Optimal condition for non-simultaneous blow-up in a reaction-diffusion system

Abstract

We study the positive blowing-up solutions of the semilinear parabolic system: $u_t-\Delta u=v^p+u^r,$$v_t-\Delta v=u^q+v^s$, where $t∊(0,T),$$x∊R^N$ and $p,$$q,$$r,$$s>1$. We prove that if $r>q+1$ or $s>p+1$ then one component of a blowing-up solution may stay bounded until the blow-up time, while if and this cannot happen. We also investigate the blow up rates of a class of positive radial solutions. We prove that in some range of the parameters $p,$$q,$$r$ and $s$, solutions of the system have an uncoupled blow-up asymptotic behavior, while in another range they have a coupled blow-up behavior.