Exponential Growth of Solutions to the Wave Equations with Boundary Fractional Derivative Damping and a Boundary Source Term

In this work, we investigate the stabilization problem for a system of coupled wave equations subject to fractional boundary damping and internal source terms. We show that, under appropriate structural assumptions on the damping and the nonlinear sources, the system may lose stability, and its energy can grow exponentially over time. The analysis is based on harmonic techniques involving the Fourier transform, together with sharp functional estimates such as the Hardy–Littlewood–Sobolev inequality, which enable us to precisely quantify the competition between the dissipative fractional boundary effects and the internal nonlinear sources. These tools allow us to rigorously establish instability criteria that lead to exponential energy growth, thereby highlighting scenarios in which the fractional damping fails to stabilize the coupled system.