Exponential stabilization of perturbed wave equations with localized internal damping: Theory and numerical validation

Kalifa Lassana Barry; Karima Laoubi; Seda İğret Araz; Salah Boulaaras; Kalifa Lassana Barry; Karima Laoubi; Seda İğret Araz; Salah Boulaaras

doi:10.3934/nhm.2026068

Networks and Heterogeneous Media

2026, Volume 21, Issue 5: 1822-1865. doi: 10.3934/nhm.2026068

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Exponential stabilization of perturbed wave equations with localized internal damping: Theory and numerical validation

1.
Department of Mathematics, M'Hamed Bougara University of Boumerdes, Boumerdes 35000, Algeria
2.
Department of Mathematics Education, Faculty of Education, Siirt University, Siirt 56100, Turkey
3.
Department of Mathematics, College of Sciences, Qassim University, Buraydah 51452, Saudi Arabia

Received: 21 January 2026 Revised: 28 July 2026 Accepted: 10 August 2026 Published: 27 August 2026

This work investigates the exponential stability of a perturbed non-linear (or linear) wave equation in a sub-critical regime with respect to both the damping and perturbation terms. Using an energy-based analysis, we first establish the decay of the total energy over a finite time interval $ T > 0 $. When the damping region satisfies the geometric control condition (GCC), and the perturbations contribute to the dissipation mechanism, we prove that the energy of the system decays exponentially by combining the multiplier method with a unique continuation principle. In addition to the theoretical analysis, we present numerical simulations for one- and two-dimensional damped wave equations with localized internal damping. Finite difference schemes are employed to approximate the dynamics and quantify the dissipation of energy, and the results confirm the theoretical predictions: Introducing a localized damping region induces strong stabilization, and the total discrete energy decays exponentially over time.
- exponential stability,
- geometric control condition,
- multiplier method,
- nonlinear equations,
- subcritical regime
Citation: Kalifa Lassana Barry, Karima Laoubi, Seda İğret Araz, Salah Boulaaras. Exponential stabilization of perturbed wave equations with localized internal damping: Theory and numerical validation[J]. Networks and Heterogeneous Media, 2026, 21(5): 1822-1865. doi: 10.3934/nhm.2026068

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Abstract

This work investigates the exponential stability of a perturbed non-linear (or linear) wave equation in a sub-critical regime with respect to both the damping and perturbation terms. Using an energy-based analysis, we first establish the decay of the total energy over a finite time interval $ T > 0 $. When the damping region satisfies the geometric control condition (GCC), and the perturbations contribute to the dissipation mechanism, we prove that the energy of the system decays exponentially by combining the multiplier method with a unique continuation principle. In addition to the theoretical analysis, we present numerical simulations for one- and two-dimensional damped wave equations with localized internal damping. Finite difference schemes are employed to approximate the dynamics and quantify the dissipation of energy, and the results confirm the theoretical predictions: Introducing a localized damping region induces strong stabilization, and the total discrete energy decays exponentially over time.

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