Stability for a wave equation with fractional structural damping subjected to an internal distributed delay

Source abstract

Abstract In this paper, we investigate the exponential stability of a wave equation in a bounded domain with fractional structural damping and an internal distributed time delay. The damping mechanism is modeled by a spectral fractional Laplacian ( - Δ ) α {(-\Delta)^{\alpha}} with α ∈ ( 0 , 1 ) {\alpha\in(0,1)} , while the delay is described by a convolution term acting on the velocity. By extending a delay-absorption technique to the genuinely fractional regime, we reformulate the system and construct an appropriate Lyapunov functional. Under explicit conditions linking the damping parameter and the size of the delay kernel, we establish exponential decay of the energy. The result quantifies the dominance relationship between fractional dissipation and memory effects, thereby filling the gap between the classical damping case ( α = 0 ) {(\alpha=0)} and the strongly damped case ( α = 1 ) {(\alpha=1)} .