Solutions of some nonlinear parabolic equations with initial blow-up

Abstract

We study the existence and uniqueness of solutions of ${\partial }_{t}u-\Delta u+u^q=0$ ($q>1$) in $\Omega \times (0,\infty )$ where $\Omega \subset {\mathbb{R}}^N$ is a domain with a compact boundary, subject to the conditions $u=f\ge 0$ on $\partial \Omega \times (0,\infty )$ and the initial condition ${\lim }_{t\to 0}u(x,t)=\infty$. By means of Brezis' theory of maximal monotone operators in Hilbert spaces, we construct a minimal solution when $f=0$, whatever is the regularity of the boundary of the domain. When $\partial \Omega$ satisfies the parabolic Wiener criterion and $f$ is continuous, we construct a maximal solution and prove that it is the unique solution which blows-up at $t=0$.