Quasi-stability and attractors for a porous-elastic system with history memory

Abstract

The paper is concerned with a porous elastic problem in a past history framework. We study its long-time behavior through the corresponding autonomous dynamical system. Instead of showing the directly the system has a bounded absorbing set, we show the system is gradient system and asymptotic smoothness, and prove the existence of a global attractor, which is characterized as unstable manifold of the set of stationary solutions. We also get the quasi-stability of the system by establishing a stabilizability inequality and therefore obtain the finite fractal dimension of the global attractor.

Keywords:

• Porous elastic system
• memory
• quasi-stable systems
• attractor

2010 Mathematics Subject Classifications:

• 35B40
• 35B41
• 74Dxx

Acknowledgments

The authors are grateful to the anonymous referees for its constructive remarks, which have enhanced the presentation of this paper.

Disclosure statement

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

Additional information

Funding

B. Feng has been supported by the National Natural Science Foundation of China, grant #11701465. D.S. Almeida Júnior thanks the CNPq for financial support through the projects ‘New guidelines for dissipative Timoshenko type systems at light of the second spectrum’, CNPq Grant 310423/2016-3 and ‘Stabilization for Timoshenko systems from the second spectrum point of view’, PNPD/CAPES/INCTMAT/LNCC 88887.351763/2019-00. A.J.A. Ramos thanks the CNPq for financial support through the projects ‘Asymptotic stabilization and numerical treatment for carbon nanotubes’ (CNPq Grant 310729/2019-0).