Dynamics of a wave equation with memory and Hardy type potentials

Abstract

This paper is concerned with the well-posedness and long-time dynamics for a wave equation with memory and Hardy type potentials. It is first proved the existence and uniqueness of weak solutions based on the Faedo-Galerkin approximation for $0\le \lambda \le (1-{\delta }_1){\lambda }^{⁎}$. Moreover, the existence of a global attractor with finite fractal dimension is shown by establishing a quasi-stability inequality. Furthermore, we also establish the asymptotic regularity of the weak solution outside arbitrarily small neighborhoods of the origin. Finally, the upper semicontinuity of attractors is established when the parameter λ goes to $0^{+}$.