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Zbl 1086.34011
Chen, Xinfu; Guo, Jong-Shenq
Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. (English)
[J] Math. Ann. 326, No. 1, 123-146 (2003). ISSN 0025-5831; ISSN 1432-1807/e

Summary: We study traveling waves of a discrete system $$\dot u_j = g(u_{j+1}) + g(u_{j-1}) - 2g(u_j) + f(u_j), \qquad j\in\Bbb Z,$$ where $f$ and $g$ are Lipschitz continuous with $g$ increasing and $f$ monostable, i.e., $f(0) = f(1) = 0$ and $f>0$ on $(0,1)$. We show that there is a positive $c_{\text{min}}$ such that a traveling wave of speed $c$ exists if and only if $c\ge c_{\text{min}}$. Also, we show that traveling waves are unique up to a translation if $f'(0) > 0 > f'(1)$ and $g'(0) > 0$. The tails of traveling waves are also investigated.

MSC 2000:: *34A35 ODE of infinite order
34C37 Homoclinic and heteroclinic solutions of ODE
35K55 Nonlinear parabolic equations
37L60 Lattice dynamics

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Zentralblatt MATH Berlin [Germany]