The coupling system of Kirchhoff and Euler-Bernoulli plates with logarithmic source terms: Strong damping versus weak damping of variable-exponent type

Adel M. Al-Mahdi; Adel M. Al-Mahdi

doi:10.3934/math.20231404

AIMS Mathematics

2023, Volume 8, Issue 11: 27439-27459. doi: 10.3934/math.20231404

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Research article

The coupling system of Kirchhoff and Euler-Bernoulli plates with logarithmic source terms: Strong damping versus weak damping of variable-exponent type

Adel M. Al-Mahdi ^{1,2
,
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1.
The Preparatory Year Program, King Fahd University of Petroleum & 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

Received: 23 July 2023 Revised: 12 September 2023 Accepted: 21 September 2023 Published: 27 September 2023
MSC : 35B37, 35L55, 74D05, 93D15, 93D20

In this paper, we study the asymptotic behavior of solutions of the dissipative coupled system where we have interactions between a Kirchhoff plate and a Euler-Bernoulli plate. We investigate the interaction between the internal strong damping acting in the Kirchhoff equation and internal weak damping of variable-exponent type acting in the Euler-Bernoulli equation. By using the potential well, the energy method (multiplier method) combined with the logarithmic Sobolev inequality, we prove the global existence and derive the stability results. We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We find explicit decay rates that depend on the weak damping of the variable-exponent type. This outcome extends earlier results in the literature.
- plate equations,
- variable-exponent,
- logarithmic nonlinearity,
- systems,
- potential well,
- multiplier method,
- stability
Citation: Adel M. Al-Mahdi. The coupling system of Kirchhoff and Euler-Bernoulli plates with logarithmic source terms: Strong damping versus weak damping of variable-exponent type[J]. AIMS Mathematics, 2023, 8(11): 27439-27459. doi: 10.3934/math.20231404

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Abstract

In this paper, we study the asymptotic behavior of solutions of the dissipative coupled system where we have interactions between a Kirchhoff plate and a Euler-Bernoulli plate. We investigate the interaction between the internal strong damping acting in the Kirchhoff equation and internal weak damping of variable-exponent type acting in the Euler-Bernoulli equation. By using the potential well, the energy method (multiplier method) combined with the logarithmic Sobolev inequality, we prove the global existence and derive the stability results. We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We find explicit decay rates that depend on the weak damping of the variable-exponent type. This outcome extends earlier results in the literature.

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