A coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities: Existence, uniqueness, blow-up and a large-time asymptotic behavior

Salim A. Messaoudi; Mohammad M. Al-Gharabli; Adel M. Al-Mahdi; Mohammed A. Al-Osta; Salim A. Messaoudi; Mohammad M. Al-Gharabli; Adel M. Al-Mahdi; Mohammed A. Al-Osta

doi:10.3934/math.2023400

AIMS Mathematics

2023, Volume 8, Issue 4: 7933-7966. doi: 10.3934/math.2023400

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A coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities: Existence, uniqueness, blow-up and a large-time asymptotic behavior

1.
Department of Mathematics and Statistics, University of Sharjah, United Arab Emirates
2.
The Preparatory Year Program, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
3.
The Interdisciplinary Research Center for Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4.
Department of Civil Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received: 13 September 2022 Revised: 18 December 2022 Accepted: 05 January 2023 Published: 31 January 2023
MSC : 35B37, 35L55, 74D05, 93D15, 93D20

In this paper, we consider a coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities. This system is supplemented with initial and mixed boundary conditions. First, we establish the existence and uniqueness results of a weak solution, under suitable assumptions on the variable exponents. Second, we show that the solutions with positive-initial energy blow-up in a finite time. Finally, we establish the global existence as well as the energy decay results of the solutions, using the stable-set and the multiplier methods, under appropriate conditions on the variable exponents and the initial data.
- biharmonic equations,
- blow-up,
- coupled system,
- global existence,
- variable exponent,
- stability
Citation: Salim A. Messaoudi, Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Mohammed A. Al-Osta. A coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities: Existence, uniqueness, blow-up and a large-time asymptotic behavior[J]. AIMS Mathematics, 2023, 8(4): 7933-7966. doi: 10.3934/math.2023400

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Abstract

In this paper, we consider a coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities. This system is supplemented with initial and mixed boundary conditions. First, we establish the existence and uniqueness results of a weak solution, under suitable assumptions on the variable exponents. Second, we show that the solutions with positive-initial energy blow-up in a finite time. Finally, we establish the global existence as well as the energy decay results of the solutions, using the stable-set and the multiplier methods, under appropriate conditions on the variable exponents and the initial data.

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