Wave equations with nonlinear damping arise naturally in the mathematical modeling of vibrating structures, wave propagation in heterogeneous media, and structural vibration control. Among the recently proposed damping mechanisms, logarithmic damping provides a realistic description of weak dissipation near equilibrium together with enhanced energy absorption at larger velocities. Motivated by these applications, this paper investigated a wave equation with nonlinear logarithmic damping and established a rigorous proof of the global existence and uniqueness of strong solutions for this class of problems. The proof was based on the Faedo–Galerkin approximation method together with suitable a priori estimates and monotonicity arguments adapted to the logarithmic nonlinearity. This result filled an important gap in the existing literature by providing the analytical foundation required for the study of logarithmically damped wave equations. Furthermore, we established a polynomial decay rate for the associated energy by developing a multiplier approach specifically tailored to the logarithmic damping term. Unlike the techniques commonly used for classical linear or polynomial damping, our analysis overcame the nonhomogeneous nature of the logarithmic dissipation through refined estimates that captured its distinct growth behavior. These results complemented and extended the existing literature on logarithmic damping by providing both the well-posedness theory and the long-time stability analysis within a unified framework.
Citation: Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Bashayer Aldossary. Global existence and energy decay for a logarithmically damped wave equation arising in structural vibration models[J]. AIMS Mathematics, 2026, 11(8): 24137-24151. doi: 10.3934/math.2026974
Wave equations with nonlinear damping arise naturally in the mathematical modeling of vibrating structures, wave propagation in heterogeneous media, and structural vibration control. Among the recently proposed damping mechanisms, logarithmic damping provides a realistic description of weak dissipation near equilibrium together with enhanced energy absorption at larger velocities. Motivated by these applications, this paper investigated a wave equation with nonlinear logarithmic damping and established a rigorous proof of the global existence and uniqueness of strong solutions for this class of problems. The proof was based on the Faedo–Galerkin approximation method together with suitable a priori estimates and monotonicity arguments adapted to the logarithmic nonlinearity. This result filled an important gap in the existing literature by providing the analytical foundation required for the study of logarithmically damped wave equations. Furthermore, we established a polynomial decay rate for the associated energy by developing a multiplier approach specifically tailored to the logarithmic damping term. Unlike the techniques commonly used for classical linear or polynomial damping, our analysis overcame the nonhomogeneous nature of the logarithmic dissipation through refined estimates that captured its distinct growth behavior. These results complemented and extended the existing literature on logarithmic damping by providing both the well-posedness theory and the long-time stability analysis within a unified framework.
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Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Bashayer Aldossary. Global existence and energy decay for a logarithmically damped wave equation arising in structural vibration models[J]. AIMS Mathematics, 2026, 11(8): 24137-24151. doi: 10.3934/math.2026974
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