Local well posedness for strongly damped wave equations with critical nonlinearities

Abstract

In this article the strongly damped wave equation is considered and a local well posedness result is obtained in the product space HQ(Q) X L 2 (il). The space of initial conditions is chosen according to the energy functional, whereas the approach used in this article is based on the theory of analytic semigroups as well as interpolation and extrapolation spaces. This functional analytic framework allows local existence results to be proved in the case of critically growing nonlinearities, which improves the existing results.

Attractors for strongly damped wave equations with critical nonlinearities

“…The existence of the ε-regular solution to (11) under the assumptions (6) has been recently discussed in [5] based on the original results reported in [2], [4]. We thus have (see [ , Y (θ) −1 and ε > 0.…”

“…We start with the results of [5] on local well-posedness and regularity for (2) with initial conditions in Y 0 and nonlinearities growing critically. Recall that (see [5,Propositions 1,4]):…”

“…The following result of [5,Corollary 2] plays an important role in the regularity of the solutions of (1) and we will refer to it later in the paper.…”

“…) (see [5]) and obtain the convergence of bounded sets from Y 0 to the attractor in the strong topology of H 1+α (Ω) × H α (Ω)-norm, α ∈ n−2 n+2 , 1 . For θ = 1 we have that the nonlinearity becomes subcritical, nevertheless, we loose compactness of the semigroup and of the nonlinearity (so that subcritical is of no help).…”

“…The paper is organized as follows. In Section 2 we briefly recall the results of [5] concerning solvability of (1). Section 3 is devoted to a discussion of the additional regularity of the solutions to (1).…”