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The behavior of the life span for solutions to u t =Δu+a(x)u p in d . (English) Zbl 0914.35056
The author of this interesting paper studies the life span T * (λ,Φ) of the positive, bounded solution u(x,t) to the Cauchy problem for the nonlinear reaction diffusion equation u t =Δu+a(x)u p (x d , t(0,T), p>1) under initial condition u(x,0)=λΦ(x), where λ>0, 0aC α ( d ), 0ΦC b ( d ). The initial function has the property Φ(x)δexp[-γ|x| 2 ] (δ,γ>0) or it is bounded. The asymptotic behavior of T * (λ,Φ) as λ0 in the case that T * (λ,Φ)<, for all λ>0 and as λ in all cases is studied accurately. The asymptotic order depends on a,Φ,p and d in case that λ0, while on the other hand in the case that λ, it depends only on whether there is a point x 0 such that a(x 0 ), Φ0, or whether the supports of a and Φ are separated by a positive distance.
MSC:
35K57Reaction-diffusion equations
35B40Asymptotic behavior of solutions of PDE
35K15Second order parabolic equations, initial value problems