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

We study the positive blowing-up solutions of the semilinear parabolic system: ut - Δu = vp + ur, vt - Δv = uq + vs, where t ∈(0, T), x ∈ RN 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. © 2004 Applied Probability Trust.