Improvements on lower bounds for the blow-up time under local nonlinear Neumann conditions

TL;DR

This paper establishes sharp lower bounds for the blow-up time of solutions to a heat equation with nonlinear Neumann boundary conditions, revealing how domain geometry and boundary conditions influence blow-up behavior.

Contribution

It provides new sharp lower bounds for blow-up time considering boundary nonlinearity and domain convexity, extending previous results to locally convex domains.

Findings

Lower bound of order (q-1)^{-1} as q→1+

Lower bound of order | ext{Γ}_1|^{-1/(n-1)} for convex domains as | ext{Γ}_1|→0

Extension of results to domains with local convexity near boundary

Abstract

This paper studies the heat equation $u_t=\Delta u$ in a bounded domain $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative $\partial u/\partial n=u^{q}$ on partial boundary $\Gamma_1\subseteq \partial\Omega$ for some $q>1$, while $\partial u/\partial n=0$ on the other part. We investigate the lower bound of the blow-up time $T^{*}$ of $u$ in several aspects. First, $T^{*}$ is proved to be at least of order $(q-1)^{-1}$ as $q\rightarrow 1^{+}$. Since the existing upper bound is of order $(q-1)^{-1}$, this result is sharp. Secondly, if $\Omega$ is convex and $|\Gamma_{1}|$ denotes the surface area of $\Gamma_{1}$, then $T^{*}$ is shown to be 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 $n=2$ as $|\Gamma_{1}|\rightarrow 0$, while the previous result is $|\Gamma_{1}|^{-\alpha}$ for any $\alpha<\frac{1}{n-1}$. Finally, we generalize the results for convex domains to the domains with only local convexity near $\Gamma_{1}$.