Global existence and global non-existence of solutions to a reaction-diffusion system
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) with \(C^2\) boundary \(\partial\Omega\). By a classification of the nonnegative real numbers, \(p_i\) and \(q_i\), \(i= 1,2\), several criteria for global existence and non-existence of positive solutions are given for the weakly coupled reaction-diffusion system \[ u_t= \Delta u+ u^{p_1} v^{q_1}\quad\text{in }\Omega\times \mathbb{R}_+, \] \[ v_t=\Delta v+ u^{p_2} v^{q_2}\quad\text{in }\Omega\times \mathbb{R}_+, \] \[ u(x,0)= \phi(x),\quad v(x,0)= \psi(x), \] \[ u=0=v\quad\text{on }\partial\Omega\times \mathbb{R}_+, \] where \(\phi\) and \(\psi\) are nonnegative functions.
scientific article; zbMATH DE number 1389530