Slow Motion Manifolds Far from the Attractor in Multistable Reaction–Diffusion Equations

Abstract

We consider a scalar reaction–diffusion equation with multistable nonlinearity with a particular symmetry. By reduction to a family of transmission problems in R, and by contraction arguments, a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed. An approximation for the associated vector field is also provided. This shows that the motion on those manifolds is exponentially slow, as in the well-known case of the bistable equation. However, in opposition to the bistable case, some of these manifolds are far from the attractor. Since these manifolds correspond to metastable patterns, this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion. It is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds.