Long-time dynamics of a problem of strain gradient porous elastic theory with nonlinear damping and source terms

Abstract

Of concern is a problem of strain gradient porous elastic theory with nonlinear damping terms, whose constitutive equations contain higher-order derivatives of the displacement in the basic postulates. The paper is based on the theory of ‘consistency’ due to Aouadi et al. [J. Therm. Stress. 43(2)(2020), 191–209] and Ieşan [American Institute of Physics, Conference Proceedings, 1329 (2011), 130–149], and contains four results. We firstly show that the system is global well posed by using maximal monotone operator. The second main result is the existence of global attractors which is proved by the method developed by Chueshov and Lasiecka [Long-time behavior of second order evolution equations with nonlinear damping. Mem. Amer. Math. Soc. vol. 195, no. 912, Providence, 2008; Von Karman evolution equations: well-posedness and long-time dynamics. Springer Monographs in Mathematics, Springer, New York, 2010]. By showing the system is gradient and asymptotically smooth, we establish the existence of global attractors with finite fractal dimension via a stabilizability inequality. Then we study the continuity of global attractors regarding the parameter in a residual dense set. The above results allow the damping terms with polynomial growth. Finally we discuss the exponential decay and global boundedness to the linear case of damping terms of the system. The assumption of equal-speed wave propagations is not needed for all of results obtained.

Keywords:

• Strain gradient
• porous elasticity
• nonlinear damping
• quasi-stable systems
• global attractor

2010 MSCs:

• 35B41
• 35B40
• 74H40

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions which improved 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 partially supported by the Fundamental Research Funds for the Central Universities [No. JBK2304084]. M. A. Jorge Silva has been partially supported by the CNPq, grant #309929/2022-9, and by Fundação Araucária Grant 226/2022.