Blow-up problems for the heat equation with a local nonlinear Neumann boundary condition

TL;DR

This paper investigates the finite-time blow-up of solutions to a heat equation with a local nonlinear Neumann boundary condition, providing bounds on blow-up time without requiring domain convexity, relevant to space shuttle insulation damage.

Contribution

It extends previous results by removing convexity assumptions and considering local nonlinear boundary conditions on parts of the boundary, offering new estimates for blow-up time.

Findings

Solutions blow up in finite time.

Derived upper and lower bounds for blow-up time.

Applicable to non-convex domains with partial boundary conditions.

Abstract

This paper estimates the blow-up time for the heat equation $u_t=\Delta u$ with a local nonlinear Neumann boundary condition: The normal derivative $\partial u/\partial n=u^{q}$ on $\Gamma_{1}$, one piece of the boundary, while on the rest part of the boundary, $\partial u/\partial n=0$. The motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects. We prove the solution blows up in finite time and estimate both upper and lower bounds of the blow-up time in terms of the area of $\Gamma_1$. In many other work, they need the convexity of the domain $\Omega$ and only consider the problem with $\Gamma_1=\partial\Omega$. In this paper, we remove the convexity condition and only require $\partial\Omega$ to be $C^{2}$. In addition, we deal with the local nonlinearity, namely $\Gamma_1$ can be just part of $\partial\Omega$.