Existence and stability results of a plate equation with nonlinear damping and source term

Mohammad M. Al-Gharabli; Adel M. Al-Mahdi; Mohammad M. Al-Gharabli; Adel M. Al-Mahdi

doi:10.3934/era.2022205

Electronic Research Archive

2022, Volume 30, Issue 11: 4038-4065. doi: 10.3934/era.2022205

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Existence and stability results of a plate equation with nonlinear damping and source term

Mohammad M. Al-Gharabli ^1,2,
Adel M. Al-Mahdi ^{1,2
,
,}

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

Academic Editor: Baowei Feng

Received: 22 June 2022 Revised: 29 August 2022 Accepted: 31 August 2022 Published: 13 September 2022

The main goal of this work is to investigate the following nonlinear plate equation

$ u_{tt}+\Delta ^2 u +\alpha(t) g(u_t) = u \vert u\vert ^{\beta}, $

which models suspension bridges. Firstly, we prove the local existence using Faedo-Galerkin method and Banach fixed point theorem. Secondly, we prove the global existence by using the well-depth method. Finally, we establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions $ \alpha $ and $ g $ and obtained without any restriction growth assumption on $ g $ at the origin. The multiplier method, properties of the convex functions, Jensen's inequality and the generalized Young inequality are used to establish the stability results.
- plate equation,
- Galerkin method,
- Banach fixed point theorem,
- general decay,
- nonlinear frictional damping
Citation: Mohammad M. Al-Gharabli, Adel M. Al-Mahdi. Existence and stability results of a plate equation with nonlinear damping and source term[J]. Electronic Research Archive, 2022, 30(11): 4038-4065. doi: 10.3934/era.2022205

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Abstract

The main goal of this work is to investigate the following nonlinear plate equation

$ u_{tt}+\Delta ^2 u +\alpha(t) g(u_t) = u \vert u\vert ^{\beta}, $

which models suspension bridges. Firstly, we prove the local existence using Faedo-Galerkin method and Banach fixed point theorem. Secondly, we prove the global existence by using the well-depth method. Finally, we establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions $ \alpha $ and $ g $ and obtained without any restriction growth assumption on $ g $ at the origin. The multiplier method, properties of the convex functions, Jensen's inequality and the generalized Young inequality are used to establish the stability results.

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