Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. (English) Zbl 1059.35049

Souplet, Philippe; Tayachi, Slim

J. Math. Soc. Japan 56, No. 2, 571-584 (2004).

Summary: We study the positive blowing-up solutions of the semilinear parabolic system: \(u_t-\Delta u=v^p+u^r\), \(v_t-\Delta v=u^q+u^s\), where \(t\in(0,T)\), \(x\in\mathbb{R}^N\) 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.

MSC:

35K45	Initial value problems for second-order parabolic systems
35K57	Reaction-diffusion equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B35	Stability in context of PDEs
35B60	Continuation and prolongation of solutions to PDEs

Keywords:

uncoupled blow-up behavior; semilinear parabolic system; coupled blow-up behavior