Uniform dynamics of partially damped semilinear Bresse systems

Abstract

This paper is concerned with the Bresse system that arises in the modeling of arched beams. It is given by a system of three coupled wave equations that reduces to the well-known Timoshenko model when the arch curvature is zero. In a context of nonlinear elastic foundation, we establish the existence of smooth finite-dimensional global attractors, by adding dissipation mechanism in only one of its equations. In addition, we study the uniform boundedness of longtime dynamics with respect to the curvature parameter. These results have not been considered for partially damped semilinear Bresse or Timoshenko systems.

Keywords:

• Smooth global attractors
• gradient systems
• quasi-stability
• Bresse–Timoshenko

MSC:

• 35B41
• 35L53
• 74K10

Acknowledgements

The authors thank the referee for his/her comments on a previous version of the paper and Professor M. A. Jorge Silva for helpful conversations. T. F. Ma was partially supported by CNPq grant 315165/2021-9 and FAPESP grant 2019/11824-0. P. N. Seminario-Huertas was fully supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior grant INCTMat 88887.507829/2020-00. This work was also partially supported by Junta de Andalucía Project FEDER P18-FR-4509.

Disclosure statement

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

Additional information

Funding

This work was supported by CAPES [INCTMat 88887.507829/2020-00], CNPQ [315165/2021-9], Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía [FEDER P18-FR-4509], São Paulo Research Foundation (FAPESP) [2019/11824-0].