On critical exponents for the heat equation with a nonlinear boundary condition

Abstract

In this paper we consider the heat equation $u_t=∆u$ in a unbounded domain $\Omega \subset R^N$ with a Neumann boundary condition $u_{\nu }=u^p$, where $p>1$ and $\nu$ is the exterior unit normal on $\partial \Omega$. It is shown for various type of domains that there exists a critical number $p_c(\Omega )\ge 1$, such that all of positive solutions blow up in a finite time when $p\in (1,p_c]$ while there exist positive global solutions if $p>p_c$ and initial data are small.