Asymptotic stability for solutions of a coupled system of quasi-linear viscoelastic Kirchhoff plate equations

Zayd Hajjej; Zayd Hajjej

doi:10.3934/era.2023176

Electronic Research Archive

2023, Volume 31, Issue 6: 3471-3494. doi: 10.3934/era.2023176

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Research article

Asymptotic stability for solutions of a coupled system of quasi-linear viscoelastic Kirchhoff plate equations

Zayd Hajjej ^,

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 07 February 2023 Revised: 16 March 2023 Accepted: 23 March 2023 Published: 19 April 2023

In this manuscript, we study the asymptotic stability of solutions of two coupled quasi-linear viscoelastic Kirchhoff plate equations involving free boundary conditions, and accounting for rotational forces

$ \begin{eqnarray*} &&\vert y_t\vert^{\rho}y_{tt}-\Delta y_{tt}+\Delta^{2}y- \int_0^t h_1(t-s)\Delta^2 y(s)\;ds+f_1(y, z) = 0,\\\\ &&\vert z_t\vert^{\rho}z_{tt}-\Delta z_{tt}+\Delta^{2}z- \int_0^t h_2(t-s)\Delta^2 z(s)\;ds+f_2(y, z) = 0. \end{eqnarray*} $

The system under study in this contribution could be seen as a model for two stacked plates. This work is motivated by previous works about coupled quasi-linear wave equations or concerning single quasi-linear Kirchhoff plate. The existence of local weak solutions is established by the Faedo-Galerkin approach. By using the perturbed energy method, we prove a general decay rate of the energy for a wide class of relaxation functions.
- Kirchhoff plate equation,
- free boundary conditions,
- viscoelastic,
- Faedo-Galerkin method,
- asymptotic stability
Citation: Zayd Hajjej. Asymptotic stability for solutions of a coupled system of quasi-linear viscoelastic Kirchhoff plate equations[J]. Electronic Research Archive, 2023, 31(6): 3471-3494. doi: 10.3934/era.2023176

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Abstract

In this manuscript, we study the asymptotic stability of solutions of two coupled quasi-linear viscoelastic Kirchhoff plate equations involving free boundary conditions, and accounting for rotational forces

$ \begin{eqnarray*} &&\vert y_t\vert^{\rho}y_{tt}-\Delta y_{tt}+\Delta^{2}y- \int_0^t h_1(t-s)\Delta^2 y(s)\;ds+f_1(y, z) = 0,\\\\ &&\vert z_t\vert^{\rho}z_{tt}-\Delta z_{tt}+\Delta^{2}z- \int_0^t h_2(t-s)\Delta^2 z(s)\;ds+f_2(y, z) = 0. \end{eqnarray*} $

The system under study in this contribution could be seen as a model for two stacked plates. This work is motivated by previous works about coupled quasi-linear wave equations or concerning single quasi-linear Kirchhoff plate. The existence of local weak solutions is established by the Faedo-Galerkin approach. By using the perturbed energy method, we prove a general decay rate of the energy for a wide class of relaxation functions.

References

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