Front motion for phase transitions in systems with memory

We consider the following partial integro-differential equation (Allen-Cahn equation with memory): epsilon (2)phi (t) = integral (t)(0) a(t - t')[epsilon (2)Delta phi + f(phi ) + epsilonh](t')dt', where epsilon is a Small parameter, h a constant, f(phi) the negative deriv ative of a double well potential and the kernel a is a piecewise continuous , differentiable at the origin, scalar-valued function on (0, infinity). Th e prototype kernels are exponentially decreasing functions of time and they reduce the integro-differential equation to a hyperbolic one, the damped K lein-Gordon equation. By means of a formal asymptotic analysis, we show tha t to the leading order and under suitable assumptions on the kernels, the i ntegro-differential equation behaves like a hyperbolic partial differential equation obtained by considering prototype kernels: the evolution of front s is governed by the extended, damped Born-Infeld equation. We also apply o ur method to a system of partial integro-differential equations which gener alize the classical phase-field equations with a non-conserved order parame ter and describe the process of phase transitions where memory effects are present: u(t) + epsilon (2)phi (t) = integral (t)(0) a(1)(t - t')Deltau(t') dt' epsi lon (2)phi (t) = integral (t)(0) a(2)(t - t')[epsilon (2)Delta phi + epsilo nu](t') dt', where epsilon is a Small parameter. In this case the functions u and phi re present the temperature field and order parameter, respectively. The kernel s a(1) and a(2) are assumed to be similar to a. For the phase-field equatio ns with memory we obtain the same result as for the generalized Klein-Gordo n equation or Allen-Cahn equation with memory. (C) 2000 Published by Elsevi er Science B.V.