Short Time Uniqueness Results for Solutions of Nonlocal and Non-monotone Geometric Equations

TL;DR

This paper introduces a method to establish short time uniqueness for viscosity solutions of complex nonlocal, non-monotone geometric equations, crucial for understanding front propagation in various systems.

Contribution

The authors develop a novel approach based on lower gradient bounds to prove short time uniqueness for a broad class of nonlocal geometric equations.

Findings

Established short time uniqueness for dislocation equations

Proved regularity properties of propagating fronts

Applied method to FitzHugh-Nagumo type systems

Abstract

We describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non-monotone second-order geometric equations arising in front propagation problems. Our method is based on some lower gradient bounds for the solution. These estimates are crucial to obtain regularity properties of the front, which allow to deal with nonlocal terms in the equations. Applications to short time uniqueness results for the initial value problems for dislocation type equations, asymptotic equations of a FitzHugh-Nagumo type system and equations depending on the Lebesgue measure of the fronts are presented.