LASSOUED, Hamza2025-10-192025-10-192025-06-03http//localhost:8080/jspui/handle/123456789/13353This thesis investigates the dynamics of a coupled nonlinear wave system with internal fractional damping, a model relevant to viscoelastic materials and engineering ap- plications. The study focuses on two key aspects: establishing the local existence and uniqueness of solutions and analyzing conditions leading to finite-time blow-up. Employing functional analysis and C0-semigroup theory, we reformulate the system as an abstract evo- lution equation in a Hilbert space, proving local well-posedness under specific conditions on the nonlinear exponents. A carefully designed Lyapunov functional is used to derive criteria for solution blow-up, highlighting the interplay between nonlinearity and fractional damp- ing. The results contribute to the theoretical understanding of fractional partial differential equations and offer insights for practical applications, such as vibration control and material stability. This work advances the mathematical framework for nonlocal systems and lays the groundwork for future research into global existence and numerical simulations.enNonlinear wave equation, Fractional damping, Local existence, Blow-up in finite time, C0-semigroup, Lyapunov functional, Viscoelastic systems.Thesis