The paper investigates the nonlinear parabolic system \(u_t=\Delta u^\mu+v^pe^{\alpha u}\), \(v_t=\Delta v^\nu+u^qe^{\beta v}\) with homogeneous Dirichlet boundary data in a parabolic domain \(\Omega\times(0,T)\).
The main result is the following. i) If \(pq\leq \mu\nu\) then: (1) For any \(\Omega\) there exists a solution that blows up in a finite time. (2) If \(\Omega\) is thin enough then the solutions are global provided the initial data are small enough. (3) If \(\Omega\) is thick enough then every nontrivial solution blows up in a finite time. ii) If \(pq>\mu\nu\) then: (1) For any \(\Omega\) the solutions are global provided the initial data are small enough. (2) For any \(\Omega\) there exists a solution that blows up in a finite time.
To prove their results, the authors construct suitable subsolutions or supersolutions.