Non-simultaneous blow-up for \(n\)-componential parabolic systems
The authors consider the system of semilinear parabolic equations \((u_i)_t=\Delta u_i+u_i^{p_i}\), \(i=1,2,\dots,n\), in a domain \(\Omega\subset\mathbb R^N\), coupled via the nonlinear boundary conditions \(\partial u_i/\partial\eta=u_{i+1}^{q_{i+1}}\), \(i=1,2,\dots n\). They first find necessary and sufficient conditions on the exponents guaranteeing blow-up of all positive solutions. Then, considering the blow-up case for \(\Omega=\{x:|x|<R\}\) and restricting themselves to radial positive solutions increasing both in \(|x|\) and \(t\), they find necessary and sufficient conditions for simultaneous and non-simultaneous blow-up. They also obtain blow-up rate estimates.
scientific article; zbMATH DE number 5586883