Regularity of invariant sets in semilinear damped wave equations

TL;DR

This paper proves that all compact invariant sets of a semilinear damped wave equation are bounded in a higher regularity space, even without dissipativeness or attractor assumptions, on possibly unbounded domains.

Contribution

It establishes regularity of invariant sets for semilinear damped wave equations under general conditions, without requiring dissipative nonlinearities or attractor properties.

Findings

Invariant sets are bounded in $D( extbf{A}) imes H^1_0( ext{Omega})$

Results hold on unbounded domains in $ extbf{R}^3$

No dissipativeness assumption needed for the nonlinearity

Abstract

Under fairly general assumptions, we prove that every compact invariant subset $\mathcal I$ of the semiflow generated by the semilinear damped wave equation \epsilon u_{tt}+u_t+\beta(x)u-\sum_{ij}(a_{ij} (x)u_{x_j})_{x_i}&=f(x,u),&& (t,x)\in[0,+\infty[\times\Omega, u&=0,&&(t,x)\in[0,+\infty[\times\partial\Omega in $H^1_0(\Omega)\times L^2(\Omega)$ is in fact bounded in $D(\mathbf A)\times H^1_0(\Omega)$. Here $\Omega$ is an arbitrary, possibly unbounded, domain in $\R^3$, $\mathbf A u=\beta(x)u-\sum_{ij}(a_{ij}(x)u_{x_j})_{x_i}$ is a positive selfadjoint elliptic operator and $f(x,u)$ is a nonlinearity of critical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an an attractor.