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

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 utt+η(−ΔD)12ut+(−ΔD)u=f(u) in H01(Ω)×L2(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 2mth order semilinear parabolic initial boundary value problem in a Hilbert space H2,{Bj}m(Ω).