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Life span and asymptotic behavior for a semilinear parabolic system with slowly decaying initial values. (English) Zbl 0906.35044

It is considered the initial value problem for the semilinear parabolic system

u t =Δu+v p ,v t =Δv+u q ,(x,t) N × + ·

At t=0, nonnegative, bounded and continuous initial values (u 0 (x),v 0 (x)) are prescribed. The main results are for the case when u 0 (λ|x| -α ) 1/(q+1) , v 0 (λ|x| -α ) 1/(p+1) with λ>0, 0a<Nmin{p+1,q+1}. The authors consider various questions of global existence and nonexistence, large time behavior or life span of the solutions in terms of simple conditions on λ,a,p,q and the space dimension N.

MSC:
35K45Systems of second-order parabolic equations, initial value problems
35K57Reaction-diffusion equations