1/2}}{T}$, $|\psi_{r}|\leq\frac{C_{\epsilon}\psi^{\epsilon}}{\rho}$ and $|\psi_{rr}|\leq\frac{C_{\epsilon}\psi^{\epsilon}}{\rho^{2}}$, where $C > 0$ is a universal constant and $C_{\epsilon} > 0$ is a constant depending only on $0 < \epsilon < 1$.

According to Souplet and Zhang's idea in ^[16], by introducing the auxiliary function $\sqrt{1+\log(N/u)}$ instead of $\log(u/N)$ used in ^[6], we prove a derivative inequality, which plays an important role in the proof of Proposition 3.1.

Lemma 2.4. Let $u$ be a smooth solution to Eq (1.2) and $0 < u\leq N$ for some constant $N$. Let $v = \sqrt{\log(P/u)}$, where $P = Ne$ and $\omega = |\nabla v|^{2}$. We then have

Proof. Since $u = Pe^{-v^{2}}$, we compute

and

Further, we get

hence

If we substitute (2.4) and (2.5) into Eq (1.2), it follows that

Using the Bochner formula (see ^[5]) for $\omega$, we have

Hence

where we used (2.6) in the second inequality.

A direct computation shows that

and

We then arrive at

which is the desired inequality (2.3).

This completes the proof of Lemma 2.4. □

3.   Elliptic gradient estimates

In this section, on the basis of the key derivative inequality, by applying maximum principle, we establish Souplet-Zhang gradient estimates for positive solutions to Eq (1.2) with a Dirichlet boundary condition. In particular, we need to use Reilly's formula to deal with the boundary case. In fact, we obtain the following result.

Proposition 3.1. Let $(\hat{M}^{n}, g, e^{-h}dv)$ be an $n$-dimensional weighted manifold with the compact boundary $\partial\hat{M}$. Assume that ${\rm Ric}_{h}\geq-(n-1)K$ and ${\rm H}_{h}\geq-L$. Here $K\geq 0$, $L\geq 0$ and $N > 0$ are some constants. Let $u\leq N$ be a positive solution to Eq (1.2) with a Dirichlet boundary condition on $\mathcal{Q}_{\rho, T}(\partial\hat{M})$. If $u_{\eta}\geq 0$ and $u_{t}\leq qu^{\alpha}$ over $\partial\hat{M}\times[-T, 0]$, then a constant $C$ depending on $n$ and $\alpha$ exists such that the following estimates hold.

(ⅰ) If $\alpha > 1$, then

where $D = 1+\log N-\log(\inf\limits_{\mathcal{Q}_{\rho, T}(\partial\hat{M})}u)$.

(ⅱ) If $\alpha\leq 1$, then

where $\bar{N} = \inf\{u(x, t)|(x, t)\in\mathcal{Q}_{\rho, T}(\partial\hat{M})\}$.

Proof. Let $\psi\omega$ reach its maximum at $(x_{1}, t_{1})\in\mathcal{Q}_{\rho/2, T/2}(\partial\hat{M})$, where $\psi$ denotes the cut-off function in Lemma 2.3 and $\omega$ is the function in Lemma 2.4. We divide the arguments into two cases.

Case 1. When $x_{1}\notin\partial\hat{M}$, without loss of generality, we may assume that $x_{1}\notin {\rm Cut}(\partial\hat{M})$ by Calabi's argument ^[17]. At $(x_{1}, t_{1})$, we know that

and

That is

A direct computation shows that

Combining (2.3) with (3.3), and using the condition of ${\rm Ric}_{h}\geq-(n-1)K$, then at $(x_{1}, t_{1})$, we get

Since $0 < \frac{2v^{2}}{2v^{2}+1}\leq 1$, $0 < \frac{1}{2v^{2}+1}\leq 1$, and $0 < \frac{v}{2v^{2}+1}\leq\frac{\sqrt{2}}{4}$, then

Next, we estimate every term on the right-hand side of (3.4) at $(x_{1}, t_{1})$.

where we used (2.1) in the first inequality.

By the Cauchy-Schwarz inequality, we have

and

By the Young's inequality, we get

where $D = 1+\log N-\log(\inf\limits_{\mathcal{Q}_{\rho, T}(\partial\hat{M})}u)$.

We now estimate the terms that contain the parameter $\alpha$ and divide the arguments into two cases.

Case (ⅰ). If $\alpha > 1$, then by using the Young's inequality

and the Cauchy-Schwarz inequality

Combining (3.5)–(3.9) with (3.10) and (3.11), for all $(x, t)\in \mathcal{Q}_{\rho, T}(\partial\hat{M})$, we have

Noting that $\psi(x, t) = 1$ in $\mathcal{Q}_{\rho/2, T/2}(\partial\hat{M})$ and $\omega = |\nabla v|^{2}$, where $v = \sqrt{\log(Ne/u)}$. It follows that

Namely, we get the desired estimate (3.1).

Case (ⅱ). If $\alpha\leq 1$, then by using the Young's inequality

where $\bar{N} = \inf\{u(x, t)|(x, t)\in\mathcal{Q}_{\rho, T}(\partial\hat{M})\}$.

Using the Cauchy-Schwarz inequality

Combining (3.5)–(3.9) with (3.12) and (3.13) for all $(x, t)\in \mathcal{Q}_{\rho, T}(\partial\hat{M})$, we get

Using $\psi(x, t) = 1$ in $\mathcal{Q}_{\rho/2, T/2}(\partial\hat{M})$ and the definition of $\omega$, we obtain

which is the desired estimate (3.2).

Case 2. When $x_{1}\in\partial\hat{M}$, we only consider the case $\alpha > 1$ because $\alpha\leq 1$ is similar. In the case, the estimate (3.1) still holds. Moreover, at $(x_{1}, t_{1})$, we get

namely,

Hence

Since $\omega = |\nabla v|^{2}$, where $v = \sqrt{\log(P/u)}$, by assumption, we know that $\omega$ also satisfies the Dirichlet boundary condition. It follows from Lemma 2.2 that

Because $u$ satisfies the Dirichlet boundary condition, then

Since

We directly compute

where we used (1.2) in the fourth equality.

Substituting (3.15) into (3.14), we arrive at

that is

at $(x_{1}, t_{1})$. The condition that $u_{t}\leq qu^{\alpha}$ over $\partial\hat{M}\times[-T, 0]$ yields

It follows from (3.16) that

Since $v\geq 1$, $2v-v^{-1}\geq 1$, and we get

at $(x_{1}, t_{1})$, which implies

or

on $\mathcal{Q}_{\rho, T}(\partial\hat{M})$, where we used the condition of ${\rm H}_{h}\geq-L$.

If (3.3) holds, then $u$ is constant and the conclusion follows.

If (3.4) holds, then for all $(x, t)\in\mathcal{Q}_{\rho/2, T/2}(\partial\hat{M})$, $\psi(x, t) = 1$, and we have

It also implies the conclusion by using

We complete the proof of Proposition 3.1. □

4.   Proof of the main theorem

In this section, applying the Souplet-Zhang gradient estimates for positive solutions to Eq (1.2) with a Dirichlet boundary condition, we complete the proof of the main theorem.

Proof. We only consider the case $\alpha\leq 1$ because $\alpha\geq 1$ is similar. The arguments can be divided into two cases.

Case 1. When $\alpha < 1$, by the estimate (3.2) in Proposition 3.1 for $K = L = 0$, we know that

for all $(x, t)\in\mathcal{Q}_{\rho/2, T/2}(\partial\hat{M})$.

Fixing $(x_{0}, t_{0})$ and using (4.1) to $u$ on $\mathcal{Q}_{\rho, \rho}(\partial\hat{M}): = B_{\rho}(\partial\hat{M})\times[t_{0}-\rho, t_{0}]$ and the assumption conditions, we get

Letting $\rho\rightarrow \infty$, we have $|\nabla u(x_{0}, t_{0})| = 0$. Because $(x_{0}, t_{0})$ is arbitrary, $\nabla u(x, t)\equiv 0$ and $u$ must be constant in space, namely, $u(x, t) = u(t)$. We now prove $u(t)$ is a constant by contradiction.

Let $\tilde{q}: = q(x)$, and thus by Eq (1.2), we have

Integrating (4.2) in the interal $(t, 0]$ with $t < 0$, we get

Using the condition of $\tilde{q} > 0$ and letting $t\rightarrow -\infty$, we have $u^{1-\alpha}(t) < 0$, which is impossible, since $u$ is a positive solution. Hence $\tilde{q} = 0$ and $u(x, t)$ is a constant.

Case 2. When $\alpha = 1$, by the same arguments as in Case 1, we easily find that $u$ must be constant in space, namely, $u(x, t) = u(t)$. Similarly, we have

for all $t < 0$.

Hence

which is a contradiction to the condition that $u(x, t) = e^{o(r_{\partial\hat{M}}(x)+|t|)}$ near infinity.

This proof is completed. □

5.   Conclusions

In this paper, we prove a Liouville-type theorem for positive ancient solutions to a weighted semilinear parabolic equation with a Dirichlet boundary condition on complete noncompact weighted manifolds with a compact boundary. The proof technique is based on Souplet-Zhang gradient estimates for positive solutions. This result can be viewed as an extension of Dung et al.'s ^[10] work on a linear heat equation.

The weight of the equation is expressed by a smooth function $q(x, t)$ in this paper. But for technical reasons, a Liouville-type theorem is obtained in the subcase $q(x)$ (i.e., it is time-independent). A natural question is whether there is a similar Liouville-type theorem when the weight is $q(x, t)$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was partially supported by the NSF of China (12161078) and the Funds for Innovative Fundamental Research Group Project of Gansu Province (24JRRA778).

Conflict of interest

The authors declare there is no conflict of interest.