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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)].

##### 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
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##### References:
 [1] Aronson, D. G.: Density dependent interaction diffusion systems, Proceedings of the adv. Seminar on dynamics and modelling of reactive systems. (1980) [2] D.G. Aronson, W.F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, Vol. 446, Springer, New York, 1975. · Zbl 0325.35050 [3] Chow, S. -N.; Mallet--Paret, J.; Shen, W.: Traveling waves in lattice dynamical systems. Journal of differential equations 149, 248 (1998) · Zbl 0911.34050 [4] Chow, S. -N.; Shen, W.: Dynamics in discrete Nagumo equation: spatial topological chaos. SIAM J. Appl. math. 55, 1764 (1995) · Zbl 0840.34012 [5] Chow, S. -N.; Shen, W.: Stability and bifurcation of traveling wave solutions in coupled map lattices. Dynamic systems appl. 4, 1 (1995) [6] De Pablo, A.; Vazquez, J. L.: Travelling waves and finite propagation in a reaction-diffusion equation. Journal of differential equations 93, 19 (1991) · Zbl 0784.35045 [7] Fife, P. C.; Mcleod, J. B.: The approach of solutions of nonlinear equations to travelling front solutions. Arch. rational mech. Anal. 65, 335 (1977) · Zbl 0361.35035 [8] Fisher, R. A.: The advance of advantageous genes. Ann. eugenics 7, 355 (1937) · Zbl 63.1111.04 [9] Hankerson, D.; Zinner, B.: Wavefronts for a cooperative tridiagonal system of differential equations. J. dyn. Differential equations 5, 359 (1993) · Zbl 0777.34013 [10] Hosono, Y.: Travelling wave solutions for some density dependent diffusion equations. Japan J. Appl. math. 3, 163 (1986) · Zbl 0612.35069 [11] C.-H. Hsu, S.-S. Lin, Travelling waves in lattice dynamical system with applications to cellular neural networks preprint. [12] C.-H. Hsu, S.-S. Lin, W. Shen, Travelling waves in cellular neural networks, Int. J. Bif. Chaos, to appear. [13] J. Mallet--Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dyn. Differential Equations, to appear. · Zbl 0927.34049 [14] J. Mallet--Paret, The global structure of traveling waves in spatial discrete dynamical systems, J. Dyn. Differential Equations, to appear. · Zbl 0921.34046 [15] Mckean, H. K.: Nagumo’s equation. Adv. math. 4, 209 (1970) · Zbl 0202.16203 [16] Wu, J.; Zou, X.: Asymtotic and periodic boundary value problems of mixed fdes and wave solutions of lattice differential equations. Journal of differential equations 135, 315 (1997) · Zbl 0877.34046 [17] Zinner, B.: Stability of travelling wavefronts for the discrete Nagumo equation. SIAM J. Math. anal. 22, 1016 (1991) · Zbl 0739.34060 [18] Zinner, B.: Existence of travelling wavefront solutions for the discrete Nagumo equation. Journal of differential equations 96, 1 (1992) · Zbl 0752.34007 [19] Zinner, B.; Harris, G.; Hudson, W.: Travelling wavefronts for the discrete Fisher’s equation. Journal of differential equations 105, 46 (1993) · Zbl 0778.34006