We study backward stability of a pullback attractor especially for a delay equation. We introduce a new concept of a backward attractor, which is defined by a compact, pullback attracting and dividedly invariant family. We then show the equivalence between existence of a backward attractor and backward stability of the pullback attractor, and present some criteria by using the backward limit-set compactness of the system. In the application part, we consider the Navier-Stokes equation with a nonuniform Lipschitz delay term and a backward tempered force. Based on the fact that the delay does not change the backward bounds of the velocity field and external forces, we establish the backward-uniform estimates and obtain a backward attractor, which leads to backward stability of the pullback attractor. Some special cases of variable delay and distributed delay are discussed.
"Backward Stability and Divided Invariance of an Attractor for the Delayed Navier-Stokes Equation." Taiwanese J. Math. 24 (3) 575 - 601, June, 2020. https://doi.org/10.11650/tjm/190603