Global Attractor for Damped Abstract Nonlinear Hyperbolic Systems

摘要

This work is concerned with the long time dynamics of a class of abstract nonlinear second order in time systems with damping. This class of systems describes nonlinear dissipative elastic models with the nonlinear term produced by neo-Hookean type stress-strain relationships. In our earlier paper it was shown that these systems give rise to a weak dynamical system and that there exists a weak compact attractor. In the present work, using a somewhat more detailed analysis based in part on the results of H.T. Banks, D.S. Gilliam and V.I. Shubov on the existence and uniqueness of the weak solutions, we show that these systems generate a 'strong' dynamical system also. More importantly, we are able to prove the existence of a compact 'strong' global attractor. Finally, we make several comments concerning the regularity of this attractor, and present two examples.