Non-simultaneous blow-up in parabolic equations coupled via localized sources

This work deals with non-simultaneous and simultaneous blow-up solutions for u t = Δ u + u m ( x 0 , t ) v p ( x 0 , t ) , v t = Δ v + u q ( x 0 , t ) v n ( x 0 , t ) , subject to homogeneous Dirichlet boundary conditions. We obtain the complete results of non-simultaneous and simultaneous blow-up solutions for any fixed point x 0 in any general bounded domain. The critical exponents of non-simultaneous blow-up are proposed.