Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity

TL;DR

This paper investigates transition fronts in nonlocal reaction-diffusion equations with time-dependent ignition nonlinearities, establishing existence, regularity, and finite speed properties through approximation and interface modification techniques.

Contribution

It introduces a novel approach using modified interface location to prove the existence and properties of transition fronts in nonlocal, time-heterogeneous ignition equations.

Findings

Existence of space-monotone transition fronts with finite speed

Uniform Lipschitz continuity of the transition fronts

Finite speed characterized by exponential decay estimates

Abstract

The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of approximating solutions in the absence of space regularity, and leads directly to uniform exponential decaying estimates.