Uniqueness of Traveling Waves for a Two-Dimensional Bistable Periodic Lattice Dynamical System

and Applied Analysis 3 This paper is organized as follows. In Section 2, we give some preliminaries including a comparison principle. Then we prove uniqueness of traveling wave with nonzero speed in Section 3. 2. Preliminaries The following lemma can be easily deduced from 1.5 and 1.6 . Lemma 2.1. Let {ui,j}i,j∈Z be a solution of 1.1 – 1.6 . If c > 0 < 0 , then ui,j t → 0 → 1 as t → ∞ and ui,j t → 1 → 0 as t → −∞ for each i, j. We can determine the sign of the speed c when c / 0 as follows. Lemma 2.2. Suppose that c / 0, then c has the same sign as − ∫1 0 f s ds . Proof. For K ≥ max{1,N/|c|}, an integration by parts gives ∫K −K [ u̇i,j t ]2 dt ∫K −K u̇i,j t { D2 [ ui,j ] t f ( ui,j t )} dt ∫K −K pi 1,j u̇i,jui 1,jdt − ∫K −K pi,jui,j u̇i−1,jdt pi,jui−1,jui,j |−K ∫K −K qi,j 1u̇i,jui,j 1dt − ∫K −K qi,jui,j u̇i,j−1dt qi,jui,j−1ui,j |−K − 1 2 di,ju 2 i,j |−K ∫K −K u̇i,jf ( ui,j ) dt for 1 ≤ i, j ≤ N. 2.1