Robust exponential attractors for a class of non-autonomous semi-linear second-order evolution equation with memory and critical nonlinearity

ABSTRACT

In this article, we investigate a class of non-autonomous semi-linear second-order evolution with memory terms, expressed by the convolution integrals, which account for the past history of one or more variables. First, the asymptotic regularity of solutions is proved, while the nonlinearity is critical and the time-dependent external forcing term is assumed to be only translation-bounded (instead of translation-compact), and then the existence of compact uniform attractors together with its structure and regularity is established. Finally, the existence of robust family of exponential attractors is constructed.

KEYWORDS:

• Nonautonomous semi-linear second-order evolution
• asymptotic regularity
• uniform attractors
• robust exponential attractors
• critical exponent

AMS SUBJECT CLASSIFICATIONS:

• 35Q35
• 35A01

Acknowledgements

The authors want to express their sincere gratitude to the anonymous reviewers for their careful reading of the paper, giving us valuable comments and suggestions. They also thank the editors for their kind help.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

F. Zhang was supported by the 2017 research funding of higher education of Gansu province project [2017A-185] and the 2018 research funding of higher education of Gansu province project [2018B-075].