Depinning asymptotics in ergodic media

TL;DR

This paper investigates the asymptotic behavior of front speeds in inhomogeneous media, establishing power-law relations near zero speed and validating assumptions through weak inhomogeneity and simulations.

Contribution

It introduces a set of assumptions to prove power-law asymptotics for front speeds in ergodic media, linking the exponent to local measure dimensions.

Findings

Power-law asymptotics for front speeds near zero

Validation of assumptions in weakly inhomogeneous media

Agreement between theoretical predictions and numerical simulations

Abstract

We study speeds of fronts in bistable, spatially inhomogeneous media at parameter regimes where speeds approach zero. We provide a set of conceptual assumptions under which we can prove power-law asymptotics for the speed, with exponent depending a local dimension of the ergodic measure near extremal values. We also show that our conceptual assumptions are satisfied in a context of weak inhomogeneity of the medium and almost balanced kinetics, and compare asymptotics with numerical simulations.