P. Butta', A. De Masi, E. Rosatelli
Slow motion and metastability for a non local evolution equation
(560K, PostScript file)
ABSTRACT. In this paper we consider a non local evolution equation in one dimension,
which describes the dynamics of a ferromagnetic system in the mean field
approximation. In the presence of a small external magnetic field, this
equation admits two stationary homogeneous solutions, which represent the
stable and metastable phases of the physical system. We prove the existence
of an invariant, one dimensional manifold connecting the stable and metastable
phases. This is the unstable manifold of a distinguished, spatially non
homogeneous, stationary solution of the evolution equation, called the
critical droplet. We show that the points on the manifold are droplets
longer or shorter than the critical one, and that their motion is very
slow in agreement with the theory of metastable patterns. The existence of
the critical droplet was firstly proved by A. De Masi, E. Olivieri,
E. Presutti in [Markov Process, Related Fields vol. 6 (1999), 439--471]
but no uniqueness result was guaranteed by that approach. However we give
here a different proof, which is also supplied with a local uniqueness result.
Finally, a detailed description of the spatial structure of the critical
droplet is given, which will be also a key tool to study the global structure
of the unstable manifold.