Local Well Posedness, Asymptotic Behavior and Asymptotic Bootstrapping for a Class of Semilinear Evolution Equations of the Second Order in Time

A class of semilinear evolution equations of the second order in time of the form utt+Au+μAut+Autt = f(u) is considered, where −A is the Dirichlet Laplacian, Ω is a smooth bounded domain in RN and f ∈ C1(R,R). A local well posedness result is proved in the Banach spacesW 1,p 0 (Ω)×W 1,p 0 (Ω) when f satisfies appropriate critical growth conditions. In the Hilbert setting, if f satisfies an additional dissipativeness condition, the nonlinear semigroup of global solutions is shown to possess a gradient-like attractor. Existence and regularity of the global attractor are also investigated following the unified semigroup approach, bootstrapping and the interpolation-extrapolation techniques. 1. Introductory notes In this article we consider a class of semilinear evolution equations of the second order in time of the form (1.1) utt +Au+ μAut +Autt = f(u), t > 0, with the initial conditions (1.2) u(0) = u0, ut(0) = v0 from a suitably chosen Banach space X. Here −A : D(A) ⊂ X → X is the generator of an exponentially decaying analytic semigroup of bounded linear operators in X and X is the domain of A with the graph norm. The equations that fall into this class are known to represent some sort of ‘propagation problems’ (see [4, 6, 22]; also [19] and the references therein), among which a specific problem is (1.3) ⎧⎪⎨ ⎪⎩ utt −∆u− μ∆ut −∆utt = f(u), t > 0, x ∈ Ω, u = 0, t ≥ 0, x ∈ ∂Ω, u(0, x) = u0(x), ut(0, x) = v0(x), x ∈ Ω, Received by the editors May 21, 2007. 2000 Mathematics Subject Classification. Primary 35G25, 35B33, 35B40, 35B41, 35B65.