Diffusion versus absorption in semilinear parabolic problems

We study the limit, when $k\to \infty$, of the solutions $u=u_k$ of (E) ${?}_{t}u?\mathrm{\Delta }u+h(t)u^q=0$ in ${\mathbb{R}}^N\times (0,\infty )$, $u_k(?,0)=k{\delta }_0$, with $q>1$, $h(t)>0$. If $h(t)={\mathrm{e}}^{?\omega (t)/t}$ where $\omega >0$ satisfies to ${\int }_0^1\sqrt{\omega (t)}{t}^{\mathrm{?1}}\mathrm{d}t<\infty$, the limit function $u_{\infty }$ is a solution of (E) with a single singularity at $(0,0)$, while if $\omega (t)\equiv 1$, $u_{\infty }$ is the maximal solution of (E). We examine similar questions for equations such as ${?}_{t}u?\mathrm{\Delta }{u}^m+h(t)u^q=0$ with $m>1$ and ${?}_{t}u?\mathrm{\Delta }u+h(t){\mathrm{e}}^u=0$. To cite this article: A. Shishkov, L. Véron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).