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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

$\frac{d{u}_{n}}{dt}=d\left[{u}_{n-1}^{m}-2{u}_{n}^{m}+{u}_{n+1}^{m}\right]+{u}_{n}\left(1-{u}_{n}\right),\phantom{\rule{2.em}{0ex}}\left(*\right)$

with $n\in ℤ$, $m\ge 1$, $d>0$. The goal is to prove the existence of a travelling wave solution to (*) with wave speed $c>0$: ${u}_{n}\left(\xi \right)=\varphi \left(n+c\xi \right)$, where $\varphi :ℝ\to \left[0,1\right]$ is differentiable and satisfies $\varphi \left(-\infty \right)=0$, $\varphi \left(+\infty \right)=1$. The authors establish such type of solution for $m=1$ and $m\ge 2$ by the method of monotone iteration (lower and upper solutions). The case $1 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:
 34B40 Boundary value problems for ODE on infinite intervals 34A35 ODE of infinite order 35K55 Nonlinear parabolic equations 34A45 Theoretical approximation of solutions of ODE