Nonlocal damping and nonlinear source terms in a system of wave equations: existence and stability analysis

In this paper, we investigate a strongly coupled system of semilinear wave equations characterized by nonlocal damping terms and highly nonlinear source functions. The model captures complex interactions between damping terms depending on space integrals of the velocity fields and nonlinearities involving mixed terms of the unknown components. We first establish the local existence and uniqueness of a weak solution using the Faedo–Galerkin method combined with the Banach Fixed-Point Theorem. Then, under appropriate conditions on the initial data and nonlinear terms, we prove the global existence of solutions by employing the potential well theory. Additionally, we derive an energy decay estimate through the use of Nakao’s inequality, which allows us to characterize the long-time behavior of the solutions. This study extends and generalizes existing results by addressing the analytical difficulties arising from the presence of both nonlocal damping and coupled nonlinear source terms, thereby contributing to the broader understanding of wave systems with nonlocal dissipation.