Local well posedness, asymptotic behavior and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time

Abstract

A class of semilinear evolution equations of the second order in time of the form u t t + A u + μ A u t + A u t t = f ( u ) u_{tt} + A u + \mu A u_t + A u_{tt}=f(u) is considered, where − A -A is the Dirichlet Laplacian, Ω \Omega is a smooth bounded domain in R N \mathbb R^N and f ∈ C 1 ( R , R ) f\in C^1(\mathbb R,\mathbb R) . A local well posedness result is proved in the Banach spaces W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) W^{1,p}_0(\Omega )\times W^{1,p}_0(\Omega ) when f f satisfies appropriate critical growth conditions. In the Hilbert setting, if f 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.