Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities

Abstract

In this paper we discuss continuation properties and asymptotic behavior of ɛ-regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions. A necessary and sufficient condition for global in time existence of ɛ-regular solutions is given. We also formulate sufficient conditions to construct a piecewise ɛ-regular solutions (continuation beyond maximal time of existence for ɛ-regular solutions). Applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed. In particular global solvability and the existence of a global attractor for u t t + η ( − Δ D ) 1 2 u t + ( − Δ D ) u = f ( u ) in H 0 1 ( Ω ) × L 2 ( Ω ) is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition. Similar result is obtained for a 2 mth order semilinear parabolic initial boundary value problem in a Hilbert space H 2 , { B j } m ( Ω ) .