Blowup time estimates for the heat equation with a nonlocal boundary condition

Abstract

We study the blowup time for the heat equation $u_{t}=\Delta u$ in a bounded domain $\Omega \subset {\mathbb {R}}^{n}(n\geqslant 2)$ with the nonlocal boundary condition, where the normal derivative $\partial u/\partial \mathbf {\eta }=\int \limits _{\Omega }u^{p}\mathrm {d}z$ on one part of boundary $\Gamma _{1}\subseteq \partial \Omega$ for some $p>1$, while $\partial u/\partial \mathbf {\eta }=0$ on the rest part of the boundary. By constructing suitable auxiliary functions and analyzing the representation formula of u, we establish the finite time blowup of the solution and get both upper and lower bounds for the blowup time in terms of the parameter p, the initial value $u_{0}(x)$ and the volume of $\Gamma _{1}$. In many other studies, they require the convexity of the domain $\Omega$ and only deal with the case $\Gamma _{1}=\partial \Omega$. In this article, we remove the convexity assumption and consider the problem with $\Gamma _{1}\subseteq \partial \Omega$.