Calculus of Variations and Geometric Measure Theory

M. Lucia - C. Muratov - M. Novaga

Existence of traveling wave solutions for Ginzburg-Landau-type problems in infinite cylinders

created on 11 Oct 2004
modified by novaga on 04 Apr 2016


Published Paper

Inserted: 11 oct 2004
Last Updated: 4 apr 2016

Journal: Arch. Rat. Mech. Anal.
Volume: 188
Number: 3
Pages: 475-508
Year: 2008


We study a class of systems of reaction-diffusion equations in infinite cylinders. These systems of equations 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 parameter. We use a novel variational characterization to study existence of traveling wave solutions under very general assumptions on the nonlinearities. These solutions are a special class of the 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. 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.