The balance between diffusion and absorption in semilinear parabolic equations

Details The balance between diffusion and absorption in semilinear parabolic equations The balance between diffusion and absorption in semilinear parabolic equations

2008-05-24

arxiv.org/abs/0805.3789v1

Rend. Lincei, Mat. Appl. 18 (2007) 59-96

Mathematics

Analysis of PDEs

Scientific paper

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 $t=0$.

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