Behavior of blow-up solutions for quasilinear parabolic equations

Abstract

We study the quasilinear parabolic equation (|u|q − 1u)t − Δpu = 0 in a multidimensional domain (0; T) × Ω under the condition u(t; x) = f(t; x) on (0; T) × 𝜕Ω, where the boundary function f blows-up at a finite time T, i.e., f(t; x) → ∞ as t → T. For p ⩾ q > 0 and the boundary function f with power-like behavior, the upper bounds of weak solutions of the problem are obtained. The behavior of solutions at the transition from the case where p > q to p = q is investigated. A general approach within the method of energy estimates to such problems is described.