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Front propagation for a two-dimensional periodic monostable lattice dynamical system. (English) Zbl 1195.34118

The authors study wave propagation for the system

u t =·(A(x)u)+f(x,u),x n ,t>0,(1)

where A and f are periodic in x. The traveling waves for the one-dimensional spatial discrete version of (1) is considered in [J.-S. Guo and F. Hamel, Math. Ann. 335, No. 3, 489–525 (2006; Zbl 1116.35063)], the results of which are extended to the two dimensional spatial discrete version in the present article. The authors investigate the following system:

u i,j ' (t)=p i+1,j u i+1,j (t)+p i,j u i-1,j (t)+q i,j+1 u i,j+1 (t)+q i,j u i,j-1 (t)-D i,j u i,j (t)+f(i,j,u i,j (t)),t,(i,j) 2 ,u i+N,j t+Nr c=u i,j (t)=u i,j+N t+Ns c,t,(i,j) 2 ,c0,lim ri+sj- u i,j (t)=1,lim ri+sj+ u i,j (t)=0,t,0u i,j (t)1,t,(i,j) 2 ,(2)

where

D i,j =p i+1,j +p i,j +q i,j+1 +q i,j ,i,j 2 ,p i+N,j =p i,j =p i,j+N ,q i+N,j =q i,j =q i,j+N ,i,j 2 ,f(i+N,j,s)=f(i,j,s)=f(i,j+N,s),i,j 2 ,s[0,1]

for some positive integer N. Here, c is the wave speed, (r,s)=(cosθ,sinθ) with θ[0,2π). Let u(·)=u i,j (·) be the wave profile. Under some assumptions, the authors show the following conclusions:

there is a c * such that (2) admits a traveling wave solution for cc * ;

if (2) admits a traveling wave solution for c0, then cc * ;

if u(·) is a traveling wave solution of (2) with c0, then u(·) is strictly increasing in t;

by constructing a discrete approximate system for (1) in the two dimensional case, the minimal wave speed of the approximate system converges to the minimal wave speed of (1).

MSC:
34K31Lattice functional-differential equations
34K10Boundary value problems for functional-differential equations
35K57Reaction-diffusion equations