Lifespan estimates via Neumann heat kernel

Abstract

This paper studies the lower bound of the lifespan $T^{*}$ for the heat equation $u_t=Δu$ in a bounded domain $Ω\subset\mathbb{R}^{n}(n\geq 2)$ with positive initial data $u_{0}$ and a nonlinear radiation condition on partial boundary: the normal derivative $\partial u/\partial n=u^{q}$ on $Γ_1\subseteq \partialΩ$ for some $q>1$, while $\partial u/\partial n=0$ on the other part of the boundary. Previously, under the convexity assumption of $Ω$, the asymptotic behaviors of $T^{*}$ on the maximum $M_{0}$ of $u_{0}$ and the surface area $|Γ_{1}|$ of $Γ_{1}$ were explored. In this paper, without the convexity requirement of $Ω$, we will show that as $M_{0}\rightarrow 0^{+}$, $T^{*}$ is at least of order $M_{0}^{-(q-1)}$ which is optimal. Meanwhile, we will also prove that as $|Γ_{1}|\rightarrow 0^{+}$, $T^{*}$ is at least of order $|Γ_{1}|^{-\frac{1}{n-1}}$ for $n\geq 3$ and $|Γ_{1}|^{-1}\big/\ln\big(|Γ_{1}|^{-1}\big)$ for $n=2$. The order on $|Γ_{1}|$ 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.