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 \(n\in\mathbb Z\), \(m\geq 1\), \(d>0\). 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 \(\phi:\mathbb R\to[0,1]\) is differentiable and satisfies \(\phi(-\infty)=0\), \(\phi(+\infty)=1\). The authors establish such type of solution for \(m=1\) and \(m\geq 2\) 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)].
Reviewer: Klaus R. Schneider (Berlin)
MSC:
| 34B40 | Boundary value problems on infinite intervals for ordinary differential equations |
| 34A35 | Ordinary differential equations of infinite order |
| 35K55 | Nonlinear parabolic equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
Citations:
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