Non-attractor elements of the rich invariant sets

Determine whether, for a general nonlinear parabolic system, the rich invariant sets consisting of trajectories with controlled backward growth contain any elements outside the global attractor.

Background

The paper introduces the sets $M_n$ of backward-extendable data whose trajectories have prescribed exponential growth as time tends to negative infinity. These sets contain the global attractor, but their possible size beyond the attractor is not generally understood.

The authors note that for the Ginzburg–Landau, viscous KdV, and Kuramoto–Sivashinsky equations the corresponding sets coincide with the global attractor, while for the two-dimensional Navier–Stokes equations their union is dense in $H$ in the weaker $V'$-topology. The unresolved issue is whether such non-attractor elements exist for a general nonlinear parabolic system.