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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 $$\frac{du_n}{dt}=d[u^m_{n-1}-2u^m_n+u^m_{n+1}]+u_n(1-u_n),\tag*$$ with $n\in\Bbb Z$, $m\ge 1$, $d>0$. The goal is to prove the existence of a travelling wave solution to (*) with wave speed $c>0$: $u_n(\xi)=\phi(n+c\xi)$, where $\phi:\Bbb R\to[0,1]$ is differentiable and satisfies $\phi(-\infty)=0$, $\phi(+\infty)=1$. 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 $1<m<2$ is treated by using the method of {\it B. Zinner, G. Harris} and {\it W. Hudson} [J. Differ. Equations 105, No. 1, 46--62 (1993; Zbl 0778.34006)].

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