Attractors met X-elliptic operators

We consider degenerate parabolic and damped hyperbolic equations involving an operator L, that is X-elliptic with respect to a family of locally Lipschitz continuous vector fields X={X1,…,Xm}. The local well-posedness is established under subcritical growth restrictions on the nonlinearity f, which are determined by the geometry and functional setting naturally associated with the family of vector fields X. Assuming additionally that f is dissipative, the global existence of solutions follows, and we can characterize their longtime behavior using methods from the theory of infinite dimensional dynamical systems.