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

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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.