Identify non-autonomous equilibria and heteroclinic connections in the non-autonomous Chafee–Infante equation

Determine whether the 2n+1 global solutions with a finite and time-constant number of spatial zeros constructed for the non-autonomous Chafee–Infante equation ut = uxx + λu − b(t)u^3 on (0,π) with zero Dirichlet boundary conditions (for n < √λ < n+1) constitute all non-autonomous equilibria, and establish the existence of non-autonomous heteroclinic connections between such solutions.

Background

For the non-autonomous Chafee–Infante equation, previous work has shown the existence of at least 2n+1 global solutions when n < √λ < n+1, with each solution having a finite and constant in time number of spatial zeros. These are conjectured to play the role of non-autonomous counterparts of hyperbolic equilibria in the autonomous case.

It is not yet established whether these solutions exhaust all non-autonomous equilibria, and there are no rigorous results demonstrating the existence of heteroclinic connections between them in the non-autonomous setting.