Well-posedness and dynamics for the von Karman equation with memory and nonlinear time-varying delay

This paper is concerned with mathematical analysis for the viscoelastic von Karman equation with memory and time-varying delay. Based on the existence and uniqueness of the strong solution established by Galerkin method, the finite dimensional global attractor for the system without time-varying delay has been shown by using the quasi-stability theory. The difficulties for our desired well-posedness are the limiting of fourth-order derivative terms and the von Karman bracket term, which need some delicate estimates to achieve. The key point for the dynamics is to obtain the quasi-stability for gradient system, which needs the higher regularity of trajectory especially the von Karman bracket.