Long-time behavior of second order evolution equations with nonlinear damping

We consider abstract nonlinear second order evolution equations with a nonlinear damping. Questions related to long time behaviour, existence and structure of global attractors are studied. Particular emphasis is put on dynamics which in addition to nonlinear dissipation - have noncompact semilinear terms and whose energy may not be necessarily decreasing. For such systems we first develop a general theory at the abstract level. This general theory is then applied to nonlinear wave and plate equations exhibiting the aforementioned characteristics. This way we are able to provide new results pertaining to several open problems in the area of structure and properties of global attractors arising in this class of PDE dynamics.