The balance between diffusion and absorption in semilinear parabolic equations

Let $h:[0,\mathrm{\infty })\mapsto [0,\mathrm{\infty })$ be continuous and nondecreasing, $h(t)>0$ if $t>0$, and $m,q$ be positive real numbers. We investigate the behavior when $k\to \mathrm{\infty }$ 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 $u^{\mathrm{\prime }}+h(t)u^q=0$ which blows-up at $t=0$.