Variable instability of a constant blow-up solution in a nonlinear heat equation

Abstract

This paper is concerned with positive solutions of the semilinear diffusion equation $u_t=∆u+u^p$ in $\Omega$ under the Neumann boundary condition, where $p>1$ is a constant and $\Omega$ is a bounded domain in $R^N$ with $C^2$ boundary. This equation has the constant solution $(p-1{)}^{-1/(p-1)}(T_0-t{)}^{-1/(p-1)}$ with the blow-up time $T_0>0$. It is shown that for any $\epsilon >0$ and open cone $\Gamma$ in $\{f∊C(\overline{\Omega }\left)\right|f\left(x\right)>0\}$, there exists a positive function $u_0\left(x\right)$ in $\overline{\Omega }$ with $\partial u_0/\partial v=0$ on $\partial \Omega$ and $\left|\right|u_0\left(x\right)-$ such that the blow-up time of the solution $u(x,t)$ with initial data $u(x,0)=u_0\left(x\right)$ is larger than $T_0$ and the function $u(x,T_0)$ belongs to the cone $\Gamma$. A theorem on the blow-up profile is also given.