## Abstract

We study a class of systems of reaction-diffusion equations in infinite cylinders which arise within the context of Ginzburg-Landau theories and describe the kinetics of phase transformation in second-order or weakly first-order phase transitions with non-conserved order parameters. We use a variational characterization to study the existence of a special class of traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation. Our main result is a simple verifiable criterion for existence of these traveling waves under the very general assumptions of non-linearities. We also prove boundedness, regularity, and some other properties of the obtained solutions, as well as several sufficient conditions for existence or non-existence of such traveling waves, and give rigorous upper and lower bounds for their speed. In addition, we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized. We give a sample application of our results using a computer-assisted approach.

Original language | English (US) |
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Pages (from-to) | 475-508 |

Number of pages | 34 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 188 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2008 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering