Front motion for phase transitions in systems with memory

We consider the following partial integro-differential equation (Allen–Cahn equation with memory):ϵ2φt=∫0ta(t−t′)[ϵ2Δφ+f(φ)+ϵh](t′) dt′,where ϵ 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+ϵ2φt=∫0ta1(t−t′)Δu(t′) dt′, ϵ2φt=∫0ta2(t−t′)[ϵ2Δφ+f(φ)+ϵu](t′) dt′,where ϵ is a small parameter. In this case the functions u and φ 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.