General decay of the wave equation with supercritical nonlinear damping and memory-type viscoelastic effects

In this paper, we investigate a viscoelastic wave model incorporating two distinct dissipation mechanisms: a nonlinear frictional damping of power type and a general memory-type viscoelastic damping. Motivated by the recent breakthrough of Haraux and Tebou [1], who established sharp decay estimates for wave equations with supercritical nonlinear damping ($ m>\frac{2N}{N-2} $) and showed that strong solutions still decay polynomially while weak solutions exhibit logarithmic decay under mild additional assumptions, we extend this line of research to a viscoelastic setting. We first establish the global existence of solutions by means of the Faedo–Galerkin approximation method. We then investigate the asymptotic behavior of these solutions. The novelty of our analysis lies in the interaction between instantaneous nonlinear frictional damping and hereditary viscoelastic dissipation, which generates additional difficulties in both the well-posedness and stability analyses. Under broad structural conditions on the damping law and the relaxation kernel, we derive general decay results for the associated energy. In particular, we show that the long-time decay rates are governed by the interplay between the nonlinear damping exponent—including the supercritical range—and the asymptotic behavior of the memory kernel. Our results extend and unify several earlier contributions on viscoelastic systems and wave equations with nonlinear damping, providing a comprehensive framework for the analysis of coupled instantaneous and hereditary dissipation mechanisms.