Diffusion versus absorption in semilinear parabolic equations

TL;DR

This paper investigates the behavior of solutions to certain semilinear parabolic equations with singular initial data as a parameter tends to infinity, revealing conditions under which solutions develop single or multiple singularities.

Contribution

It characterizes the limiting behavior of solutions with concentrated initial data for a class of semilinear parabolic equations, identifying conditions that lead to single or multiple singularities.

Findings

Limit solutions depend on the decay rate of h(t)

Single singularity at (0,0) under specific conditions on gw(t)

Maximal solutions arise when gw(t) is constant

Abstract

We study the limit, when $k\to\infty$, of the solutions $u=u_{k}$ of (E) $\prt_{t}u-\Delta u+ h(t)u^q=0$ in $\BBR^N\ti (0,\infty)$, $u_{k}(.,0)=k\delta_{0}$, with $q>1$, $h(t)>0$. If $h(t)=e^{-\gw(t)/t}$ where $\gw>0$ satisfies to $\int_{0}^1\sqrt{\gw(t)}t^{-1}dt<\infty$, the limit function $u_{\infty}$ is a solution of (E) with a single singularity at $(0,0)$, while if $\gw(t)\equiv 1$, $u_{\infty}$ is the maximal solution of (E). We examine similar questions for equations such as $\prt_{t}u-\Gd u^m+ h(t)u^q=0$ with $m>1$ and $\prt_{t}u-\Gd u+ h(t)e^{u}=0$.