Pullback attractors for a weakly damped wave equation with delays and sup-cubic nonlinearity

Kaixuan Zhu; Yongqin Xie; Xinyu Mei

doi:10.3934/dcdsb.2020294

Article Contents

2021, Volume 26, Issue 8: 4433-4458. Doi: 10.3934/dcdsb.2020294

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Pullback attractors for a weakly damped wave equation with delays and sup-cubic nonlinearity

1.
Hunan Province Cooperative Innovation Center for the Construction and, Development of Dongting Lake Ecological Economic Zone & , College of Mathematics and Physics Science, Hunan University of Arts and Science, Changde 415000, China
2.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China
3.
School of Mathematical and Statistics, Shenzhen University, Shenzhen 518060, China

^* Corresponding author: Kaixuan Zhu
^* Corresponding author: Kaixuan Zhu

Received: May 2020

Revised: August 2020

Early access: October 2020

Published: August 2021

This work is supported by the Natural Science Foundation of Hunan Province (Grants No. 2018JJ2416)

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Abstract

In this paper, we consider the weakly damped wave equations with hereditary effects and the nonlinearity $ f $ satisfying sup-cubic growth. Based on the recent extension of the Strichartz estimates to the case of bounded domains, we establish the global well-posedness of the Shatah-Struwe solutions for the non-autonomous case. Then, we prove the existence of the pullback $ \mathcal{D} $-attractors in $ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $ for the solutions process $ \{U(t,\tau)\}_{t\geq\tau} $ by applying the idea of contractive functions.

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Mathematics Subject Classification: 35L05, 35B40, 35B41.

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