Strong attractors for the strongly damped wave equations on time-dependent spaces

Kaixuan Zhu; Tao Sun; Yongqin Xie

doi:10.3934/dcdsb.2022207

Article Contents

2023, Volume 28, Issue 5: 3160-3171. Doi: 10.3934/dcdsb.2022207

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Strong attractors for the strongly damped wave equations on time-dependent spaces

1.
College of Mathematics and Physics Science, Hunan University of Arts and Science, Changde, 415000, China
2.
Department of Mathematics, Wenzhou University, Wenzhou, 325035, China
3.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410114, China

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

Received: July 2022

Revised: September 2022

Early access: October 28, 2022

Published online: October 28, 2022

This work is partially supported by the NSFC (Grants No. 11901194), the Hunan Province Natural Science Foundation of China (Grants No. 2022JJ30417) and the Scientific Research Fund of Hunan Provincial Education Department (Grants Nos. 21B0617, 20C1263).

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Abstract

We examine the long-term behavior of strong solutions for the wave equation $ \varepsilon(t)u_{tt}-\Delta u_{t}-\Delta u+f(u) = g(x) $ and obtain the time-dependent attractor in $ \mathcal{H}_{t}^{1} = [H^{2}(\Omega)\cap H_{0}^{1}(\Omega)]\times H_{0}^{1}(\Omega) $. Moreover, the attractor is bounded in $ [H^{2}(\Omega)\cap H_{0}^{1}(\Omega)]\times[H^{2}(\Omega)\cap H_{0}^{1}(\Omega)] $.

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

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