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PreprintCC BY 4.0arXiv

The Global Existence and Energy Decay of a Partially Logarithmically Damped Wave System

  • Salim A. Messaoudi
  • Suleyman Ulusoy
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Abstract

We investigate the global existence and energy decay properties of a weakly coupled evolution system consisting of two wave equations in a bounded domain. The system exhibits a unique structural asymmetry: a nonlinear logarithmic damping term acts exclusively as a friction-like feedback mechanism on the first wave equation, while the second equation remains completely undamped and relies solely on linear cross-coupling to transfer and dissipate its energy. We establish two primary objectives. First, by utilizing the Faedo-Galerkin approximation scheme in conjunction with the structural monotonicity properties of the non-Lipschitz logarithmic nonlinearity, we prove a rigorous global existence result for strong solutions. Second, we provide a rigorous mathematical proof demonstrating the stability and asymptotic energy decay of the solutions under suitable constraints on the cross-coupling parameter. These results provide new insights into the stabilization mechanisms of coupled hyperbolic systems under degenerate nonlinear damping.

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Cite

BibTeXRIS
BibTeX
@misc{messaoudi2026global,
  title = {{The Global Existence and Energy Decay of a Partially Logarithmically Damped Wave System}},
  author = {Salim A. Messaoudi and Suleyman Ulusoy},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05268},
  eprint = {2610.05268},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05268},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Global Existence and Energy Decay of a Partially Logarithmically Damped Wave System
AU  - Salim A. Messaoudi
AU  - Suleyman Ulusoy
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05268
UR  - https://arxiv.org/abs/2610.05268
AB  - We investigate the global existence and energy decay properties of a weakly coupled evolution system consisting of two wave equations in a bounded domain. The system exhibits a unique structural asymmetry: a nonlinear logarithmic damping term acts exclusively as a friction-like feedback mechanism on the first wave equation, while the second equation remains completely undamped and relies solely on linear cross-coupling to transfer and dissipate its energy. We establish two primary objectives. First, by utilizing the Faedo-Galerkin approximation scheme in conjunction with the structural monotonicity properties of the non-Lipschitz logarithmic nonlinearity, we prove a rigorous global existence result for strong solutions. Second, we provide a rigorous mathematical proof demonstrating the stability and asymptotic energy decay of the solutions under suitable constraints on the cross-coupling parameter. These results provide new insights into the stabilization mechanisms of coupled hyperbolic systems under degenerate nonlinear damping.
ER  -