Characterize the non-autonomous attractor of the Chafee–Infante equation

Characterize the full structure of the global non-autonomous attractor for the non-autonomous Chafee–Infante equation ut = uxx + λu − b(t)u^3 on (0,π) with zero Dirichlet boundary conditions, as defined in equation (1.1), including the identification and organization of non-autonomous equilibria and their connections, for given parameter λ > 0 and time-dependent forcing b(t).

Background

In the autonomous Chafee–Infante equation (with constant b(t)), the global attractor is completely described: it consists of equilibria arising through pitchfork bifurcations and heteroclinic connections between them, with the structure determined by the parameter λ.

For the non-autonomous version (time-dependent b(t)), equilibria are replaced by non-constant trajectories, termed non-autonomous equilibria. Despite existence results for non-autonomous attractors, their detailed structure analogous to the autonomous case has not been established.