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): ${\partial }_{t}u-\Delta u+u^q=0$ ($q>1$) 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)

16 pages