On uniqueness of semi-wavefronts

Abstract

Motivated by the uniqueness problem for monostable semi-wave-fronts, 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), nonlocal 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 Am Math Soc 132: 2433-2439, 2004), and other more recent studies.