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

In this paper we discuss continuation properties and asymptotic behavior of ε-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 ε-regular solutions is given. We also formulate sufficient conditions to construct a piecewise ε-regular solutions (continuation beyond maximal time of existence for ε-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)1/2ut +(−ΔD)u = f (u) in H1 0 (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 Hm 2,{Bj }(Ω).  2005 Elsevier Inc. All rights reserved.