Quasi‐stability and continuity for wave equation with a fractional damping and a small delay

Abstract

In this paper, we shall deal with the long‐time dynamics of a nonlinear system of wave equation with a fractional damping and a small delay. Inspired by the works of Chueshov and Lasiecka on the property of quasi‐stability of dynamic systems, we show first the quasi‐stability property holds for the problem considered here and then prove the existence of global and exponential attractors. We shall also prove the upper semicontinuities of global attractors with respect to the fractional exponents. Finally, we shall show the continuities of global attractors with respect to a small parameter multiplying the delay term in a residual dense set and their upper semicontinuities in a complete metric space.