Blow-up profile of a solution for a nonlinear heat equation with small diffusion

Abstract

This paper is concerned with positive solutions of semilinear diffusion equations $u_t={\epsilon }^2∆u+u^p$ in $\Omega$ with small diffusion under the Neumann boundary condition, where $p>1$ is a constant and $\Omega$ is a bounded domain in $R^N$ with $C^2$ boundary. For the ordinary differential equation $u_t=u^p$, the solution $u^0$ with positive initial data $u_0∊C\left(\overline{\Omega }\right)$ has a blow-up set $S^0=\{x∊\overline{\Omega }|u_0\left(x\right)={\max }_{y∊\overline{\Omega }}{u}_0\left(y\right)\}$ and a blowup profile $$u_{*}^0\left(x\right)=\left(u_0\right(x{)}^{-(p-1)}-\left(\underset{y∊\overline{\Omega }}{\max }{u}_0\right(y){)}^{-(p-1)}{)}^{-1/(p-1)}$$ outside the blow-up set $S^0$. For the diffusion equation $u_t={\epsilon }^2∆u+u^p$ in $\Omega$ under the boundary condition $\partial u/\partial v=0$ on $\partial \Omega$, it is shown that if a positive function $u_0∊C^2\left(\overline{\Omega }\right)$ satisfies $\partial u_0/\partial v=0$ on $\partial \Omega$, then the blow-up profile $u_{*}^{\epsilon }\left(x\right)$ of the solution $u^{\epsilon }$ with initial data $u_0$ approaches $u_{*}^0\left(x\right)$ uniformly on compact sets of $\overline{\Omega }\setminus S^0$ as $\mathrm{\epsilon }\to +0$.