Pullback exponential attractors in nonlocal Mindlin's strain gradient porous elasticity

ABSTRACT

In this paper, we study the long-time dynamics of a pullback exponential attractors for non-autonomous one dimensional system in nonlocal Mindlin's strain gradient porous elastic theory recently developed by Aouadi [Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin's strain gradient porous elasticity. Z Angew Math Phys. 2022;73:111]. In this theory, the second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution. Then we establish a Lipschitz stability result. We also prove the existence of pullback exponential attractors which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. Finally, we prove the upper-semicontinuity of minimal pullback attractors with respect to the perturbations parameter in natural energy space.

KEYWORDS:

• Nonlocal strain gradient porous elasticity
• non-autonomous
• pullback exponential attractors
• minimal pullback attractor
• upper semicontinuity

AMS CLASSIFICATIONS:

• 35B41
• 37B55
• 35L75
• 74K9

Acknowledgments

The author would like to thank the referees for their critical review and valuable comments that allowed to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the author.