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 \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.