ON THE SLOWNESS OF PHASE-BOUNDARY MOTION IN ONE SPACE DIMENSION

We study the limiting behavior of the solution of u(t) - epsilon-2u(xx) + u3 - u = 0, a < x < b, with a Neumann boundary condition or an appropriate Dirichlet condition. The analysis is based on "energy methods". We assume that the initial data has a "transition layer structure", i.e., u(epsilon) almost-equal-to +/- 1 except near finitely many transition points. We show that, in the limit as epsilon --> 0, the solution maintains its transition layer structure, and the transition points move slower than any power of epsilon.