Long Time Behavior of Shear beam Mode Subject to a Delay

Abstract

In this work, we investigate a one-dimensional system known as the Shear Beam Model without rotary inertia, where the transverse displacement equation incorporates an internal delay. Under appropriate assumptions on the delay weight, we establish the global wellposedness of the system using the classical Faedo–Galerkin method combined with three a priori estimates. Furthermore, we analyze the asymptotic behavior of the solutions through an energy-based approach, proving the exponential decay of the system's energy under suitable stability conditions. This study contributes to a deeper theoretical understanding of delayed dynamic systems and lays the groundwork for future developments in more complex or nonlinear models.