Attractors and their properties for a class of Kirchhoff models with integro-differential damping

ABSTRACT

In this paper, we investigate a class of Kirchhoff models with integro-differential damping given by a possibly vanishing memory term in a past history framework and a nonlinear nonlocal strong dissipation $\begin{array}{rl}& u_{tt}+{\alpha }_{\mu }{\mathrm{△}}^{2}u-{\mathrm{△}}_{p}u-{\int }_{-\mathrm{\infty }}^t\mu (t-s){\mathrm{△}}^{2}u\left(s\right)\mathrm{d}s\\ & -N\left({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u\left(t\right){|}^2\mathrm{d}x\right)\mathrm{△}{u}_t+f\left(u\right)=h,\end{array}$defined in a bounded Ω of ${\mathbb{R}}^N$. Our main goal is to show the well-posedness and the long-time behavior through the corresponding autonomous dynamical system by regarding the relative past history. More precisely, under the assumptions that the exponent p and the growth of $f\left(u\right)$ are up to the critical range, the well-posedness and the existence of a global attractor with its geometrical structure are established. Furthermore, in the subcritical case, such a global attractor has finite fractal dimensions as well as regularity of trajectories. A result on generalized fractal exponential attractor is also proved. These results are presented for a wide class of nonlocal damping coefficient $N(\cdot )$ and possibly degenerate memory term $(\mu \equiv 0)$, which deepen and extend earlier results on the subject.

Keywords:

• p-Laplacian
• nonlocal damping
• memory
• long-time dynamics
• global attractor

2010 Mathematics Subject Classifications:

• 35B40
• 35B41
• 37L30
• 74H40

Acknowledgments

The authors would like to thank the referees for the careful reading of this paper and for the valuable suggestions to improve the presentation and the style of the paper. This project is supported by NSFC (No.11801145), Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China (Grant No.19A110004), the Fund of Young Backbone Teacher in Henan Province (2018GGJS068) and CNPq, Grant 301116/2019-9.

Availability of data and materials

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Authors contributions

All authors contributed equally to this work.

Disclosure statement

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

Additional information

Funding

This project is supported by the National Natural Science Foundation of China (NSFC) [grant number 11801145]; the Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China [grant number 19A110004]; and the Fund of Young Backbone Teacher in Henan Province [2018GGJS068]. Supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) [grant number 301116/2019-9].