Stability analysis of a coupled system with viscoelastic and supercritical nonlinear dampings

In this paper, we study a viscoelastic coupled system that incorporates two distinct mechanisms of energy dissipation: a nonlinear frictional damping of power type and a memory-type viscoelastic damping. Our investigation is inspired by the recent breakthrough of Haraux and Tebou (Energy decay estimates for the wave equation with supercritical nonlinear damping. Math Anal Appl. 2025;550:129622. doi: 10.1016/j.jmaa.2025.129622), who derived sharp decay estimates for wave equations endowed with supercritical nonlinear damping (m>2NN−2), showing that strong solutions decay at a polynomial rate while weak solutions exhibit logarithmic decay under mild additional conditions. Extending this framework, we examine a viscoelastic system with memory effects, where the interaction between the instantaneous nonlinear damping and the hereditary viscoelastic dissipation introduces new analytical challenges concerning stability and long-time behavior. Under general structural hypotheses on the damping function and the relaxation kernel, we establish general decay results that capture the influence of both mechanisms. In particular, we show that the asymptotic decay rate of the system energy is governed by the interplay between the nonlinear damping exponent, covering the supercritical regime, and the asymptotic profile of the memory kernel.