Lifespan estimates via Neumann heat kernel

Abstract

This paper studies the lower bound of the lifespan $T^{\ast }$ for the heat equation $u_t=\Delta u$ in a bounded domain $\Omega \subset {\mathbb{R}}^n(n\ge 2)$ with positive initial data $u_0$ and a nonlinear radiation condition on partial boundary: the normal derivative $\partial u/\partial n=u^q$ on ${\Gamma }_1\subseteq \partial \Omega$ for some $q>1$, while $\partial u/\partial n=0$ on the other part of the boundary. Previously, under the convexity assumption of $\Omega$, the asymptotic behaviors of $T^{\ast }$ on the maximum $M_0$ of $u_0$ and the surface area $|{\Gamma }_1|$ of ${\Gamma }_1$ were explored. In this paper, without the convexity requirement of $\Omega$, we will show that as $M_0\to 0^{+}$, $T^{\ast }$ is at least of order $M_0^{-(q-1)}$ which is optimal. Meanwhile, we will also prove that as $|{\Gamma }_1|\to 0^{+}$, $T^{\ast }$ is at least of order $|{\Gamma }_1{|}^{-\frac{1}{n-1}}$ for $n\ge 3$ and $|{\Gamma }_1{|}^{-1}/\ln (|{\Gamma }_1{|}^{-1})$ for $n=2$. The order on $|{\Gamma }_1|$ when $n=2$ is almost optimal. The proofs are carried out by analyzing the representation formula of $u$ in terms of the Neumann heat kernel.

31 pages, 7 figures