# The balance between diffusion and absorption in semilinear parabolic equations

### Laurent Véron

Université François Rabelais, Tours, France### Andrey Shishkov

Academy of Sciences of Ukraine, Donetsk, Ukraine

## 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 $\prt_{t} u-\Gd u^m+h(t)u^q=0$ in $\Gw\ti (0,T)$ satisfying $u_{k}(x,0)=k\gd_0$. 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 $t=0$.

## Cite this article

Laurent Véron, Andrey Shishkov, The balance between diffusion and absorption in semilinear parabolic equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 1, pp. 59–96

DOI 10.4171/RLM/481