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Exponential Stabilization of the Wave Equation on Hyperbolic Spaces with Nonlinear Locally Distributed Damping

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Abstract

In this article, we consider the wave equation on hyperbolic spaces \(\mathbb {H}^n(n\ge 2)\) with nonlinear locally distributed damping as follow:

$$\begin{aligned} {\left\{ \begin{array}{ll}u_{tt}-\Delta _g u+a(x)g(u_t)=0\qquad (x,t)\in \mathbb {H}^n\times (0,+\infty ), \\ u(x,0)=u_0(x),\quad u_0(x,0)=u_1(x)\qquad x\in \mathbb {H}^n. \end{array}\right. } \end{aligned}$$
(1)

It is well-known that the energy of the system (1) is of polynomial decay in the Euclidean space. However, on hyperbolic spaces, owing to the following inequality

$$\begin{aligned} \int _{\mathbb {H}^n} u^2 dx_g \le C \int _{\mathbb {H}^n} |\nabla _g u|_g^2dx_g , \quad for \quad u\in H^1(\mathbb {H}^n), \end{aligned}$$
(2)

we prove the exponential stabilization of the wave equation by multiplier methods and compactness-uniqueness arguments.

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Acknowledgements

The authors would like to thank the referee and editor for their very helpful comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Jiacheng Wang

  2. Key Laboratory of Systems and Control,Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Jiacheng Wang

  3. Faculty of Information Technology, Beijing University of Technology, Beijing, 100124, China

    Zhen-Hu Ning

  4. School of Sciences, Beijing Forestry University, Beijing, 100083, China

    Fengyan Yang

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  1. Jiacheng Wang
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  2. Zhen-Hu Ning
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  3. Fengyan Yang
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Correspondence to Fengyan Yang.

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This work is supported by the National Natural Science Foundation of China under Grant No. 61573342 the Key Research Program of Frontier Sciences, Chinese Academy of Sciences, No. QYZDJ-SSW-SYS011 and the Fundamental Research Funds for the Central Universities (NO.BLX201924).

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Wang, J., Ning, ZH. & Yang, F. Exponential Stabilization of the Wave Equation on Hyperbolic Spaces with Nonlinear Locally Distributed Damping. Appl Math Optim 84, 3437–3449 (2021). https://doi.org/10.1007/s00245-021-09751-1

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