Slow motion manifolds far from the attractor in multistable reaction-diffusion equations

We consider a scalar reaction-diffusion equation with multistable nonlinear ity with a particular symmetry. By reduction to a family of transmission pr oblems 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 provi ded. This shows that the motion on those manifolds is exponentially slow, a s in the well-known case of the bistable equation. However, in opposition t o the bistable case, some of these manifolds are far from the attractor. Si nce these manifolds correspond to metastable patterns, this shows the impor tance of the transient motion toward the attractor and the importance of th ese manifolds in organizing that motion. It is also shown that by a suitabl e perturbation we can obtain new equilibria on those manifolds. (C) 2001 Ac ademic Press.