THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING

doi:10.1007/s10473-024-0317-6

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (3): 1064-1095.doi: 10.1007/s10473-024-0317-6

THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING

Han Wang, Yinghui Zhang^*

School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China

收稿日期:2021-11-03 修回日期:2023-10-14 出版日期:2024-06-25 发布日期:2024-05-21

THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING

Han Wang, Yinghui Zhang^*

School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China

Received:2021-11-03 Revised:2023-10-14 Online:2024-06-25 Published:2024-05-21
Contact: *Yinghui Zhang yinghuizhang@mailbox.gxnu.edu.cn
About author:Han Wang, E-mail:wanghan85@stu.gxnu.edu.cn
Supported by:
National Nature Science Foundation of China (12271114),the Guangxi Natural Science Foundation (2023JJD110009, 2019JJG110003, 2019AC20214), the Innovation Project of Guangxi Graduate Education (JGY2023061), the Key Laboratory of Mathematical Model and Application (Guangxi Normal University) and the Education Department of Guangxi Zhuang Autonomous Region.

摘要/Abstract

摘要： We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping. The main novelty of this paper is two-fold. First, we prove the optimal decay rates of the second and third order spatial derivatives of the solution, which are the same as those of the heat equation, and in particular,are faster than ones of previous related works. Second, for well-chosen initial data, we also show that the lower optimal $L^2$ convergence rate of the $k$ ($\in [0, 3]$)-order spatial derivatives of the solution is $(1+t)^{-\frac{3+2k}{4}}$. Therefore, our decay rates are optimal in this sense. The proofs are based on the Fourier splitting method, low-frequency and high-frequency decomposition, and delicate energy estimates.

关键词: quasilinear hyperbolic equations, large time behavior, optimal decay rates

Abstract: We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping. The main novelty of this paper is two-fold. First, we prove the optimal decay rates of the second and third order spatial derivatives of the solution, which are the same as those of the heat equation, and in particular,are faster than ones of previous related works. Second, for well-chosen initial data, we also show that the lower optimal $L^2$ convergence rate of the $k$ ($\in [0, 3]$)-order spatial derivatives of the solution is $(1+t)^{-\frac{3+2k}{4}}$. Therefore, our decay rates are optimal in this sense. The proofs are based on the Fourier splitting method, low-frequency and high-frequency decomposition, and delicate energy estimates.

Key words: quasilinear hyperbolic equations, large time behavior, optimal decay rates

中图分类号:

35B40

Han Wang, Yinghui Zhang. THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING[J]. 数学物理学报(英文版), 2024, 44(3): 1064-1095.

Han Wang, Yinghui Zhang. THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING[J]. Acta mathematica scientia,Series B, 2024, 44(3): 1064-1095.

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THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING

THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING

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