Localized peaking regimes for quasilinear parabolic equations

This paper deals with the asymptotic behavior as (Formula presented.) of all weak (energy) solutions of a class of equations with the following model representative: (Formula presented.) with prescribed global energy function (Formula presented.) Here (Formula presented.), (Formula presented.), (Formula presented.), Ω is a bounded smooth domain, (Formula presented.). Particularly, in the case (Formula presented.) it is proved that the solution u remains uniformly bounded as t → T in an arbitrary subdomain Ω0 ⊂ Ω : Ω0 ⊂ Ω and the sharp upper estimate of 𝑢(t, x) when t → T has been obtained depending on µ > 0 and µ = dist(𝑥x, ∂Ω). In the case b(t𝑡, x) > 0 for all (t, x) ϵ (0, T) × Ω, sharp sufficient conditions on degeneration of b(t𝑡, 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. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim