Universal selection of pulled fronts

TL;DR

This paper proves the universal selection and convergence of critical pulled fronts in invasion processes, including those with compactly supported initial data, using spectral stability and sharp decay estimates.

Contribution

It introduces a new, robust approach to establish front selection without relying on comparison principles or probabilistic methods, applicable to open classes of systems.

Findings

Convergence to pulled fronts with logarithmic shift for broad initial data classes.

Validation of assumptions for open classes of spatially extended systems.

Development of sharp linear decay estimates for nonlinear matching.

Abstract

We establish selection of critical pulled fronts in invasion processes. Our result shows convergence to a pulled front with a logarithmic shift for open sets of steep initial data, including one-sided compactly supported initial conditions. We rely on robust, conceptual assumptions, namely existence and marginal spectral stability of a front traveling at the linear spreading speed. We demonstrate that the assumptions hold for open classes of spatially extended systems. Previous results relied on comparison principles or probabilistic tools with implied non-open conditions on initial data and structure of the equation. Technically, we describe the invasion process through the interaction of a Gaussian leading edge with the pulled front in the wake. Key ingredients are sharp linear decay estimates to control errors in the nonlinear matching and corrections from initial data.