Exponential Decay Results for Semilinear Parabolic PDE with $C^0$ Potentials: A “Mean Value” Approach

Abstract

The asymptotic behavior of some semilinear parabolic PDEs is analyzed by means of a “mean value” property. This property allows us to determine, by means of appropriate a priori estimates, some exponential decay results for suitable global solutions. We also apply the method to investigate a well-known finite time blow-up result. An application is given to a one-dimensional semilinear parabolic PDE with boundary degeneracy. Our results shed further light onto the problem of determining initial data for which the corresponding solution is guaranteed to exponentially decay to zero or blow-up in finite time.

Appendix

Here we report the Mean Value Theorem for Integrals that is used throughout this paper. The interested reader can also see [10, Chapter 7].

Theorem 4.1

Let $T>0$ and $\Omega$ be a bounded domain (open and connected) in $\mathbb {R}^N$. Let $f,\varphi \in C([0,T];L^1(\Omega ))$. Suppose $\varphi (t)$ is nonnegative a.e. on $\Omega$. Then there is $\xi (t)\in C([0,T];\Omega )$ in which, for all $t\in [0,T]$,

$$\begin{aligned} \int _\Omega f(x,t)\varphi (x,t) \mathrm{{d}} x = f(\xi (t),t)\int _\Omega \varphi (x,t) \mathrm{{d}} x, \end{aligned}$$

that is,

$$\begin{aligned} \langle f(t),\varphi (t) \rangle _{L^2(\Omega )} = f(\xi (t),t) |\varphi (t)|_{L^1(\Omega )}. \end{aligned}$$

Proof

Denote by $C_c(\Omega )$ the continuous functions on $\Omega$ that are compactly supported in $\Omega$ (recall that such functions are dense in $L^p(\Omega )$, $1\le p<\infty$). Fix $t'\in [0,T]$ and let $\tilde{f}(t'),\tilde{\varphi }(t')\in C_c(\Omega )$, where $\tilde{\varphi }(t')$ is nonnegative on $\Omega$. By the Extreme Value Theorem for continuous functions, there exist $x_m$ and $x_M$ in $\Omega$ so that

$$\begin{aligned} m(t'):=\min _{x\in \Omega }\{\tilde{f}(x,t')\} = \tilde{f}(x_m,t') \end{aligned}$$

and

$$\begin{aligned} M(t'):=\max _{x\in \Omega }\{\tilde{f}(x,t')\} = \tilde{f}(x_M,t'). \end{aligned}$$

Hence, there holds,

$$\begin{aligned} m(t') \int _\Omega \tilde{\varphi }(x,t') \mathrm{{d}} x \le \int _\Omega \tilde{f}(x,t')\tilde{\varphi }(x,t') \mathrm{{d}} x \le M(t') \int _\Omega \tilde{\varphi }(x,t') \mathrm{{d}} x. \end{aligned}$$

Since $\tilde{\varphi }(t')$ is nonnegative, $|\tilde{\varphi }(t')|_{L^1(\Omega )}=\int _\Omega \tilde{\varphi }(x,t') \mathrm{{d}} x$. In the case when $\tilde{\varphi }(t')=0$, we obtain equality and identity. So in the case when $\tilde{\varphi }(t')$ is nonzero, we obtain

$$\begin{aligned} m(t') \le \frac{1}{|\tilde{\varphi }(t')|_{L^1(\Omega )}}\int _\Omega \tilde{f}(x,t')\tilde{\varphi }(x,t') \mathrm{{d}} x \le M(t'). \end{aligned}$$

Since $\tilde{f}(x,t')$ is continuous and $\tilde{f}(x,t')\in [m(t'),M(t')]$, then the Intermediate Value Theorem provides an $\xi (t')\in \Omega$ in which

$$\begin{aligned} \tilde{f}(\xi (t'),t') = \frac{1}{|\tilde{\varphi }(t')|_{L^1(\Omega )}}\int _\Omega \tilde{f}(x,t')\tilde{\varphi }(x,t') \mathrm{{d}} x. \end{aligned}$$

Thus,

$$\begin{aligned} \int _\Omega \tilde{f}(x,t')\tilde{\varphi }(x,t') \mathrm{{d}} x = \tilde{f}(\xi (t'),t') \int _\Omega \tilde{\varphi }(x,t') \mathrm{{d}} x. \end{aligned}$$

By the density of $C_c(\Omega )\subset L^1(\Omega )$, we also have for $f(t'),\varphi (t')\in L^1(\Omega )$,

$$\begin{aligned} \int _\Omega f(x,t')\varphi (x,t') \mathrm{{d}} x = f(\xi (t'),t') \int _\Omega \varphi (x,t') \mathrm{{d}} x, \end{aligned}$$

or

$$\begin{aligned} \langle f(t'),\varphi (t') \rangle _{L^2(\Omega )} = f(\xi (t'),t') |\varphi (t')|_{L^1(\Omega )}. \end{aligned}$$

(41)

So far we have shown that, for each fixed $t'\in [0,T]$, there exists $\xi (t') \in \Omega$ in which (41) holds. It remains to show that the map $[0,T]\ni t\mapsto \xi (t)\in \Omega$ is continuous. Let $t^*\in [0,T]$ and $(t_n)_{n\in \mathbb {N}_{>0}}$ be such that $t_n\rightarrow t^*$ as $n\rightarrow \infty$. Then with the continuity of f and $\varphi$ on [0, T],

$$\begin{aligned}\begin{aligned} \lim _{n\rightarrow \infty } f(\xi (t_n),t_n)&= \lim _{n\rightarrow \infty } \left( \frac{1}{|\varphi (t_n)|_{L^1(\Omega )}} \langle f(t_n),\varphi (t_n) \rangle _{L^2(\Omega )} \right) \\&= \frac{1}{|\varphi (t^*)|_{L^1(\Omega )}} \langle f(t^*),\varphi (t^*) \rangle _{L^2(\Omega )} = f(\xi (t^*),t^*). \end{aligned}\end{aligned}$$

Therefore, $\xi \in C([0,T];\Omega )$. This finishes the proof. $\square$