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 $u_{tt} + A u + \mu A u_t + A u_{tt}=f(u)$ is considered, where $-A$ is the Dirichlet Laplacian, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ and $f\in C^1(\mathbb{R},\mathbb{R})$. A local well posedness result is proved in the Banach spaces $W^{1,p}_0(\Omega)\times W^{1,p}_0(\Omega)$ 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.