Asymptotic Behaviour of Nonlinear Viscoelastic Wave Equations with Boundary Feedback

Abstract

This study investigates a nonlinear viscoelastic wave equation subject to acoustic boundary conditions and a nonlinear distributed delay feedback acting on the boundary. The analysis of the asymptotic behavior of such systems is of paramount importance for both theoretical advancements in the field of partial differential equations and for practical applications in science and engineering. Viscoelastic materials, which exhibit both elastic and viscous properties, are modeled by equations that incorporate memory effects, often represented by integral terms. The inclusion of nonlinear distributed delay in the boundary feedback introduces additional complexity, reflecting more realistic physical scenarios where system responses are not instantaneous but occur over a range of times. Furthermore, the consideration of acoustic boundary conditions enhances the model's applicability to problems involving wave interactions at material interfaces. In this work, we establish a framework for analyzing the long-term behavior of solutions to this complex system. By employing the multiplier method and constructing a suitable Lyapunov functional, we derive general decay results for the energy of the system. The analysis is carried out under a general set of assumptions on the memory kernel and the nonlinear functions that characterize the boundary feedback and delay. We demonstrate that the energy of the system decays to zero as time tends to infinity, and we provide explicit decay rates that depend on the properties of the memory kernel and the nonlinearities in the system. Our findings contribute to the fundamental understanding of energy dissipation and stability in viscoelastic systems with time-delayed boundary controls. This research not only advances the mathematical theory but also provides valuable insights for the design and analysis of materials and structures where viscoelasticity, acoustic effects, and delayed feedback are significant factors.