The balance between diffusion and absorption in semilinear parabolic equations

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 $t=0$.