Global attractor for degenerate damped hyperbolic equations

The aim of this paper is to consider a class of degenerate damped hyperbolic equations with the critical nonlinearity involving an operator L that is X-elliptic with respect to a family of vector fields X. We prove the global existence of solutions and characterize their long time behavior. In particular, we show the semigroup generated by the equation has a compact, connected and invariant attractor.