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Mathematical Methods in Engineering pp 235–256Cite as

Energy Decay in a Quasilinear System with Finite and Infinite Memories

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Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 23))

Abstract

In this paper, we consider the following quasilinear system of two coupled nonlinear equations with both finite and infinite memories

$$\displaystyle \left \{ \begin {array}{l} \left \vert u_{t}\right \vert ^{\rho }u_{tt}-\Delta u-\Delta u_{tt}+\int _{0}^{t}g_{1}(s)\Delta u(t-s)ds+f_{1}(u,v)=0 \\ \left \vert v_{t}\right \vert ^{\rho }v_{tt}-\Delta v-\Delta v_{tt}+\int _{0}^{\infty }g_{2}(s)\Delta v(t-s)ds+f_{2}(u,v)=0 \end {array} \right . $$

and investigate the asymptotic behavior of this system. We use the multiplier method to establish an explicit energy decay formula. Our result allows a wider class of relaxation functions and provides more general decay rates for which the usual exponential and polynomial rates are only special cases. AMS Classification35B40, 74D99, 93D15, 93D20

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Acknowledgements

The author thanks University of Sharjah for its continuous support.

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Authors and Affiliations

  1. Department of Mathematics, University of Sharjah, Sharjah, United Arab Emirates

    Muhammad I. Mustafa

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  1. Muhammad I. Mustafa
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Correspondence to Muhammad I. Mustafa .

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Editors and Affiliations

  1. Department of Mathematics, Çankaya University, Ankara, Turkey

    Kenan Taş

  2. Department of Mathematics, Çankaya University, Ankara, Turkey

    Dumitru Baleanu

  3. Instituto Superior de Engenharia do Porto, Porto, Portugal

    J. A. Tenreiro Machado

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Mustafa, M.I. (2019). Energy Decay in a Quasilinear System with Finite and Infinite Memories. In: Taş, K., Baleanu, D., Machado, J. (eds) Mathematical Methods in Engineering. Nonlinear Systems and Complexity, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-91065-9_12

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  • DOI: https://doi.org/10.1007/978-3-319-91065-9_12

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  • Online ISBN: 978-3-319-91065-9

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