Regularity of invariant sets in semilinear damped wave equations

Abstract

Under fairly general assumptions, we prove that every compact invariant subset I of the semiflow generated by the semilinear damped wave equation ϵ u t t + u t + β ( x ) u − ∑ i j ( a i j ( x ) u x j ) x i = f ( x , u ) , ( t , x ) ∈ [ 0 , + ∞ [ × Ω , u = 0 , ( t , x ) ∈ [ 0 , + ∞ [ × ∂ Ω , in H 0 1 ( Ω ) × L 2 ( Ω ) is in fact bounded in D ( A ) × H 0 1 ( Ω ) . Here Ω is an arbitrary, possibly unbounded, domain in R 3 , A u = β ( x ) u − ∑ i j ( a i j ( 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 I needs not to be an attractor.