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Study of a Viscoelastic Wave Equation with a Strong Damping and Variable Exponents

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Abstract

The goal of the present paper is to study the viscoelastic wave equation with variable exponents

$$\begin{aligned} u_{tt}-\Delta _{p(x)}u-\Delta u+\int _0^tg(t-s)\Delta u(s)\mathrm{{d}}s-\Delta u_t=|u|^{q(x)-2}u \end{aligned}$$

under initial-boundary value conditions, where the exponents of nonlinearity p(x) and q(x) are given functions. To be more precise, blow-up in finite time is proved, upper and lower bounds of the blow-up time are obtained as well. The global existence of weak solutions is presented, moreover, a general stability of solutions is obtained. This work generalizes and improves earlier results in the literature.

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Acknowledgements

The author would like to express her sincere gratitude to Professor Wenjie Gao and Professor Bin Guo for their support and constant encouragement.

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Menglan Liao

  2. School of Mathematics, Jilin University, Changchun, 130012, Jilin, People’s Republic of China

    Menglan Liao

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Liao, M. Study of a Viscoelastic Wave Equation with a Strong Damping and Variable Exponents. Mediterr. J. Math. 18, 186 (2021). https://doi.org/10.1007/s00009-021-01826-1

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  • DOI: https://doi.org/10.1007/s00009-021-01826-1

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