## Abstract

We consider the following partial integro-differential equation (Allen-Cahn equation with memory):ε^{2}φ_{t}=∫0ta(t-t′)[ε ^{2}Δφ+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:u_{t}+ε^{2}φ_{t}=∫0ta _{1}(t-t′)Δu(t′) dt′,ε^{2}φ_{t}=∫0ta _{2}(t-t′)[ε ^{2}Δφ+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 a_{1} and a_{2} 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.

Original language | English (US) |
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Pages (from-to) | 137-149 |

Number of pages | 13 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 146 |

Issue number | 1-4 |

DOIs | |

State | Published - Nov 15 2000 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

## Keywords

- Allen-Cahn equation memory
- Born-Infeld equation
- Front motion
- Integro-differential equations
- Phase transition dynamics with memory