A note on maximal solutions of nonlinear parabolic equations with absorption

TL;DR

This paper investigates the existence and uniqueness of maximal solutions to nonlinear parabolic equations with absorption in bounded domains, linking these properties to the stationary problem under superlinear growth conditions.

Contribution

It establishes conditions under which existence and uniqueness of solutions are equivalent to those of the stationary problem for equations with superlinear absorption.

Findings

Existence and uniqueness depend on the stationary problem in most cases.

Superlinear growth at infinity influences solution behavior.

Results connect parabolic and stationary problem properties.

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 $\Omega$.