Regularity of invariant sets in semilinear damped wave equations

Abstract

Under fairly general assumptions, we prove that every compact invariant subset $\mathcal{I}$ of the semiflow generated by the semilinear damped wave equation εu_{tt}+u_t+β(x)u-\sum_{ij}(a_{ij} (x)u_{x_j})_{x_i}&=f(x,u),&& (t,x)\in[0,+\infty[\timesΩ, u&=0,&&(t,x)\in[0,+\infty[\times\partialΩin $H_0^1(\Omega )\times L^2(\Omega )$ is in fact bounded in $D(\mathbf{A})\times H_0^1(\Omega )$. Here $\Omega$ is an arbitrary, possibly unbounded, domain in ${\mathbb{R}}^3$, $\mathbf{A}u=\beta (x)u-{\sum }_{ij}(a_{ij}(x)u_{x_j}{)}_{x_i}$ 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.

23 pages