Asymptotic properties of a semilinear heat equation with strong absorption and small diffusion (Q914947)

We consider the following family of Cauchy problems \[ (1)\quad u_ t- \epsilon \Delta u+u^ p=0,\quad x\in {\mathbb{R}}^ N,\quad t>0;\quad u(x,0)=u_ 0(x),\quad x\in {\mathbb{R}}^ N \] where \(N\geq 1\), \(0<p<1\), \(u_ 0(x)\) is continuous, nonnegative and bounded, and \(0<\epsilon \ll 1\). We discuss the behaviour of the corresponding solutions \(u_{\epsilon}(x,t)\) as \(\epsilon\downarrow 0\). In particular, we derive an asymptotic expansion for the difference \((u_{\epsilon}(x,t)-\bar u(x,t)),\) where \(\bar u(x,t)\) solves \[ (2)\quad u_ t+u^ p=0,\quad x\in {\mathbb{R}}^ N,\quad t>0;\quad u(x,0)=u_ 0(x),\quad x\in {\mathbb{R}}^ N. \] Solutions of (1) and (2) vanish in a finite time. We use the previous asymptotic result to study the convergence as \(\epsilon\downarrow 0\) of the local and global extinction times of (1) to the corresponding quantities for (2), these last being explicitly written in terms of \(u_ 0(x)\) and p.