Global Attractor for Damped Abstract Nonlinear Hyperbolic Systems

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 i...

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Bibliographic Details
Main Author: Pinter, Gabriella A
Format: Report
Language:English
Subjects:
ALGORITHMS
GLOBAL ATTRACTOR
HILBERT SPACE
HYPERBOLIC DIFFERENTIAL EQUATIONS
MATHEMATICAL MODELS
NONLINEAR DISSIPATIVE ELASTIC MODELS
NONLINEAR SYSTEMS
Numerical Mathematics
Theoretical Mathematics
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Summary: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.