Regularity of invariant sets in semilinear damped wave equations

Under fairly general assumptions, we prove that every compact invariant subset of the semiflow generated by the semilinear damped wave equation in is in fact bounded in . Here Ω is an arbitrary, possibly unbounded, domain in , Au=β(x)u−∑ij(aij(x)uxj)xi 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 needs not to be an attractor.