1
1. Introduction
We study the existence and uniqueness of traveling waves to the following two-dimensional (2-D) lattice dynamical system:
˙u
i,j= u
i+1,j+ u
i−1,j+ u
i,j+1+ u
i,j−1− 4u
i,j+ f (u
i,j), i, j ∈ Z, (1.1)
where f is monostable: f (0) = f (1) = 0 < f (u), ∀u ∈ (0, 1). The equation (1.1) is a spatial discrete version of the following reaction-diffusion equation
(1.2) u
t= ∆u + f (u), x ∈ R
N, t ∈ R,
for N = 2. When f (u) = u(1 − u), the equation (1.2) is called Fisher’s equation [9] or KPP equation [11] which arises in the study of gene development or population dynamics.
A solution {u
i,j}
i,j∈Zis called a traveling wave with speed c, if there exists a θ ∈ [0, 2π) and a differentiable function U : R → [0, 1] such that U (−∞) = 1, U (+∞) = 0, and u
i,j(t) = U (ip + jq − ct) for all i, j ∈ Z, t ∈ R, where p := cos θ and q := sin θ. The parameter θ represents the direction of movement of wave and U is called the wave profile.
Set ξ := ip + jq − ct. Then it is easy to see that (1.1) has a traveling wave with speed c if and only if the equation
c U
0(ξ) + D
2[U ](ξ) + f (U (ξ)) = 0, ξ ∈ R, (1.3)
has a solution U defined on R with 0 ≤ U ≤ 1, U (−∞) = 1, and U (+∞) = 0, where D
2[U ](ξ) := U (ξ + q) + U (ξ + p) + U (ξ − q) + U (ξ − p) − 4U (ξ).
In particular, if θ = 0, then the problem (1.1) is reduced to a one-dimensional (1-D) lattice dynamical system on Z.
In Cahn, Mallet-Paret, and van Vleck [1], they studied a two-dimensional (2-D) lattice dynamical system with bistable nonlinearity (f is called a bistable nonlinearity, if there is a ∈ (0, 1) such that f (0) = f (a) = f (1) = 0, f
0(0) < 0, and f
0(1) < 0). They obtained the existence and non-existence (so-called propagation failure) of traveling waves for the studied lattice dynamical system. The purpose of this paper is to study a 2-D lattice dynamical system with monostable nonlinearity.
We shall make the following assumptions.
(A) f ∈ C
1([0, 1]), f (0) = f (1) < f (u), ∀u ∈ (0, 1) and f
0(0) > 0.
(B) There exists M
0= M
0(f ) > 0 and α ∈ (0, 1] such that
(1.4) f
0(0)u − M
0u
1+α≤ f (u) ≤ f
0(0)u, ∀u ∈ [0, 1].
(C) f
0(1) < 0 and f (u) − f
0(1)(u − 1) = O(|u − 1|
1+α) as u → 1
−.
By the symmetry of D
2[U ], we may only consider θ ∈ [0, π/2). Since we are dealing with a 2-D problem, we shall always assume that θ ∈ (0, π/2). Therefore, for a given θ ∈ (0, π/2), our problem is to find (c, U ) ∈ R × C
1(R) such that
c U
0(ξ) + D
2[U ](ξ) + f (U (ξ)) = 0, ξ ∈ R, U (+∞) = 0, U (−∞) = 1,
0 ≤ U (ξ) ≤ 1 ∀ξ ∈ R.
(1.5)
Note that, by integrating (1.3) from −∞ to +∞, we have
(1.6) c =
Z
∞−∞
f (U (ξ))dξ
2
for any solution (c, U ) of (1.5). Hence c > 0 for any solution (c, U ) of (1.5).
We now state our main results as follows.
Theorem 1. Assume (A) and (B). Then the following hold:
(i) The problem (1.5) admits a solution if and only if c ≥ c
∗, where c
∗:= min
λ>0
