Stabilization of a viscoelastic wave equation with boundary damping and variable exponents: Theoretical and numerical study

Adel M. Al-Mahdi; Mohammad M. Al-Gharabli; Maher Nour; Mostafa Zahri; Adel M. Al-Mahdi; Mohammad M. Al-Gharabli; Maher Nour; Mostafa Zahri

doi:10.3934/math.2022842

AIMS Mathematics

2022, Volume 7, Issue 8: 15370-15401. doi: 10.3934/math.2022842

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Stabilization of a viscoelastic wave equation with boundary damping and variable exponents: Theoretical and numerical study

1.
The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2.
The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3.
DCC-Math, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4.
Department of Mathematics, College of Sciences, Research Groups MASEP and BioInformatics FG, University of Sharjah, Sharjah, United Arab Emirates

Received: 31 March 2022 Revised: 31 May 2022 Accepted: 06 June 2022 Published: 20 June 2022
MSC : 35B40, 35L70, 35B35, 93D20

In this work, we consider a viscoelastic wave equation with boundary damping and variable exponents source term. The damping terms and variable exponents are localized on a portion of the boundary. We first, prove the existence of global solutions and then we establish optimal and general decay estimates depending on the relaxation function and the nature of the variable exponent nonlinearity. Finally, we run two numerical tests to demonstrate our theoretical decay results. This study generalizes and enhances existing literature results, and the acquired results are thus of significant importance when compared to previous literature results with constant or variable exponents in the domain.
- variable exponent,
- Lebesgue and Sobolev spaces,
- boundary feedback,
- viscoelasticity,
- relaxation functions,
- general decay,
- finite difference method
Citation: Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Maher Nour, Mostafa Zahri. Stabilization of a viscoelastic wave equation with boundary damping and variable exponents: Theoretical and numerical study[J]. AIMS Mathematics, 2022, 7(8): 15370-15401. doi: 10.3934/math.2022842

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Abstract

In this work, we consider a viscoelastic wave equation with boundary damping and variable exponents source term. The damping terms and variable exponents are localized on a portion of the boundary. We first, prove the existence of global solutions and then we establish optimal and general decay estimates depending on the relaxation function and the nature of the variable exponent nonlinearity. Finally, we run two numerical tests to demonstrate our theoretical decay results. This study generalizes and enhances existing literature results, and the acquired results are thus of significant importance when compared to previous literature results with constant or variable exponents in the domain.

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