## 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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