On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited)
arXiv:1011.3749 · doi:10.1007/s00208-011-0722-8
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.
32 pages, submitted
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- Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction
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- A simple approach to the wave uniqueness problem
- A note on the existence of non-monotone non-oscillating wavefronts
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- An estimation of level sets for non local KPP equations with delay
- On the geometric diversity of wavefronts for the scalar Kolmogorov ecological equation
- Nonlinearly determined wavefronts of the Nicholson's diffusive equation: when small delays are not harmless
- On pushed wavefronts of monostable equation with unimodal delayed reaction
- Propagation dynamics of Fisher-KPP equation with time delay and free boundaries