Local structure of solutions of the Dirichlet problem for $N$-dimensional reaction-diffusion equations in bounded domains

Abstract

We consider the problem \begin{alignat}{2} u_t-\Delta u^m+bu^\beta & =0, &\quad& (x,t)\ \epsilon\ Q_T =\Omega\times(0,T]\\ u(x,t) & =0, &\quad& (x,t)\ \epsilon\ S_T=\partial\Omega\times(0,T]\\ u(x,0) & =u_0(x), &\quad& x\ \epsilon\ \Omega, \end{alignat} where $m\geq 1,$ $ b \in R^1,$ $ \beta>0, $ $ T>0;$ $ \Omega$ is a bounded, connected domain in $R^N$ with compact boundary, which is assumed to be piecewise smooth and to satisfy the exterior sphere condition. Let $u_0 \in C(\bar\Omega),$ $ u_0>0$ for $x \in \Omega$ and $u_0=0$ for $x\in \partial\Omega$. Assume also that $u_0$ is smooth near $\partial\Omega$. We show that the small time behaviour of the solution near the boundary $\partial\Omega$ depends on the relative strength of the diffusion and reaction terms near the boundary, that is to say on the function $$ \gamma(\bar x)=\lim\limits_{x\to\bar x} \bigl(\Delta u_0^m\big/bu^\beta_0\bigr),\quad\bar x \in \partial\Omega,\quad x\in \Omega. $$ We are essentially interested in the initial development of the interface between the dead core and the positive $u(x,t)$ field. In all possible cases, the small time behaviour of the interface is found, together with the local solution.