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

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
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