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 ${\partial }_{t}u-\Delta u^m+h(t)u^q=0$ in $\Omega \times (0,T)$ 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 $u^{\prime }+h(t)u^q=0$ which blows up at $t=0$.