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

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)