Lifespan estimates via Neumann heat kernel

TL;DR

This paper establishes lower bounds for the lifespan of solutions to a heat equation with nonlinear boundary conditions, removing convexity assumptions and analyzing the influence of initial data and boundary measure.

Contribution

It provides optimal lower bounds for the lifespan without convexity assumptions, extending previous results to more general domains.

Findings

Lower bound of order $M_0^{-(q-1)}$ as initial data $M_0 o 0^+$

Lower bound of order $| abla_1|^{-rac{1}{n-1}}$ for boundary measure as $| abla_1| o 0^+$ in $n eq 2$

Almost optimal order $| abla_1|^{-1} / ext{ln}(| abla_1|^{-1})$ for $n=2$

Abstract

This paper studies the lower bound of the lifespan $T^{*}$ for the heat equation $u_t=\Delta u$ in a bounded domain $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ with positive initial data $u_{0}$ and a nonlinear radiation condition on partial boundary: the normal derivative $\partial u/\partial n=u^{q}$ on $\Gamma_1\subseteq \partial\Omega$ 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 $T^{*}$ 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 $M_{0}\rightarrow 0^{+}$, $T^{*}$ is at least of order $M_{0}^{-(q-1)}$ which is optimal. Meanwhile, we will also prove that as $|\Gamma_{1}|\rightarrow 0^{+}$, $T^{*}$ is at least of order $|\Gamma_{1}|^{-\frac{1}{n-1}}$ for $n\geq 3$ and $|\Gamma_{1}|^{-1}\big/\ln\big(|\Gamma_{1}|^{-1}\big)$ 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.