Optimal conditions of non-simultaneous blow-up and uniform blow-up profiles in localized parabolic equations

This paper deals with localized parabolic equations ut=Δu+um(x0,t)epv(x0,t), vt=Δv+uq(x0,t)env(x0,t) with homogeneous Dirichlet boundary conditions, where x0x0 is any fixed point in a bounded domain of RNRN. The optimal classification of non-simultaneous and simultaneous blow-up phenomena is proposed for all of the nonnegative exponents. Moreover, uniform blow-up profiles are obtained for all kinds of simultaneous blow-up solutions.