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-Δ u=v^p+u^r, v_t-Δ v=u^q+v^s, where t∈(0, T), x∈ \bm{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 r<q+1 and s<p+1 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.