Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders

M. Lucia, Cyrill Muratov, M. Novaga

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

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 languageEnglish (US)
Pages (from-to)475-508
Number of pages34
JournalArchive for Rational Mechanics and Analysis
Volume188
Issue number3
DOIs
StatePublished - Jun 1 2008

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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