A reaction-diffusion system with mixed-type coupling

This paper deals with asymptotic behavior of solutions to a reaction-diffusion system coupled via localized and local sources: ut=Δu+vp(x∗(t),t),vt=Δv+uq. Both the initial-boundary problem with null Dirichlet boundary condition and the Cauchy problem are considered to study the interaction between the two kinds of sources. For the initial-boundary problem we prove that the nonglobal solutions blow up everywhere in the bounded domain with uniform blow-up profiles. In addition, it is interesting to observe that the Cauchy problem admits an infinity Fujita exponent, namely, the solutions blow up under any nontrivial and nonnegative initial data whenever pq>1. All these imply that the blow-up behavior of solutions is governed by the localized source for the two problems with mixed-type coupling.