A note on maximal solutions of nonlinear parabolic equations with absorption

Abstract

If $\Omega$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-\Delta u+f(u)=0$ in $Q_\infty^\Omega:=\Omega\times (0,\infty)$, $u=\infty$ on the parabolic boundary $\partial_{p}Q$. We prove that in most cases, the existence and uniqueness is reduced to the same property for the associated stationary equation in $\Omega$.