The balance between diffusion and absorption in semilinear parabolic equations

TL;DR

This paper studies the asymptotic behavior of fundamental solutions to a class of semilinear parabolic equations with absorption and diffusion, focusing on the limit as initial data intensity grows infinitely large.

Contribution

It analyzes whether the solutions converge to a PDE solution with a singularity or to an ODE solution that blows up at initial time as initial data increases.

Findings

Determines conditions under which solutions tend to PDE or ODE limits.

Provides insights into the balance between diffusion and absorption effects.

Characterizes the nature of singularities in the solutions.

Abstract

Let $h:[0,\infty)\mapsto [0,\infty)$ be continuous and nondecreasing, $h(t)>0$ if $t>0$, and $m,q$ be positive real numbers. We investigate the behavior when $k\to\infty$ of the fundamental solutions $u=u_{k}$ of $\prt_{t} u-\Delta u^m+h(t)u^q=0$ in $\Omega\ti (0,T)$ satisfying $u_{k}(x,0)=k\delta_0$. The main question is wether the limit is still a solution of the above equation with an isolated singularity at $(0,0)$, or a solution of the associated ordinary differential equation $ u'+h(t)u^q=0$ which blows-up at $t=0$.