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 \( \prt_{t} u-\Gd u^m+h(t)u^q=0 \) 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 $u^{\prime }+h(t)u^q=0$ which blows-up at $t=0$.