Transition fronts and stretching phenomena for a general class of reaction-dispersion equations

TL;DR

This paper investigates reaction-dispersion equations with non-local dispersal and local reactions, establishing conditions for the non-existence of transition fronts and describing stretching phenomena in solutions over time.

Contribution

It provides new theoretical results on the non-existence of transition fronts and stretching behavior for a broad class of reaction-dispersion equations with various dispersal operators.

Findings

Non-existence of transition fronts under certain conditions.

Stretching phenomena observed at large times.

Applicable to equations with fractional Laplacian, fat-tailed kernels, or nonlinear fast diffusion.

Abstract

We consider a general form of reaction-dispersion equations with non-local dispersal and local reaction. Under some general conditions, we prove the non-existence of transition fronts, as well as some stretching properties at large time for the solutions of the Cauchy problem. These conditions are satisfied in particular when the reaction is monostable and when the dispersal operator is either the fractional Laplacian, a convolution operator with a fat-tailed kernel or a nonlinear fast diffusion operator.