Dynamics of a damped quintic wave equation with time-dependent coefficients

Feng Zhou; Hongfang Li; Kaixuan Zhu; Xin Li; Feng Zhou; Hongfang Li; Kaixuan Zhu; Xin Li

doi:10.3934/math.20241202

AIMS Mathematics

2024, Volume 9, Issue 9: 24677-24698. doi: 10.3934/math.20241202

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Dynamics of a damped quintic wave equation with time-dependent coefficients

1.
College of Science, China University of Petroleum, Qingdao, Shandong 266580, China
2.
School of Mathematics and Physics Science, Hunan University of Arts and Science, Changde, Hunan 415000, China
3.
School of Science, Yanshan University, Qinhuangdao, Hebei 066004, China

Received: 11 June 2024 Revised: 30 July 2024 Accepted: 02 August 2024 Published: 22 August 2024
MSC : 37L30, 35L70, 47H20

We present a comprehensive investigation of the long-term dynamics generated by a semilinear wave equation with time-dependent coefficients and quintic nonlinearity on a bounded domain subject to Dirichlet boundary conditions. By employing rescaling techniques for time and utilizing the Strichartz estimates applicable to bounded domains, we initially study the global well-posedness of the Shatah–Struwe (S–S) solutions. Subsequently, we establish the existence of a uniform weak global attractor consisting of points on complete bounded trajectories through an approach based on evolutionary systems. Finally, we prove that this uniformly weak attractor is indeed strong by means of a backward asymptotic a priori estimate and the so-called energy method. Moreover, the smoothness of the obtained attractor is also shown with the help of a decomposition technique.
- wave equation,
- time-dependent coefficients,
- quintic nonlinearity,
- attractor,
- evolutionary systems
Citation: Feng Zhou, Hongfang Li, Kaixuan Zhu, Xin Li. Dynamics of a damped quintic wave equation with time-dependent coefficients[J]. AIMS Mathematics, 2024, 9(9): 24677-24698. doi: 10.3934/math.20241202

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Abstract

We present a comprehensive investigation of the long-term dynamics generated by a semilinear wave equation with time-dependent coefficients and quintic nonlinearity on a bounded domain subject to Dirichlet boundary conditions. By employing rescaling techniques for time and utilizing the Strichartz estimates applicable to bounded domains, we initially study the global well-posedness of the Shatah–Struwe (S–S) solutions. Subsequently, we establish the existence of a uniform weak global attractor consisting of points on complete bounded trajectories through an approach based on evolutionary systems. Finally, we prove that this uniformly weak attractor is indeed strong by means of a backward asymptotic a priori estimate and the so-called energy method. Moreover, the smoothness of the obtained attractor is also shown with the help of a decomposition technique.

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