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Blow-up of positive solutions of semilinear heat equations. (English) Zbl 0576.35068

The following problem is discussed:

u t =Δu+f(u) in Ω ×(0,T), fC 1 , f(s)>0 if s>0, ΩR n ; u(x,0)=ϕ(x) if xΩ, ϕC 1 (Ω ¯), ϕ0, ϕ=0 on Ω, u(x,t)=0, xΩ, 0<t<T·

As U(t)=max xΩ u(x,t) grows with t, it is assumed that T< is the supremum of all σ such that the solution to the problem above exists for t<σ, and that U(T-)=+:xΩ is a blow-up point if there is {(x m t m )}, t m T, x m X, and u(x m ,t m ), m·

A partial outline of the results can be divided into:

Case (i): Ω is a ball, u(.,t) are radial functions, ϕ r 0. The authors prove that the only blow-up point is x=0. For the case f(u)=(u+λ) p , p>1, λ0, they obtain: |u(r,t)|C/r 2/(λ-1) , any γ<p, lim tT supu(·,t) L q (Ω) <, q<n(p-1)/2, lim tT infu(·,t) L q (Ω) =, q>n(p-1)/2; if moreover Δϕ+f(ϕ)0 and n=1,2; or n3 and p(n+2)/n-2, (T-t) 1/(p-1) u(r,t)(1/p-1) 1/p-1 as tT provided rC(T-t) 1/2 , some C>0·

Case (ii): Non symmetric, Ω is a convex domain: the blow-up points lie in a compact subset of Ω. For the particular f(u)=(u+λ) p , u(x,t) C /(T-t) 1/(p-1) , all xΩ, and lim tT infu(·,t) L q (Ω) = if q>n(p-1)/2·

In the one-dimensional case n=1, if ϕ ’ changes sign just once, then the solution blow-up at a single point. The authors extend some of the results to the case of a boundary condition u/ν+βu=0, xΩ, 0<t<T, β0 and ν the outward normal vector.

Reviewer: J.E.Bouillet

MSC:
35K60Nonlinear initial value problems for linear parabolic equations
35B40Asymptotic behavior of solutions of PDE
35K20Second order parabolic equations, initial boundary value problems
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)