Stability of invariant manifolds in one and two dimensions

We consider the gradient flow associated with a nonlocal free energy functional and extend to such a case results obtained for the Allen–Cahn equation on 'slow motions on invariant manifolds'. The manifolds in question are time-invariant one-dimensional curves in an L2 space which connect the two ground states (interpreted as the pure phases of the system) to the first excited state (interpreted as a diffuse interface). Local and structural stability of the manifolds are proved and applications to the characterization of optimal tunnelling are discussed