The existence and stability of the solution for a viscoelastic Petrovsky equation with a nonlinear strong damping term

This paper investigates a nonlinear viscoelastic Petrovsky equation defined in a bounded domain, which incorporates a strong damping term and a delay feedback mechanism. We focus on establishing the global existence of solutions by utilizing Sobolev spaces and combining the energy method with the Faedo-Galerkin approximation technique. This approach is particularly effective in dealing with nonlinear terms and ensures the existence of weak solutions.The delay feedback term plays a crucial role in the dynamics of the system, and we impose conditions on the relative weights of the delay and non-delay terms in the feedback to guarantee the well-posedness of the problem. Specifically, the conditions ensure that the solutions depend continuously on the initial data, preventing issues such as instability or non-existence that could arise from improper choice of delay parameters.In addition to existence, we derive decay estimates for the energy of the system. These estimates are obtained using the multiplier method, which is a powerful tool in the analysis of stability for differential equations, and by exploiting the properties of convex functions. The decay estimates offer insights into how quickly the energy dissipates over time, which is a critical aspect of understanding the long-term behavior of the system.Our findings extend previous work on viscoelastic systems by incorporating delay feedback terms. These results refine the understanding of the interplay between the damping mechanisms and the delay feedback, particularly in terms of their impact on the energy stability of the system. This study contributes to a deeper understanding of the behavior of nonlinear dynamic systems, especially in scenarios where delay feedback plays a significant role in the system's stability and energy dissipation.