Lower semicontinuity of attractors of gradient systems and applications

Abstract

For 0⩽e⩽e0, let Te(t), t⩾0, be a family of semigroups on a Banach space X with local attractors Ae. Under the assumptions that T0(t) is a gradient system with hyperbolic equilibria and Te(t) converges to T0(t) in an appropriate sense, it is shown that the attractors {Ae, 0⩽e⩽e0} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between Ae and A0, in some examples.