Global existence and finite time blow up for a degenerate reaction-diffusion system
The paper deals with the blow up and global existence of the nonnegative solutions to the degenerate parabolic system \[ u_t = \Delta u^m + u^{\alpha} v^p, \qquad v_t = \Delta v^n + u^q v^{\beta}, \qquad (x,t) \in \Omega \times (0,T) \] with homogeneous Dirichlet boundary data and nonnegative bounded initial functions. Here \(\Omega \subset \mathbb{R}^N\) is a bounded domain with smooth boundary and \(m,n > 1\), \( \alpha, \beta \geq 0\), \(p,q >0\). The system above describes processes of heat diffusion and combustion in two-component continua. Its special form \[ u_t = \Delta u^{\nu + 1} + v^p, \qquad \quad v_t = \Delta u^{\mu+1} + u^q \] was investigated by many authors (detailed quotations are given). The purpose of the paper under review is to extend these results.
scientific article; zbMATH DE number 2136698
scientific article