Well-Posedness, Sharp Energy Dissipation, and Validated Numerical Dynamics for a Coupled Viscoelastic Wave System with Logarithmic Nonlinear Sources

This paper studies a strongly coupled viscoelastic wave system driven by logarithmic source terms in a bounded domain $$\Omega \subset \mathbb {R}^N$$. The model describes the interaction between memory effects, internal damping, and non-polynomial nonlinear sources, which introduces additional analytical and numerical difficulties compared with classical polynomial models. We establish well-posedness results for the system by using the Faedo–Galerkin method together with suitable a priori energy estimates adapted to logarithmic growth, which yield the local existence of weak solutions. The potential well framework is then employed in the logarithmic setting in order to obtain the global existence of solutions for sufficiently small initial data. We also investigate the long-time behavior of solutions and construct a Lyapunov functional that allows us to derive general energy decay estimates under appropriate assumptions on the viscoelastic relaxation kernels, leading to exponential or polynomial decay depending on the properties of the memory kernels. Finally, a fully discrete finite difference scheme is implemented and numerical simulations are presented to illustrate the theoretical decay properties and to provide further insight into the influence of logarithmic source terms and memory effects on the asymptotic dynamics of the system.