Global existence and polynomial stability for a nonlinear clamped plate equation with logarithmic velocity damping

Iqra Kanwal; Jianghao Hao; Muhammad Fahim Aslam; Mohamed Balegh; Zayd Hajjej; Iqra Kanwal; Jianghao Hao; Muhammad Fahim Aslam; Mohamed Balegh; Zayd Hajjej

doi:10.3934/math.20261081

AIMS Mathematics

2026, Volume 11, Issue 8: 26943-26985. doi: 10.3934/math.20261081

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Global existence and polynomial stability for a nonlinear clamped plate equation with logarithmic velocity damping

1.
School of Mathematics and Statistics, Shanxi University, Taiyuan, China
2.
Department of Mathematics, University of Kotli Azad Jammu and Kashmir(UOKAJK), Kotli, Pakistan
3.
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia
4.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 05 July 2026 Revised: 03 August 2026 Accepted: 14 August 2026 Published: 27 August 2026
MSC : 74K20, 49K40, 93D20, 65M50

This paper investigates a nonlinear clamped plate equation with logarithmic velocity damping and a polynomial source. The model combines the biharmonic operator, a nonstandard logarithmic damping mechanism and an energy-producing source term. Since the logarithmic damping is weak near the origin and the source may destabilize the system, the analysis requires a combination of monotonicity, potential well, and multiplier techniques. We first establish the local well-posedness of weak solutions by the Faedo–Galerkin method and compactness arguments. We then prove global existence for the initial data in the stable set. Finally, using logarithmic dissipation estimates and a Haraux-type integral inequality, we derive a polynomial decay estimate for the energy. To demonstrate the theoretical decay behavior and the impact of the damping and source factors on the plate's energy, numerical simulations are also provided.
- clamped plate equation,
- logarithmic velocity damping,
- polynomial source,
- well-posedness,
- global existence,
- potential well,
- polynomial stability,
- numerical simulation,
- energy decay
Citation: Iqra Kanwal, Jianghao Hao, Muhammad Fahim Aslam, Mohamed Balegh, Zayd Hajjej. Global existence and polynomial stability for a nonlinear clamped plate equation with logarithmic velocity damping[J]. AIMS Mathematics, 2026, 11(8): 26943-26985. doi: 10.3934/math.20261081

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Abstract

This paper investigates a nonlinear clamped plate equation with logarithmic velocity damping and a polynomial source. The model combines the biharmonic operator, a nonstandard logarithmic damping mechanism and an energy-producing source term. Since the logarithmic damping is weak near the origin and the source may destabilize the system, the analysis requires a combination of monotonicity, potential well, and multiplier techniques. We first establish the local well-posedness of weak solutions by the Faedo–Galerkin method and compactness arguments. We then prove global existence for the initial data in the stable set. Finally, using logarithmic dissipation estimates and a Haraux-type integral inequality, we derive a polynomial decay estimate for the energy. To demonstrate the theoretical decay behavior and the impact of the damping and source factors on the plate's energy, numerical simulations are also provided.

References

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