Asymptotic analysis and upper semicontinuity to a system of coupled nonlinear wave equations

Abstract

In this paper we study the long-time behaviour of a system consisting of two nonlinear wave equations under the action of three competing forces, damping forces, strong source and external force. It is of great interest to know how the relationship between these forces acts on the behaviour of the solutions of the system. In this sense, we investigate the well-posedness of system, as well as the existence of global and exponential attractors. In addition, we consider the upper semicontinuity of the global attractor when the coupling parameter of the system tends to zero. Once proved the existence of global solutions (in time), to obtain the existence of global and exponential attractors results, we prove that the dynamical system associated to solutions of the model is quasi-stable and gradient.

Keywords:

• Nonlinear wave equations system
• global attractors
• exponential attractor
• upper continuity

MSC (2010):

• 35B40
• 35B41
• 37L30

Acknowledgements

The authors are grateful to the anonymous reviewers who contributed greatly to the improvement of this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

A. J. A. Ramos thanks the CNPQ for financial support through the project CNPq Grant No. 310729/2019-0. D. S. Almeida Júnior thanks the CNPQ for financial support through the project: Impact of the second spectrum of frequency on the stabilization of partially dissipative Timoshenko type systems', Grant No. 314273/2020-4].