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Traveling wave solutions for some discrete quasilinear parabolic equations. (English) Zbl 1092.34012

Consider the class of lattice ordinary differential equations

du n dt=d[u n-1 m -2u n m +u n+1 m ]+u n (1-u n ),(*)

with n, m1, d>0. The goal is to prove the existence of a travelling wave solution to (*) with wave speed c>0: u n (ξ)=ϕ(n+cξ), where ϕ:[0,1] is differentiable and satisfies ϕ(-)=0, ϕ(+)=1. The authors establish such type of solution for m=1 and m2 by the method of monotone iteration (lower and upper solutions). The case 1<m<2 is treated by using the method of B. Zinner, G. Harris and W. Hudson [J. Differ. Equations 105, No. 1, 46–62 (1993; Zbl 0778.34006)].

MSC:
34B40Boundary value problems for ODE on infinite intervals
34A35ODE of infinite order
35K55Nonlinear parabolic equations
34A45Theoretical approximation of solutions of ODE