| Abstract: |
| This paper investigates a wave equation in the presence of nonlinear logarithmic damping. We present, for the first time, a comprehensive and rigorous proof of the global existence of weak solutions for this class of problems. The analysis is carried out using the Faedo-Galerkin approximation method. This result establishes a robust foundational framework for future studies involving logarithmic dampings. In addition, we prove that the energy of the system decays at a polynomial rate as time progresses, by employing a suitable multiplier method adapted to the structure of the logarithmic dissipation. Our findings contribute to the deeper understanding of stabilization mechanisms in nonlinear wave models and offer new insights into the role of logarithmic damping in the long-time behavior of solutions. |
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