On uniqueness of large solutions of nonlinear parabolic equations in nonsmooth domains

TL;DR

This paper investigates the existence, uniqueness, and properties of large solutions to a nonlinear parabolic PDE in bounded, possibly nonsmooth domains, establishing conditions for uniqueness and maximal solutions.

Contribution

It introduces new conditions under which large solutions are unique or maximal in nonsmooth domains for the nonlinear parabolic equation.

Findings

Maximal solutions exist for the PDE in bounded domains.

Uniqueness holds when the boundary is regular or has the local graph property.

Large solutions are unique if the boundary satisfies certain geometric conditions.

Abstract

We study the existence and uniqueness of the positive solutions of the problem (P): $\partial_tu-\Delta u+u^q=0$ ($q>1$) in $\Omega\times (0,\infty)$, $u=\infty$ on $\partial\Omega\times (0,\infty)$ and $u(.,0)\in L^1(\Omega)$, when $\Omega$ is a bounded domain in $\mathbb R^N$. We construct a maximal solution, prove that this maximal solution is a large solution whenever $q<N/(N-2)$ and it is unique if $\partial\Omega=\partial\bar\Omega^c$. If $\partial\Omega$ has the local graph property, we prove that there exists at most one solution to problem (P)