Regularity and stability of transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity

TL;DR

This paper studies the regularity, steepness, and stability of transition fronts in nonlocal equations with time-heterogeneous ignition nonlinearities, providing new insights into their mathematical properties.

Contribution

It is the first to analyze the regularity and stability of transition fronts in nonlocal equations with heterogeneous media.

Findings

Transition fronts are continuously differentiable in space.

Transition fronts are uniformly steep.

Transition fronts exhibit exponential asymptotic stability.

Abstract

The present paper is devoted to the investigation of various properties of transition fronts in nonlocal equations in heterogeneous media of ignition type, whose existence has been established by the authors of the present paper in a previous work. It is first shown that the transition front is continuously differentiable in space with uniformly bounded and uniformly Lipschitz continuous space partial derivative. This is the first time that regularity of transition fronts in nonlocal equations is ever studied. It is then shown that the transition front is uniformly steep. Finally, asymptotic stability, in the sense of exponentially attracting front like initial data, of the transition front is studied.