On a nonlinear system of plate equations with variable exponent nonlinearity and logarithmic source terms: Existence and stability results

Adel M. Al-Mahdi; Mohammad M. Al-Gharabli; Nasser-Eddine Tatar; Adel M. Al-Mahdi; Mohammad M. Al-Gharabli; Nasser-Eddine Tatar

doi:10.3934/math.20231018

AIMS Mathematics

2023, Volume 8, Issue 9: 19971-19992. doi: 10.3934/math.20231018

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Research article

On a nonlinear system of plate equations with variable exponent nonlinearity and logarithmic source terms: Existence and stability results

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
3.
Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4.
IRC for Intelligent Manufacturing and Robotics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received: 25 March 2023 Revised: 13 May 2023 Accepted: 21 May 2023 Published: 15 June 2023
MSC : 35B37, 35L55, 74D05, 93D15, 93D20

In this paper, we consider a coupling non-linear system of two plate equations with logarithmic source terms. First, we study the local existence of solutions of the system using the Faedo-Galerkin method and Banach fixed point theorem. Second, we prove the global existence of solutions of the system by using the potential wells. Finally, using the multiplier method, we establish an exponential decay result for the energy of solutions of the system. Some conditions on the variable exponents that appear in the coupling functions and the involved constants that appear in the source terms are determined to ensure the existence and stability of solutions of the system. A series of lemmas and theorems have been proved and used to overcome the difficulties caused by the variable exponent and the logarithmic nonlinearities. Our result generalizes some earlier related results in the literature from the case of only constant exponent of the nonlinear internal forcing terms to the case of variable exponent and logarithmic source terms, which is more useful from the physical point of view and needed in several applications.
- plate equations,
- variable-exponent,
- logarithmic nonlinearity,
- logarithmic Sobolev inequality,
- Galerkin method,
- Banach fixed point theorem,
- multiplier method,
- exponential decay
Citation: Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Nasser-Eddine Tatar. On a nonlinear system of plate equations with variable exponent nonlinearity and logarithmic source terms: Existence and stability results[J]. AIMS Mathematics, 2023, 8(9): 19971-19992. doi: 10.3934/math.20231018

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Abstract

In this paper, we consider a coupling non-linear system of two plate equations with logarithmic source terms. First, we study the local existence of solutions of the system using the Faedo-Galerkin method and Banach fixed point theorem. Second, we prove the global existence of solutions of the system by using the potential wells. Finally, using the multiplier method, we establish an exponential decay result for the energy of solutions of the system. Some conditions on the variable exponents that appear in the coupling functions and the involved constants that appear in the source terms are determined to ensure the existence and stability of solutions of the system. A series of lemmas and theorems have been proved and used to overcome the difficulties caused by the variable exponent and the logarithmic nonlinearities. Our result generalizes some earlier related results in the literature from the case of only constant exponent of the nonlinear internal forcing terms to the case of variable exponent and logarithmic source terms, which is more useful from the physical point of view and needed in several applications.

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