Regularity of invariant sets in semilinear damped wave equations

Under fairly general assumptions, we prove that every compact invariant subset II of the semiflow generated by the semilinear damped wave equationϵutt+ut+β(x)u−∑ij(aij(x)uxj)xi=f(x,u),(t,x)∈[0,+∞[×Ω,u=0,(t,x)∈[0,+∞[×∂Ω, in H01(Ω)×L2(Ω) is in fact bounded in D(A)×H01(Ω). Here Ω   is an arbitrary, possibly unbounded, domain in R3R3, Au=β(x)u−∑ij(aij(x)uxj)xiAu=β(x)u−∑ij(aij(x)uxj)xi is a positive selfadjoint elliptic operator and f(x,u)f(x,u) is a nonlinearity of critical growth. The nonlinearity f(x,u)f(x,u) needs not to satisfy any dissipativeness assumption and the invariant subset II needs not to be an attractor.