Front motion for phase transitions in systems with memory

We consider the following partial integro-differential equation (Allen–Cahn equation with memory): φt = ∫ t 0 a(t − t ′)[ ∆φ + f (φ)+ h](t ′) dt ′, where is a small parameter, h a constant, f (φ) the negative derivative of a double well potential and the kernel a is a piecewise continuous, differentiable at the origin, scalar-valued function on (0,∞). The prototype kernels are exponentially decreasing functions of time and they reduce the integro-differential equation to a hyperbolic one, the damped Klein–Gordon equation. By means of a formal asymptotic analysis, we show that to the leading order and under suitable assumptions on the kernels, the integro-differential equation behaves like a hyperbolic partial differential equation obtained by considering prototype kernels: the evolution of fronts is governed by the extended, damped Born–Infeld equation. We also apply our method to a system of partial integro-differential equations which generalize the classical phase-field equations with a non-conserved order parameter and describe the process of phase transitions where memory effects are present: ut + φt = ∫ t 0 a1(t − t ′)∆u(t ′) dt ′, φt = ∫ t 0 a2(t − t ′)[ ∆φ + f (φ)+ u](t ′) dt ′, where is a small parameter. In this case the functionsu andφ represent the temperature field and order parameter, respectively. The kernels a1 and a2 are assumed to be similar to a. For the phase-field equations with memory we obtain the same result as for the generalized Klein–Gordon equation or Allen–Cahn equation with memory. © 2000 Published by Elsevier Science B.V.