Prevention of blowup via Neumann heat kernel

TL;DR

This paper investigates how the decay rate of a shrinking boundary with superlinear radiation law affects the global boundedness of solutions to the heat equation, establishing polynomial decay conditions for global existence.

Contribution

It introduces a decay condition on the shrinking boundary's surface area using Neumann heat kernel estimates to ensure global bounded solutions.

Findings

Polynomial decay of boundary area with rate > n-1 guarantees global bounded solutions.

Neumann heat kernel techniques are effective in analyzing boundary decay effects.

The results are relevant for temperature control in practical scenarios.

Abstract

Consider the heat equation $u_t-\Delta u=0$ on a bounded $C^2$ domain $\Omega$ in $\mathbb{R}^{n}(n\geq 2)$ with any positive initial data. If a superlinear radiation law $\frac{\partial u}{\partial n}=u^{q}$ with $q>1$ is imposed on a partial boundary $\Gamma_1\subseteq\partial\Omega$ which has a positive surface area, then it has been known that the solution $u$ blows up in finite time. However, if the partial boundary, on which the superlinear radiation law is prescribed, is shrinking and is denoted as $\Gamma_{1,t}$ at time $t$, then the solution may exist globally as long as the surface area $|\Gamma_{1,t}|$ of $\Gamma_{1,t}$ decays fast enough. This paper asks the question that how fast should $|\Gamma_{1,t}|$ decay in order to have a bounded global solution? This question is of significant importance in realistic situations, such as the temperature control within a certain safe range. By taking advantage of the Neumann heat kernel, we conclude that a polynomial decay $|\Gamma_{1,t}|\sim |\Gamma_1|(1+Ct)^{-\beta}$ with any $\beta>n-1$ suffices to ensure a bounded global solution.