  We consider the Allen-Cahn equations with memory (a partial
integro-differential convolution equation). The prototype kernels are
exponentially decreasing functions of time and they reduce the
integrodifferential 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 behave 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.
