Global Attractors For Wave Equations With Nonlinear Damping And State-dependent Delay

The wave equation with nonlinear damping has an important physical background,and there are still many open problems in the study of its long-time behavior.Compared to fixed time-dependent delays,the solutions of state-dependent delay differential equations have less smoothness,and the current research methods for infinite dimensional state-dependent delay equations are more inadequate.In this paper,we consider the long-time behavior of wave equations with nonlinear damping and state-dependent delay.First,we proved the well-posedness of the solution in a certain appropriate C1-type space;secondly,we used the Lyapunov method to obtain the dissipativity,the delay term requires modifications to the standard Lyapunov functional;finally,the quasi-stability method is used to prove the existence of finite fractal dimensional global attractors for the case where the nonlinear term has critical growth.In addition,in the summary and outlook section,we have provided and commented on some important unresolved issues closely related to the present paper.