Admissible initial growth for diffusion equations with weakly superlinear absorption

TL;DR

This paper investigates the conditions under which initial data for certain diffusion equations with mildly superlinear absorption can grow at infinity, revealing when solutions behave like solutions to an ordinary differential equation due to dominant absorption effects.

Contribution

It establishes the threshold for initial data growth that allows diffusion to persist versus when absorption dominates, leading to ODE-like behavior.

Findings

If initial data grows too fast, diffusion is suppressed and solutions follow an ODE.

The paper characterizes admissible growth rates for initial data in diffusion equations with superlinear absorption.

Solutions with excessively fast-growing initial data do not exhibit diffusion and satisfy an ODE.

Abstract

We study the admissible growth at infinity of initial data of positive solutions of $\prt\_t u-\Gd u+f(u)=0$ in $\BBR\_+\ti\BBR^N$ when $f(u)$ is a continuous function, {\it mildly} superlinear at infinity, the model case being $f(u)=u\ln^\ga (1+u)$ with $1\textless{}\ga\textless{}2$. We prove in particular that if the growth of the initial data at infinity is too strong, there is no more diffusion and the corresponding solution satisfies the ODE problem $\prt\_t \gf+f(\gf)=0$ on $\BBR\_+$ with $\gf(0)=\infty$.