Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations

Abstract

In this paper, we study the Cauchy problem of semilinear heat equations. By introducing a family of potential wells, we first prove the invariance of some sets and isolating solutions. Then we obtain a threshold result for the global existence and nonexistence of solutions. Finally we discuss the asymptotic behavior of the solution.

Introduction

On the global existence and blowup of solutions for the Cauchy problem of the semilinear heat equation

$$u_t-\Delta u=u^{\gamma }$$

there have been many results [1], [2], [4], [5], [6], [7], [8], [9], [10], [12], [13], [14], [18], [19], [20], [21]. In 1992, Wang and Ding [17] studied the Cauchy problem

$$u_t-\Delta u=u^{\gamma }-u\text{,}x\in R^n,t>0\text{,}$$$$u(x,0)=\phi \left(x\right)\ge 0$$

where $1<\gamma <\infty$ if $n=1,2$; $1<\gamma \le \frac{n+2}{n-2}$ if $n\ge 3$. Letting $\bar{u}\left(x\right)$ be the steady state solution of problem (1.1), (1.2), they proved that

• If $\phi \left(x\right)\le \bar{u}\left(x\right)$, then problem (1.1), (1.2) admits a global $L^p$ solution.

• If $\phi \left(x\right)\le \bar{u}\left(x\right)$ and $\phi \left(x\right)\ne \bar{u}\left(x\right)$, then the $L^p$ solution $u\left(t\right)\to 0$ as $t\to +\infty$.

• If $\phi \left(x\right)\ge \bar{u}\left(x\right)$ and $\phi \left(x\right)\ne \bar{u}\left(x\right)$, then the $L^p$ solution blows up in finite time.

In [3], the Cauchy problem

$$u_t-\Delta u={\left|u\right|}^{p-1}u-u\text{,}x\in R^n,t>0\text{,}$$$$u(x,0)=\phi \left(x\right)$$

was studied, and similar results were derived.

In [11] problem (1.1), (1.2), for the general case $\phi \left(x\right)$ not satisfying $\phi \left(x\right)\le \bar{u}\left(x\right)$ or $\phi \left(x\right)\ge \bar{u}\left(x\right)$, was studied again. And some sufficient conditions for the global existence and nonexistence of $L^p$ solutions were given.

Throughout the present paper, the following notations are used for precise statements: $L^p\left(R^n\right)(1\le p\le \infty )$ denotes the usual space of all $L^p$-functions on $R^n$ with norm ${\Vert u\Vert }_{L^p\left(R^n\right)}={\Vert u\Vert }_p$ and ${\Vert u\Vert }_{L^2\left(R^n\right)}=\Vert u\Vert$; $H^1\left(R^n\right)$ denotes the usual Sobolev space on $R^n$ with norm ${\Vert u\Vert }_{H^1\left(R^n\right)}^2={\Vert u\Vert }^2+{\Vert \nabla u\Vert }^2$.

In the present paper, we consider the Cauchy problem

$$u_t-\Delta u={\left|u\right|}^{p-1}u-u\text{,}x\in R^n,t>0\text{,}$$$$u(x,0)=u_0\left(x\right)\text{,}x\in R^n$$

where $p$ satisfies

• $1<p<\infty$ if $n=1,2$; $1<p\le \frac{n+2}{n-2}$ if $n\ge 3$.

First, a family of potential wells and its corresponding sets are introduced. Then, we give a series of their properties. By using them, we not only prove the invariance of some sets under the flow of (1.3), (1.4), but also get isolate solutions. Further, we obtain a threshold result for the global existence and nonexistence of solutions. Finally, the asymptotic behavior of solutions is discussed.

Although there has been a lot of work using the potential well method, almost all of this is on the initial boundary value problem on a bounded domain. And there are few results dealing with the Cauchy problem. In [15], Nakao and Ono studied the Cauchy problem of the semilinear wave equation by using the potential well method, but they required the initial function $u_0\left(x\right)$ to possess a bounded support, which means this problem can be resolved by using the potential well method on a bounded domain. In particular, for the global existence, finite time blowup and asymptotic behavior of solutions for the Cauchy problem of nonlinear parabolic equations with critical initial conditions, to our knowledge, there is no any result up to now. In this paper we obtain the corresponding threshold results on the global existence, nonexistence and asymptotic behavior of solutions of problem (1.3), (1.4) under some conditions.

We define some functionals and sets as follows

$$J\left(u\right)=\frac{1}{2}({\Vert \nabla u\Vert }^2+{\Vert u\Vert }^2)-\frac{1}{p+1}{\Vert u\Vert }_{p+1}^{p+1}\text{,}$$$$I\left(u\right)={\Vert \nabla u\Vert }^2+{\Vert u\Vert }^2-{\Vert u\Vert }_{p+1}^{p+1}\text{,}$$$$W=\{u\in H^1\left(R^n\right)\mid I\left(u\right)>0,J\left(u\right)<d\}\cup \left\{0\right\}\text{,}$$

where

$$d=\underset{u\in \mathcal{N}}{inf}J\left(u\right)\text{,}$$$$\mathcal{N}=\{u\in H^1\left(R^n\right)\mid I\left(u\right)=0,{\Vert u\Vert }_{H^1}\ne 0\}\text{.}$$

Then the main results obtained in this paper can be described as follows.

Assume that $p$ satisfies (H), $u_0\left(x\right)\in H^1\left(R^n\right)$ and $J\left(u_0\right)\le d$, where $d$ is the depth of potential well $W$. Then

• Problem (1.3), (1.4) admits a global weak solution $u\left(t\right)\in L^{\infty }(0,\infty ;H^1\left(R^n\right))$ with $u_t\left(t\right)\in L^2(0,\infty ;L^2\left(R^n\right))$ and $u\left(t\right)\in W$ for $0\le t<\infty$, provided $I\left(u_0\right)\ge 0$.

• The global weak solution of problem (1.3), (1.4) decays to zero exponentially as $t\to +\infty$, provided $I\left(u_0\right)>0$.

• The weak solution of problem (1.3), (1.4) blows up in finite time, provided $I\left(u_0\right)<0$.