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

Ugur G. Abdulla

#### 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.

#### Article information

Source
Adv. Differential Equations, Volume 4, Number 2 (1999), 197-224.

Dates
First available in Project Euclid: 18 April 2013

Abdulla, Ugur G. Local structure of solutions of the Dirichlet problem for $N$-dimensional reaction-diffusion equations in bounded domains. Adv. Differential Equations 4 (1999), no. 2, 197--224. https://projecteuclid.org/euclid.ade/1366291413