On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited)

TL;DR

This paper revises the Diekmann-Kaper theory to address the uniqueness of monostable semi-wavefronts, enabling analysis of more complex models including nonlocal, delayed, and anisotropic equations, and considers the critical slowest wavefronts.

Contribution

The authors develop a generalized version of the Diekmann-Kaper theory that includes new model types, the critical case, and relaxes previous restrictions, advancing the understanding of semi-wavefront uniqueness.

Findings

Unified framework for nonlocal and delayed models

Inclusion of critical wavefront cases

Comparison with classical and recent results

Abstract

Motivated by the uniqueness problem for monostable semi-wavefronts, we propose a revised version of the Diekmann and Kaper theory of a nonlinear convolution equation. Our version of the Diekmann-Kaper theory allows 1) to consider new types of models which include nonlocal KPP type equations (with either symmetric or anisotropic dispersal), non-local lattice equations and delayed reaction-diffusion equations; 2) to incorporate the critical case (which corresponds to the slowest wavefronts) into the consideration; 3) to weaken or to remove various restrictions on kernels and nonlinearities. The results are compared with those of Schumacher (J. Reine Angew. Math. 316: 54-70, 1980), Carr and Chmaj (Proc. Amer. Math. Soc. 132: 2433-2439, 2004), and other more recent studies.