02-421 P. Butta', A. De Masi, E. Rosatelli

Slow motion and metastability for a non local evolution equation (560K, PostScript file) Oct 13, 02

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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.

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