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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 $$u_t=\nabla\cdot(A(x)\nabla u)+f(x,u),\quad x\in \mathbb{R}^n,t>0,\tag1$$ where $A$ and $f$ are periodic in $x$. The traveling waves for the one-dimensional spatial discrete version of (1) is considered in [{\it J.-S. Guo} and {\it 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: \aligned u^{\prime}_{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)),\quad t\in \mathbb{R},(i,j)\in \mathbb{Z}^2,\\ u_{i+N,j}\left(t+\frac{Nr}{c}\right)=u_{i,j}(t)=u_{i,j+N}\left(t+\frac{Ns}{c}\right),\quad t\in \mathbb{R},(i,j)\in \mathbb{Z}^2,c\not=0,\\ \lim\limits_{ri+sj\to -\infty}u_{i,j}(t)=1,\quad \lim\limits_{ri+sj\to +\infty}u_{i,j}(t)=0,\quad t\in \mathbb{R},\\ 0\leq u_{i,j}(t)\leq 1, \quad t\in \mathbb{R},(i,j)\in \mathbb{Z}^2,\endaligned\tag2 where \aligned D_{i,j}=p_{i+1,j}+p_{i,j}+q_{i,j+1}+q_{i,j}, \quad {i,j}\in \mathbb{Z}^2,\\ p_{i+N,j}=p_{i,j}=p_{i,j+N},\quad q_{i+N,j}=q_{i,j}=q_{i,j+N}, \quad {i,j}\in \mathbb{Z}^2,\\ f(i+N,j,s)=f(i,j,s)=f(i,j+N,s), \quad {i,j}\in \mathbb{Z}^2,s\in [0,1]\endaligned for some positive integer $N$. Here, $c$ is the wave speed, $(r,s)=(\cos\theta,\sin\theta)$ with $\theta \in [0,2\pi)$. Let $u(\cdot)=\left\{u_{i,j}(\cdot)\right\}$ be the wave profile. Under some assumptions, the authors show the following conclusions: {\parindent8mm \item{(a)} there is a $c_*$ such that (2) admits a traveling wave solution for $c\geq c_*$; \item{(b)} if (2) admits a traveling wave solution for $c\not=0$, then $c\geq c_*$; \item{(c)} if $u(\cdot)$ is a traveling wave solution of (2) with $c\not=0$, then $u(\cdot)$ is strictly increasing in $t$; \item{(d)} 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).\par}

##### MSC:
 34K31 Lattice functional-differential equations 34K10 Boundary value problems for functional-differential equations 35K57 Reaction-diffusion equations
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