Wave equations with viscoelastic and logarithmic damping: existence, decay rates, and numerical study

We study a viscoelastic wave equation incorporating a general viscoelastic damping term together with a nonlinear logarithmic damping. The focus is on the interaction between these two damping mechanisms and its influence on the system long-time behavior. Using the Faedo–Galerkin approximation method and compactness arguments adapted to this coupled framework, we prove the global existence of strong solutions. A suitably constructed multiplier method, combined with some suitable logarithmic inequalities, yields a general decay rate that depends explicitly on the combined effects of both dampings. The results provide a rigorous theoretical foundation for wave models with mixed viscoelastic–logarithmic dissipation and reveal new aspects of their stabilization and asymptotic dynamics. Numerical experiments are conducted to illustrate and validate the theoretical findings.