Separation dichotomy and wavefronts for a nonlinear convolution equation

TL;DR

This paper investigates the behavior of solutions to a nonlinear convolution equation related to traveling waves, establishing a dichotomy principle and demonstrating the existence of various wavefronts in non-symmetric models.

Contribution

It introduces a dichotomy principle for solutions and proves the existence of semi-wavefronts in nonlinear convolution equations, including non-symmetric cases with multiple wave types.

Findings

Bounded positive solutions are either asymptotically separated from zero or decay exponentially to zero.

Theorems guarantee the existence of semi-wavefront solutions under certain conditions.

Non-symmetric models can have stationary, expansion, and extinction waves simultaneously.

Abstract

This paper is concerned with a scalar nonlinear convolution equation which appears naturally in the theory of traveling waves for monostable evolution models. First, we prove that each bounded positive solution of the convolution equation should either be asymptotically separated from zero or it should converge (exponentially) to zero. This dichotomy principle is then used to establish a general theorem guaranteeing the uniform persistence and existence of semi-wavefront solutions to the convolution equation. Finally, we apply our abstract results to several well-studied classes of evolution equations with asymmetric non-local and non-monotone response. We show that, contrary to the symmetric case, these equations can possess at the same time the stationary, the expansion and the extinction waves.