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Zbl 1092.34012
Fu, Sheng-Chen; Guo, Jong-Shenq; Shieh, Shang-Yau
Traveling wave solutions for some discrete quasilinear parabolic equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 48, No. 8, A, 1137-1149 (2002). ISSN 0362-546X

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)].
[Klaus R. Schneider (Berlin)]
MSC 2000:
*34B40 Boundary value problems on infinite intervals
34A35 ODE of infinite order
35K55 Nonlinear parabolic equations
34A45 Theoretical approximation of solutions of ODE

Citations: Zbl 0778.34006

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