Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system

Abstract

We study simultaneous and nonsimultaneous blow-up for solutions of the following system $$\left\{\begin{array}{l} u_t = \Delta u + u^r + v^p, \\ v_t = \Delta v + v^s + u^q, \end{array} \quad\mbox{ in } \Omega \times (0,T), \right. $$ with Dirichlet boundary conditions. We show that, in the range of exponents where either component may blow up alone, there also exist initial data for which both components blow up simultaneously. The proof is based on a continuity argument, which requires upper and lower blow-up estimates, independent of initial data, and continuous dependence of the existence time. In turn, we prove a result of continuous dependence of the existence time, under the assumption of uniform upper blow-up estimates, in the framework of general abstract semiflows.