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Variable-coefficient viscoelastic wave equation with acoustic boundary conditions: global existence, blowup and energy decay rates

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Abstract

This paper deals with the following initial acoustic boundary value problem for a variable-coefficient wave equation with memory term

$$\begin{aligned}{} & {} u_{tt}-\Delta u-\Delta u_{t}-\Delta u_{tt}+\int _{0}^{t}b(t-s){\textrm{div}} (a_{1}(x)\nabla u(s)){\textrm{d}}s\\{} & {} \quad +\,a_{2}(x)u_{t}|u_{t}|^{q-2}=u|u|^{p-2}. \end{aligned}$$

Firstly, we get the global existence of solutions by energy estimates under some advisable assumptions on the kernel function b and variable coefficients \(a_{1}(x),\) \(a_{2}(x).\) In addition, we discuss the global nonexistence of solutions, and then derive the evaluate for the lifespan of weak solutions under some certain conditions. Finally, we establish the polynomial or exponential energy decay rates for global solutions by using perturbed energy method.

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Acknowledgements

The authors would like to express their sincere thanks to the anonymous referees for useful comments and suggestions. This work was completed with the support of the Natural Science Foundation of China (no. 11801108), Guangdong Basic and Applied Basic Research Foundation (nos. 2021A1515010314, 2023A1515030107), the Natural Science Research Project of the Educational Department of Liaoning Province (no. JDL2020027), the Science and Technology Planning Project of Guangzhou City (no. 202201010111), the Tertiary Education Scientific Research Project of Guangzhou Municipal Education Bureau (no. 202235103).

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Authors and Affiliations

  1. School of Science, Dalian Jiaotong University, Dalian, 116028, Liaoning, People’s Republic of China

    Jiali Yu

  2. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, Guangdong, People’s Republic of China

    Huafei Di

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  1. Jiali Yu
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Correspondence to Huafei Di.

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Communicated by Tom ter Elst.

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Yu, J., Di, H. Variable-coefficient viscoelastic wave equation with acoustic boundary conditions: global existence, blowup and energy decay rates. Banach J. Math. Anal. 17, 68 (2023). https://doi.org/10.1007/s43037-023-00292-z

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