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

In this paper we discuss continuation properties and asymptotic behavior of epsilon-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 epsilon-regular solutions is given. We also formulate sufficient conditions to construct a piecewise epsilon-regular solutions (continuation beyond maximal time of existence for epsilon-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(tt) + eta (-Delta D)(1/2)ut + (-Delta D)u = f (u) in H-0(1) (Omega) x L-2 (Omega) 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 H-2(m).(B-j) ((Omega)). (c) 2005 Elsevier Inc. All rights reserved.