Blow-Up Rate Estimates and Liouville Type Theorems for a Semilinear Heat Equation with Weighted Source

Abstract

We study the Liouville-type theorem for the semilinear parabolic equation $u_t-\Delta u =|x|^a u^p$ with $p>1$ and $a\in {\mathbb R}$. Relying on the recent result of Quittner (Math Ann, doi:10.1007/s00208-015-1219-7, 2015), we establish the optimal Liouville-type theorem in dimension $N=2$, in the class of nonnegative bounded solutions. We also provide a partial result in dimension $N\ge 3$. As applications of Liouville-type theorems, we derive the blow-up rate estimates for the corresponding Cauchy problem.

Appendix

Lemma 4.1

Assume $N\ge 2$ and $a>-2$, we consider problem

$$\begin{aligned} w_s=\Delta w-\frac{1}{2}y.\nabla w-\beta w+|y|^aw^p, \quad (y,s)\in {\mathbb R}^N\times [0,T), \end{aligned}$$

(19)

with nonnegative initial datum $w_0\in L^\infty ({\mathbb R}^N)$. We define the weighted energy functional by

$$\begin{aligned} E(w)=\int _{{\mathbb R}^N}\Big (\frac{1}{2}|\nabla w|^2+\frac{\beta }{2}w^2-\frac{1}{p+1}|y|^aw^{p+1}\Big )\rho dy \end{aligned}$$

(20)

where $\rho (y)=e^{-|y|^2/4}$. Assume $E(w_0)<\infty$, then for any $t_1, t_2\in [0,T)$, $t_1<t_2$ we have

$$\begin{aligned} \int _{t_1}^{t_2}\int _{{\mathbb R}^N}w_s^2(y,s)\rho (y)dyds\le E(w(t_1))-E(w(t_2)). \end{aligned}$$

(21)

The proof of Lemma 4.1 is standard when $a\ge 0$. For general case, we refer the reader to [30, Lemma 3.1] for the similar arguments.

Lemma 4.2

Assume $N\ge 2$ and $a>-2$. Let w be solution of the problem (19) in ${\mathbb R}^N\times [0,T_{\max })$ with nonnegative initial datum $w_0\in L^\infty ({\mathbb R}^N)$. We define the weighted energy functional (20). If $E(w_0)<0$ then $T_{\max }<\infty$.

Proof

We follow the concavity method in [24] (see also [39, Theorem 17.6]). Assume that $T_{\max }(u_0)=\infty$. Let $M(s)=\frac{1}{2}\int _{0}^{s}\int _{{\mathbb R}^N}w^2(y,\tau )\rho (y)dy d\tau$ then

$$\begin{aligned} M''(s)&=\int _{{\mathbb R}^N}w(s)w_s(s)\rho dy\\&=-\int _{{\mathbb R}^N}|\nabla w(s)|^2\rho dy-\beta \int _{{\mathbb R}^N}w^2(s)\rho dy +\int _{{\mathbb R}^N}|y|^aw^{p+1}(s)\rho dy\\&=-(p+1)E(w(s))+\frac{p-1}{2}\int _{{\mathbb R}^N}|\nabla w(s)|^2\rho dy+\beta \frac{p-1}{2}\int _{{\mathbb R}^N}w^2(s)\rho dy\\&\ge -(p+1)E(w_0)>0. \end{aligned}$$

Consequently, $M'(s)\rightarrow \infty$ and $M(s)\rightarrow \infty$ as $s\rightarrow \infty$. Moreover,

$$\begin{aligned} M''(s)&\ge -(p+1)E(w(s))\ge -(p+1)E(w(s))+(p+1)E(w_0)\\&\ge (p+1)\int _{0}^{s}\int _{{\mathbb R}^N}w_s^2(y,\tau )\rho (y) dyd\tau , \end{aligned}$$

hence

$$\begin{aligned} M(s)M''(s)&\ge \frac{p+1}{2}\left( \int _{0}^{s}\int _{{\mathbb R}^N}w_s^2(y,\tau )\rho (y) dyd\tau \right) \left( \int _{0}^{s}\int _{{\mathbb R}^N}w^2(y,\tau )\rho (y) dyd\tau \right) \\&\ge \frac{p+1}{2}\left( \int _{0}^{s}\int _{{\mathbb R}^N}w(y,\tau )w_s(y,\tau )\rho (y)dyd\tau \right) ^2\\&=\frac{p+1}{2}\left( M'(s)-M'(0)\right) ^2. \end{aligned}$$

Since $M'(s)\rightarrow \infty$ as $s\rightarrow \infty$, there exist $\alpha , s_0>0$ such that

$$\begin{aligned} M(s)M''(s)\ge (1+\alpha )(M'(s))^2, \quad s\ge s_0. \end{aligned}$$

This guarantees that the nonincreasing function $s\mapsto M^{-\alpha }(s)$ is concave on $[s_0,\infty )$ which contradicts the fact that $M^{-\alpha }(s)\rightarrow 0$ as $s\rightarrow \infty$. $\square$

Lemma 4.3

Assume $N\ge 2$ and $-2<a\le 0$. Consider the problem (6) with $u_0$ radial nonincreasing. Then u is radial nonincreasing in ${\mathbb R}^N\times [0,T)$.

Proof

The proof is similar to [39, Proposition 52.17]. We recall that (see [30, Lemma A.2]) u satisfies the integral equation

$$\begin{aligned} u(t)=e^{t\Delta }u_0+\int _{0}^{t}e^{(t-s)\Delta }(|.|^au^p(s))ds \end{aligned}$$

The conclusion follows from the fact that $e^{t\Delta }$ preserves the radial nonincreasing property and $a\le 0$. $\square$