Local and global properties of solutions of heat equation with superlinear absorption

TL;DR

This paper investigates the asymptotic behavior of solutions to a heat equation with superlinear absorption, classifying solutions based on the integrability of inverse functions related to the absorption term.

Contribution

It provides a classification of solution behaviors depending on the properties of the absorption function and establishes results on initial trace existence and non-uniqueness for unbounded initial data.

Findings

Three main types of solution behavior identified.

Conditions for initial trace existence established.

Non-uniqueness results for solutions with unbounded initial data.

Abstract

We study the limit, when $k\to\infty$ of solutions of $u_t-\Delta u+f(u)=0$ in $R^N\times(0,\infty)$ with initial data $k\gd$, when $f$ is a positive increasing function. We prove that there exist essentially three types of possible behaviour according $f^{-1}$ and $F^{-1/2}$ belong or not to $L^1(1,\infty)$, where $F(t)=\int_0^t f(s)ds$. We emphasize the case where f(u)=u((\ln u+1))^{\alpha}. We use these results for giving a general result on the existence of the initial trace and some non-uniqueness results for regular solutions with unbounded initial data.