Asymptotic study of critical wave fronts for parameter-dependent Born-Infeld models: physically predicted behaviors and new phenomena

TL;DR

This paper analyzes parameter-dependent Born-Infeld reaction-diffusion models, revealing how critical wave front speeds behave under various electric field and diffusion conditions, including new phenomena like non-zero limits and sharp profiles.

Contribution

It provides the first rigorous analysis of critical wave front speeds and profiles for Born-Infeld models, uncovering novel behaviors and explicit limit characterizations.

Findings

Critical speeds converge to linear diffusion limits for large electric fields.

New phenomena where the limit front profile becomes sharp on one side.

Explicit determination of the gluing point and limit speed for profiles.

Abstract

In this paper, we study some parameter-dependent reaction-diffusion models governed by the Born-Infeld (or Minkowski) operator. In dependence on two parameters $a, b > 0$, related to the field strength and to the diffusivity, we investigate the limit critical speed for traveling fronts, together with the limit behavior of the associated critical profiles. As the two main results, on the one hand we rigorously show that for arbitrarily large electric fields the critical speed and the critical profiles converge to the ones of the linear diffusion problem, agreeing with the well-known physical prediction that, in this case, Born's law and Maxwell's law coincide. Such a result is accompanied by a counterpart for arbitrarily small electric fields and a complete analysis for vanishing/large diffusion. On the other hand, we prove the onset of a new and unexpected phenomenon for the singular perturbation problem: the critical speed of propagation does not converge to zero and, for KPP or combustion-type reactions, the limit front profile becomes sharp on one side only. In fact, this latter coincides with the $C^1$-gluing of a piecewise linear profile having slope equal to 1 when nonconstant, and of a regular inviscid profile propagating at the limit speed. The gluing point and the value of the limit speed are determined explicitly. The outcomes of the whole discussion, based on a careful analysis of the first-order reduction associated with the original equation, show substantial differences and peculiarities of the Born- Infeld operator with respect to linear or saturating diffusions, and are supported by several numerical experiments.