Diffusion versus absorption in semilinear parabolic equations

We study the limit, when $k\to\infty$, of the solutions $u=u_{k}$ of (E) $\prt_{t}u-\Delta u+ h(t)u^q=0$ in $\BBR^N\ti (0,\infty)$, $u_{k}(.,0)=k\delta_{0}$, with $q>1$, $h(t)>0$. If $h(t)=e^{-\gw(t)/t}$ where $\gw>0$ satisfies to $\int_{0}^1\sqrt{\gw(t)}t^{-1}dt<\infty$, the limit function $u_{\infty}$ is a solution of (E) with a single singularity at $(0,0)$, while if $\gw(t)\equiv 1$, $u_{\infty}$ is the maximal solution of (E). We examine similar questions for equations such as $\prt_{t}u-\Gd u^m+ h(t)u^q=0$ with $m>1$ and $\prt_{t}u-\Gd u+ h(t)e^{u}=0$.