Localized peaking regimes for quasilinear parabolic equations

This paper deals with the asymptotic behavior as t→T 1,with prescribed global energy function E(t):=∫Ω|u(t,x)|p+1dx+∫0t∫Ω|∇xu(τ,x)|p+1dxdτ→∞ast→T.Here Δp(u)=∑i=1n|∇xu|p−1uxixi, p>0, λ>p, Ω is a bounded smooth domain, b(t,x)≥0. Particularly, in the case E(t)≤Fμ(t)=expω(T−t)−1p+μforallt 0,ω>0,it is proved that the solution u remains uniformly bounded as t→T in an arbitrary subdomain Ω0⊂Ω:Ω¯0⊂Ω and the sharp upper estimate of u(t,x) when t→T has been obtained depending on μ>0 and s=dist(x,∂Ω). In the case b(t,x)>0 for all (t,x)∈(0,T)×Ω, sharp sufficient conditions on degeneration of b(t,x) near t=T that guarantee the above mentioned boundedness for an arbitrary (even large) solution have been found and the sharp upper estimate of a final profile of the solution when t→T has been obtained.