Global exponential convergence to variational traveling waves in cylinders

TL;DR

This paper proves that in infinite cylinders, a specific variational traveling wave acts as a globally attracting solution for reaction-diffusion equations, with convergence happening exponentially fast, under generic conditions.

Contribution

It establishes exponential convergence to a variational traveling wave in infinite cylinders, extending previous results on front propagation and selection.

Findings

Exponential convergence to the variational traveling wave

The result relies on the gradient flow structure in weighted spaces

The convergence is independent of detailed problem specifics

Abstract

We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strengthens our earlier generic propagation and selection result for "pushed" fronts.