Nonlinear Instability for a Leslie-Gower Predator-Prey Model with Cross Diffusion

Abstract

A rigorous mathematical characterization for early-stage spatial and temporal patterns formation in a Leslie-Gower predator-prey model with cross diffusion is investigated. Given any general perturbation near an unstable constant equilibrium, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of the fastest growing modes.

1. Introduction

Since Turing proposed the striking idea of “diffusion-driven instability” in 1952 [1], reaction-diffusion systems are often employed to investigate chemical and biological pattern formations and have received much attention from the scientists [2–7]. However, most of the works concentrate on pattern formation in the case of linear instability, and there is a little discussion about the nonlinear effect of a reaction-diffusion system on the evolution of a nonuniform pattern.

In general, nonlinear instability is treated with great delicacy and difficulty. At first, nonlinear instability was established for nondissipative systems [8–11]. In 2004, Guo et al. [12] established nonlinear instability for an unstable Kirchhoff ellipse. Based upon a precise linear analysis, they found that the dynamics of general perturbation can be characterized by the linear dynamics of the fastest growing modes. This marks a beginning of a quantitative description of instability. Subsequently, Guo and Hwang dealt with nonlinear stability for a Keller-Segel model in [13] and described the early-stage pattern formation in that model.

Recently, Guo and Hwang considered the following reaction-diffusion system [14] 𝜕𝑈𝐷𝜕𝑡=∇⋅₁(𝑈,𝑉)∇𝑈+𝑓(𝑈,𝑉),𝜕𝑉𝐷𝜕𝑡=∇⋅₂(𝑈,𝑉)∇𝑉+𝑔(𝑈,𝑉), (1) in a box 𝕋^𝑁=(0,𝜋)^𝑁⊂ℝ^𝑁(𝑁≤3) with the homogeneous Neumann boundary conditions. In system (1), 𝑈(𝑥,𝑡), 𝑉(𝑥,𝑡) denote the densities of two interactive species at time 𝑡, the functions 𝐷₁, 𝐷₂ are their diffusion rates, and 𝑓, 𝑔 are the reaction functions. The classical Turing instability and Turing patterns were studied under some suitable conditions. Their result shows that the nonlinear evolution of patterns is dominated by the corresponding linear dynamics along a fixed finite number of the fastest growing modes over a time period.

In this paper, we consider the following Leslie-Gower predator-prey model with cross diffusion: 𝑢_𝑡−𝑑₁Δ𝑢=𝜆𝑢−𝑢²−𝛽𝑢𝑣,𝑥∈𝕋^𝑁𝑣,𝑡>0,_𝑡−𝑑₂Δ1+𝑑₃𝑢𝑣𝑣=𝑣𝜇−𝑚+𝑢,𝑥∈𝕋^𝑁,𝑡>0,𝜕𝑢𝜕𝑥_𝑖=𝜕𝑣𝜕𝑥_𝑖=0,𝑥_𝑖𝑢()=0,𝜋,𝑖=1,…,𝑁,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈𝕋^𝑁, (2) where 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) represent the densities of the species prey and predator, respectively. The parameters 𝜆, 𝛽, 𝜇, 𝑚, 𝑑₁, 𝑑₂, and 𝑑₃ are all positive constants, where 𝜆 and 𝜇 are the intrinsic growth rates of the prey and predator, 𝛽 is the predation rate, and the term 𝑣/(𝑚+𝑢) is a modified Leslie-Gower term [15]. The constants 𝑑₁, 𝑑₂, called diffusion coefficients, represent the natural tendency of each species to diffuse to areas of smaller population concentration, and 𝑑₃, called a cross-diffusion coefficient, expresses the population flux of the predator resulting from the presence of the prey species. For more ecological backgrounds about this model, one can refer to [15–17].

System (2) and its variants were studied widely for pattern formation by applying the bifurcation theory and the degree theory [6, 18–20] in the case of linear instability. Inspired by the works [13, 14], in this paper, we attempt to study the nonlinear instability for this system and give a rigorous mathematical characterization for the nonlinear evolution of pattern by using a bootstrap technique. The mathematical approach in this paper is similar in spirit to that of [13, 14]. However, our problem (2) is much more complex. Notice that the diffusion term of the predator equation in the model (2) is 𝑑₂Δ1+𝑑₃𝑢𝑣=∇⋅𝑑₂𝑑₃𝑣∇𝑢+𝑑₂1+𝑑₃𝑢∇𝑣. (3) In some sense, the coupled degree in (2) is stronger than that in (1). As a result, our analysis here, especially in establishing 𝐻² estimates for nonlinear terms 𝑑₂𝑑₃∇(𝑣∇𝑢) and 𝑑₂∇[(1+𝑑₃𝑢)∇𝑣], is much more difficult and requires some techniques beyond those of [13, 14].

It is obvious that (2) has a unique positive equilibrium (𝑢,𝑣)̃ if and only if 𝜆>𝛽𝜇𝑚, where 𝑢=𝜆−𝛽𝜇𝑚,̃𝜇()1+𝛽𝜇𝑣=𝑚+𝜆.1+𝛽𝜇 (4) Let 𝑢=𝑢(𝑥,𝑡)−𝑢, 𝑣̂=𝑣(𝑥,𝑡)−𝑣̃ be the perturbation around (𝑢,𝑣)̃ and still denote it by (𝑢,𝑣). Then, the perturbation (𝑢,𝑣) satisfies the following strongly coupled equations: 𝑢_𝑡−𝑑₁Δ𝑢=𝑔₁(𝑢+𝑢,𝑣+𝑣)̃,𝑥∈𝕋^𝑁𝑣,𝑡>0,_𝑡−𝑑₂Δ1+𝑑₃(𝑢+𝑢)(𝑣+𝑣)̃=𝑔₂(𝑢+𝑢,𝑣+𝑣)̃,𝑥∈𝕋^𝑁,𝑡>0,𝜕𝑢𝜕𝑥_𝑖=𝜕𝑣𝜕𝑥_𝑖=0,𝑥_𝑖𝑢()=0,𝜋,𝑖=1,…,𝑁,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈𝕋^𝑁, (5) where 𝑔₁(𝑢,𝑣)=𝜆𝑢−𝑢²−𝛽𝑢𝑣,𝑔₂()𝑣𝑢,𝑣=𝑣𝜇−.𝑚+𝑢 (6)

This paper is organized as follows. In Section 2, the growing modes in the linearized system are studied, which are important for our later discussions. Section 3 gives some estimates for the perturbation. The key is to control the nonlinear growth of high-order energy. In Section 4, the nonlinear instability is obtained.

2. Growing Modes in the Linearized System

The corresponding linearized system of (5) takes the form of 𝑢_𝑡−𝑑₁𝑣Δ𝑢=−𝑢𝑢−𝛽𝑢𝑣,_𝑡−𝑑₂𝑑₃𝑣Δ̃𝑢−𝑑₂1+𝑑₃𝑢Δ𝑣=𝜇²𝑢−𝜇𝑣. (7) We use [⋅,⋅] to denote a column vector and let 𝐰(𝑥,𝑡)=[𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)], 𝐪=(𝑞₁,…,𝑞_𝑁)∈ℕ^𝑁. Then, 𝐪²=∑^𝑁_𝑖=1𝑞_𝑖² are eigenvalues of −Δ on 𝕋^𝑁 under the homogeneous Neumann boundary condition, and the corresponding normalized eigenfunctions are given by 𝑒_𝐪(𝑥)=1𝜋^𝑁/22,𝐪=𝟎,𝜋^𝑁𝑁/2_𝑖=1cos𝑞_𝑖𝑥_𝑖,𝐪≠𝟎. (8) This set of eigenfunctions forms an orthonormal basis in 𝐿²(𝕋^𝑁).

We look for a normal mode to be the linear system (7) of the following form: 𝐰(𝑥,𝑡)=𝐫_𝐪𝑒^𝜆𝐪𝑡𝑒_𝐪(𝑥), (9) where 𝜆_𝐪 is a complex number and 𝐫_𝐪 is a vector; they depend on 𝐪. Substituting (9) into (7), we have 𝜆_𝐪𝐫_𝐪=−𝑢−𝑑₁𝐪²𝑢𝜇−𝛽₂−𝑑₂𝑑₃𝑣𝐪̃²−𝜇−𝑑₂1+𝑑₃𝑢𝐪²𝐫_𝐪∶=𝐿𝐫_𝐪. (10) System (7) possesses a nontrivial normal mode if and only if 𝜆det^𝐪+𝑢+𝑑₁𝐪²𝛽𝑢−𝜇²+𝑑₂𝑑₃𝑣𝐪̃²𝜆_𝐪+𝜇+𝑑₂1+𝑑₃𝑢𝐪²=0, (11) which is equivalent to 𝜆²_𝐪+𝑢+𝜇+𝑑₁+𝑑₂1+𝑑₃𝑢𝐪²𝜆_𝐪+𝑑₁𝑑₂1+𝑑₃𝑢𝐪⁴+𝑑₁𝜇+𝑑₂1+𝑑₃𝑢𝑢−𝛽𝑑₂𝑑₃𝑢𝑣̃𝐪²+𝜇𝑢+𝛽𝜇²𝑢=0. (12) Thus, we deduce the following well-known aggregation (i.e., linear instability) criterion by requiring that there exists a 𝐪, such that the constant term in (12) is 𝑑₁𝑑₂1+𝑑₃𝑢𝐪⁴+𝑑₁𝜇+𝑑₂1+𝑑₃𝑢𝑢−𝛽𝑑₂𝑑₃𝑢𝑣̃𝐪²+𝜇𝑢+𝛽𝜇²𝑢<0. (13) In this paper, we always assume that there exists a 𝐪, such that (13) holds. Then, the discriminant of (12) is Δ=[𝑑₁−𝑑₂(1+𝑑₃𝑢)]²𝐪⁴+2𝑢+𝜇𝑑₁+𝑑₂1+𝑑₃𝑢−4𝑑₁𝜇+𝑑₂1+𝑑₃𝑢𝑢+4𝛽𝑑₂𝑑₃𝑢𝑣̃𝐪²+(𝑢−𝜇)²−4𝛽𝜇²𝑢, (14) and the coefficient of 𝐪² is positive, since (13) implies 𝑑₁𝜇+𝑑₂1+𝑑₃𝑢𝑢−𝛽𝑑₂𝑑₃𝑢𝑣̃<0. (15)

For given 𝐪, we denote the corresponding eigenvalues by 𝜆^±_𝐪 and eigenvectors by 𝐫_±(𝐪). We split it into three cases for the linear analysis.(1)Δ>0. Let Λ₁={𝐪∣Δ>0}, and let 𝜆^±_𝐪 be two distinct real roots with 𝜆⁺_𝐪>𝜆⁻_𝐪, 𝜆^±_𝐪 being the corresponding (linearly independent) real eigenvectors. It is easy to see that 𝐫_±(𝐪)=𝜆1,−^±𝐪+𝑢+𝑑₁𝐪²𝛽𝑢. (16) Denote Λ_∗=𝐪∣𝑑₁𝑑₂1+𝑑₃𝑢𝐪⁴+𝑑₁𝜇+𝑑₂1+𝑑₃𝑢𝑢−𝛽𝑑₂𝑑₃𝑢𝑣̃𝐪²+𝜇𝑢+𝛽𝜇²𝑢<0. (17) Clearly, 𝜆⁺_𝐪>0 for 𝐪∈Λ_∗. Note that there are only finitely many 𝐪 in Λ_∗ and Λ_∗⊂Λ₁. Therefore, there are only finitely many linear growing modes, such that the constant equilibrium (0,0) of (5) is unstable. Furthermore, we define 𝜆_max=max_𝜆+𝐪>0𝜆⁺_𝐪,Λ_max=𝐪∣𝜆⁺_𝐪=𝜆_max. (18) Then, Λ_max⊂Λ_∗⊂Λ₁.(2)Δ=0. Let Λ₂={𝐪∣Δ=0}. In this case, (12) possesses repeated real eigenvalues. Consider 𝜆_𝐪=𝜆⁺_𝐪=𝜆⁻_𝐪1=−2𝑢+𝜇+𝑑₁+𝑑₂1+𝑑₃𝑢𝐪²<0. (19) The corresponding eigenvectors are 𝐫(𝐪)=𝐫₊(𝐪)=𝐫₋(𝐪)=𝜆1,−^𝐪+𝑢+𝑑₁𝐪²𝛽𝑢, (20) and we can find another independent vector 𝐫^′(𝐪)=[0,−1/(𝛽𝑢)], satisfying 𝐿−𝜆_𝐪𝐼𝐫^′(𝐪)=𝐫(𝐪). (21)(3)Δ<0. The complex case is where (12) possesses a pair of complex eigenvalues with a negative real part. Denote Λ₃={𝐪Δ<0}, and for any 𝐪∈Λ₃, denote 𝜆⁺_𝐪=Re𝜆_𝐪+𝑖Im𝜆_𝐪,𝐫₊(𝐪)=Re𝐫(𝐪)+𝑖Im𝐫(𝐪). (22) Then, 𝜆⁻_𝐪=Re𝜆_𝐪−𝑖Im𝜆_𝐪,𝐫₋(𝐪)=Re𝐫(𝐪)−𝑖Im𝐫(𝐪), (23) where Re𝐫(𝐪) and Im𝐫(𝐪) are linearly independent vectors.

Given any initial perturbation 𝐰(𝑥,0), we can expand it as follows: 𝐰(𝑥,0)=_𝐪∈Λ1𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝑥)+_𝐪∈Λ2𝑤_𝐪𝐫(𝐪)+𝑤^′_𝐪𝐫^′(𝐪)𝑒_𝐪(𝑥)+_𝐪∈Λ3𝑤_𝐪^ReRe𝐫(𝐪)+𝑤_𝐪^ImIm𝐫(𝐪)𝑒_𝐪(𝑥)∶=_{𝐪∈ℕ𝑁}𝐰_𝐪𝑒_𝐪(𝑥), (24) where 𝑤⁻_𝐪, 𝑤⁺_𝐪, 𝑤_𝐪, 𝑤^′_𝐪, 𝑤_𝐪^Re, and  𝑤_𝐪^Im are constants, and 𝐰_𝐪=𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪),𝐪∈Λ₁,𝐰_𝐪=𝑤_𝐪𝐫(𝐪)+𝑤^′_𝐪𝐫^′(𝐪),𝐪∈Λ₂,𝐰_𝐪=𝑤_𝐪^ReRe𝐫(𝐪)+𝑤_𝐪^ImIm𝐫(𝐪),𝐪∈Λ₃. (25) The unique solution 𝐰(𝑥,𝑡)=[𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)] to (7) is given by 𝐰(𝑥,𝑡)=_𝐪∈Λ1𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}+𝑤⁺_𝐪𝐫₊(𝐪)𝑒^{𝜆+𝐪𝑡}𝑒_𝐪(𝑥)+_𝐪∈Λ2𝑤_𝐪𝐫(𝐪)+𝑤^′_𝐪𝐫^′(𝐪)+𝑤^′_𝐪𝐫(𝐪)𝑡𝑒^𝜆𝐪𝑡𝑒_𝐪(𝑥)+_𝐪∈Λ3𝑤_𝐪^ReRe𝐫(𝐪)cosIm𝜆_𝐪𝑡−Im𝐫(𝐪)sinIm𝜆_𝐪𝑡+𝑤_𝐪^ImRe𝐫(𝐪)sinIm𝜆_𝐪𝑡+Im𝐫(𝐪)cosIm𝜆_𝐪𝑡×𝑒^{(Re𝜆𝐪)𝑡}𝑒_𝐪(𝑥)∶=𝑒^𝐿𝑡𝐰(𝑥,0). (26)

In the sequel, the constant 𝐶₀ will only depend on the domain 𝕋^𝑁 and the dimension 𝑁, and the generic constants 𝐶₁, 𝐶₂, 𝐶₁, 𝐶₂, and so forth will depend on 𝕋^𝑁, 𝑁, and the parameters 𝜆, 𝛽, 𝑚, 𝜇, 𝑑₁, 𝑑₂,  and  𝑑₃. Our main result of this section is the following lemma.

Lemma 1. Assume that the instability criterion (13) is valid. Suppose that 𝐰(𝑥,𝑡)=[𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)]∶=𝑒^𝐿𝑡𝐰(𝑥,0) (27) is a solution to the linearized system (7) with the initial condition 𝐰(𝑥,0). Then, there exists a constant 𝐶₁≥1, such that ‖𝐰(⋅,𝑡)‖_𝐿²_(𝕋^𝑁₎≤𝐶₁𝑒^𝜆max𝑡‖𝐰(⋅,0)‖_𝐿²_(𝕋^𝑁₎, (28) for all 𝑡≥0.

Proof. We first consider the case for 𝑡>1. For any 𝐪∈Λ₁, det𝐫₋(𝐪),𝐫₊(𝐪=𝜆)^+𝐪−𝜆⁻_𝐪𝛽𝑢=√Δ𝛽𝑢, (29) where Δ is given by (14). Applying Cramer’s rule to (25), we have 𝑤^±_𝐪=det𝐫_∓(𝐪),𝐰_𝐪det𝐫₋(𝐪),𝐫₊(𝐪≤𝐫)^∓(𝐪)×𝐰_𝐪det𝐫₋(𝐪),𝐫₊(𝐪=𝛽𝑢𝐫_∓)(𝐪)×𝐰_𝐪√Δ, (30) where 𝐫_±(𝐪)=1+(𝜆^±_𝐪+𝑢+𝑑₁𝐪²)²(𝛽𝑢)². (31) It follows from (14) that there exist positive constants 𝑀₁ and 𝐶₁=𝑢+𝜇𝑑₁+𝑑₂1+𝑑₃𝑢−2𝑑₁𝜇+𝑑₂1+𝑑₃𝑢𝑢+2𝛽𝑑₂𝑑₃𝑢𝑣,̃ (32) such that Δ>𝐶₁𝐪² for all |𝐪|>𝑀₁. Hence, for any |𝐪|>𝑀₁, 𝑤^±_𝐪≤𝛽𝑢𝐫_∓(𝐪)×𝐰_𝐪𝐶₁|𝐪|, (33) and by (12), lim_|𝐪|→∞𝜆^±_𝐪𝐪²=−𝑑₁,−𝑑₂1+𝑑₃𝑢. (34) Thus, lim_|𝐪|→∞𝐫_±(𝐪)𝐪²=lim_|𝐪|→∞𝜆^±_𝐪+𝑢+𝑑₁𝐪²𝛽𝑢𝐪²=1𝛽𝑢lim_|𝐪|→∞𝜆^±_𝐪𝐪²+𝑑₁≤𝑑₁+𝑑₂1+𝑑₃𝑢𝛽𝑢. (35) Consequently, there exists a positive constant 𝑀₂>𝑀₁, such that 𝐫_±(𝐪≤𝑑)¹+𝑑₂1+𝑑₃𝑢𝛽𝑢+1𝐪² (36) for any |𝐪|>𝑀₂. Substituting this into (33) yields 𝑤^±_𝐪≤𝐶₂|𝐪|𝐰_𝐪 (37) for any |𝐪|>𝑀₂, where 𝐶₂=(𝑑₁+𝑑₂(1+𝑑₃𝑢)+𝛽𝑢)/𝐶₁. We thus obtain 𝑤^±_𝐪𝐫_±(𝐪)𝑒^{𝜆±𝐪𝑡}≤𝐶₂|𝐪|𝐰_𝐪𝐫_±(𝐪)𝑒^{𝜆±𝐪𝑡}. (38) Since lim_|𝐪|→∞|𝐪|𝐫_±(𝐪)𝑒^{𝜆±𝐪𝑡}≤lim_|𝐪|→∞|𝐪|𝜆1+^±𝐪+𝑢+𝑑₁𝐪²²𝛽𝑢²×𝑒^{−min{𝑑1/2,𝑑2(1+𝑑3}𝑢_)/2}𝐪²_𝑡=0, (39) there exists a constant 𝑀₃>𝑀₂, such that 𝑤^±_𝐪𝐫_±(𝐪)𝑒^{𝜆±𝐪𝑡}≤𝐶₂𝐰_𝐪 (40) for any |𝐪|>𝑀₃.
For any 𝐪∈Λ₁ and |𝐪|≤𝑀₃, as Δ is an increasing function of |𝐪|², we denote 𝑀_∗=min{|𝐪|Δ(𝐪)>0}. (41) With the help of (30) and (31), we get 𝑤^±_𝐪𝐫_±(𝐪)𝑒^{𝜆±𝐪𝑡}≤𝐶₃𝐰_𝐪𝑒^𝜆max𝑡, (42) where 𝐶₃ only depends on 𝜆, 𝛽, 𝜇, 𝑚, 𝑀_∗, and 𝑀₃. Hence, we conclude that, for any 𝐪∈Λ₁, there exists a positive constant 𝐶₄=max{𝐶₂,𝐶₃}, such that 𝑤^±_𝐪𝐫_±(𝐪)𝑒^{𝜆±𝐪𝑡}≤𝐶₄𝐰_𝐪𝑒^𝜆max𝑡. (43)
For all 𝐪∈Λ₂ and 𝐪∈Λ₃, by some similar arguments as above we can show that there exist positive constants 𝐶₅ and 𝐶₆, such that 𝑤_𝐪𝐫(𝐪),𝑤^′_𝐪𝐫^′(𝐪),𝑤^′_𝐪𝐫(𝐪)≤𝐶₅𝐰_𝐪,𝑡𝑒^𝜆𝐪𝑡≤𝐶₅,𝑤_𝐪^ReRe𝐫(𝐪),𝑤_𝐪^ReIm𝐫(𝐪),𝑤_𝐪^ImRe𝐫(𝐪),𝑤_𝐪^ImIm𝐫(𝐪)≤𝐶₆𝐰_𝐪. (44)
Next, we derive the energy estimate in 𝐿² for 𝐰(𝑥,𝑡). Recall that {𝑒_𝐪(𝑥)}_𝐪∈ℕ^𝑁 is an orthonormal basis in 𝐿²(𝕋^𝑁). Then, ‖𝐰(⋅,𝑡)‖²_{𝐿2(𝕋𝑁)}=𝐴₁+𝐴₂+𝐴₃, (45) where 𝐴₁=_𝐪∈Λ1{𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}+𝑤⁺_𝐪𝐫₊(𝐪)𝑒^{𝜆+𝐪𝑡}}²,𝐴₂=_𝐪∈Λ2{𝑤_𝐪𝐫(𝐪)+𝑤^′_𝐪𝐫^′(𝐪)+𝑤^′_𝐪𝐫(𝐪)𝑡}²𝑒^{2𝜆𝐪𝑡},𝐴₃=_𝐪∈Λ3𝑤_𝐪^ReRe𝐫(𝐪)cosIm𝜆_𝐪𝑡−Im𝐫(𝐪)sinIm𝜆_𝐪𝑡+𝑤_𝐪^ImRe𝐫(𝐪)sinIm𝜆_𝐪𝑡+Im𝐫(𝐪)cosIm𝜆_𝐪𝑡²𝑒^{2(Re𝜆𝐪)𝑡}. (46) From (43) and (44), we obtain 𝐴₁≤_𝐪∈Λ1{|𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}|+|𝑤⁺_𝐪𝐫₊(𝐪)𝑒^{𝜆+𝐪𝑡}|}²≤4𝐶²_4𝐪∈Λ1|𝐰_𝐪|²𝑒^2𝜆max𝑡,𝐴₂≤_𝐪∈Λ2𝑤_𝐪𝐫(𝐪)+𝑤^′_𝐪𝐫^′(𝐪)+𝑤^′_𝐪𝐫(𝐪)𝑡²𝑒^{2𝜆𝐪𝑡}≤8𝐶²_5𝐪∈Λ2𝐰_𝐪²+𝑤^′_𝐪𝐫(𝐪)²𝑡²𝑒^{2𝜆𝐪𝑡}≤8𝐶²_5𝐪∈Λ2𝐰_𝐪²𝑒^{2𝜆𝐪𝑡}+𝐶²₅𝑤^′_𝐪𝐫(𝐪)²≤8𝐶²_5𝐪∈Λ2𝐰_𝐪²𝑒^{2𝜆𝐪𝑡}+𝐶⁴₅𝐰_𝐪²≤8𝐶²₅1+𝐶⁴_5𝐪∈Λ2|𝐰_𝐪|²𝑒^2𝜆max𝑡,𝐴₃≤_𝐪∈Λ3(|Re𝐫(𝐪)|+|Im𝐫(𝐪)|)𝑤_𝐪^Re+𝑤_𝐪^Im2𝑒^{2(Re𝜆𝐪)𝑡}≤16𝐶²_6𝐪∈Λ3|𝐰_𝐪|²𝑒^{2(Re𝜆𝐪)𝑡}≤16𝐶²_6𝐪∈Λ3|𝐰_𝐪|²𝑒^2𝜆max𝑡. (47) Thus, ‖𝐰(⋅,𝑡)‖²_{𝐿2(𝕋𝑁)}≤𝐶²_7𝐪∈Λ1|𝐰_𝐪|²+_𝐪∈Λ2|𝐰_𝐪|²+_𝐪∈Λ3|𝐰_𝐪|²𝑒^2𝜆max𝑡=𝐶²₇‖𝐰(⋅,0)‖²_{𝐿2(𝕋𝑁)}𝑒^2𝜆max𝑡, (48) where 𝐶²₇=max{4𝐶²₄,8𝐶²₅(1+𝐶⁴₅),16𝐶²₆}. Finally, for any 𝑡>1, we have ‖𝐰(⋅,𝑡)‖_𝐿²_(𝕋^𝑁₎≤𝐶₇‖𝐰(⋅,0)‖_𝐿²_(𝕋^𝑁₎𝑒^𝜆max𝑡. (49)
For finite time 𝑡≤1, we multiply the first and second equations of (7) by 𝑢 and 𝐴𝑣, respectively, then add them and use the integration by parts to get 12𝑑𝑑𝑡_𝕋^𝑁|𝑢|²+𝐴|𝑣|²+𝑑𝑥_𝕋𝑁𝑑₁|∇𝑢|²+𝐴𝑑₂1+𝑑₃𝑢|∇𝑣|²+𝐴𝑑₂𝑑₃̃+𝑣∇𝑢∇𝑣𝑑𝑥_𝕋𝑁𝑢𝑢²+𝐴𝜇𝑣²=𝑑𝑥_𝕋𝑁𝐴𝜇²−𝛽𝑢𝑢𝑣𝑑𝑥. (50) Firstly, we claim that the integrand of the second integral in (50) satisfies 𝑑₁|∇𝑢|²+𝐴𝑑₂1+𝑑₃𝑢|∇𝑣|²+𝐴𝑑₂𝑑₃̃≥1𝑣∇𝑢∇𝑣2𝑑₁|∇𝑢|²+𝐴𝑑₂1+𝑑₃𝑢|∇𝑣|² (51) for some positive constant 𝐴. Obviously, it suffices to require that (𝐴𝑑₂𝑑₃𝑣)̃²−𝐴𝑑₁𝑑₂1+𝑑₃𝑢<0. (52) This is equivalent to 𝑑𝐴<¹1+𝑑₃𝑢𝑑₂𝑑²₃𝑣²̃. (53) Denote 𝐴₀=𝑑₁1+𝑑₃𝑢2𝑑₂𝑑²₃𝑣²̃. (54)
On the other hand, the term on the right of (50) is _𝕋𝑁𝐴𝜇²−𝛽𝑢𝑢𝑣𝑑𝑥≤_𝕋𝑁𝐴𝜇²𝑢|𝑢|⋅|𝑣|≤+𝛽𝑑𝑥_𝕋𝑁𝐴𝜇²𝑢𝑢+𝛽²+𝑣²2=𝑑𝑥𝐴𝜇²+𝛽𝑢2_𝕋^𝑁𝑢²+𝑣²𝑑𝑥. (55) Taking 𝐴=𝐴₀, and substituting (51) and (55) into (50), we get 𝑑𝑑𝑡_𝕋^𝑁|𝑢|²+𝐴₀|𝑣|²𝑑𝑥≤𝐴₀𝜇²+𝛽𝑢_𝕋𝑁𝑢²+𝑣²𝑑𝑥. (56) Integrating (56) from 0 to 𝑡 leads to |𝑢_𝕋𝑁(𝑥,𝑡)|²+𝐴₀|𝑣(𝑥,𝑡)|²≤𝑑𝑥_𝕋𝑁|𝑢(𝑥,0)|²+𝐴₀|𝑣(𝑥,0)|²+𝐴𝑑𝑥₀𝜇²+𝛽𝑢^𝑡_0𝕋𝑁𝑢²+𝑣²𝑑𝑥𝑑𝑡. (57) If 𝐴₀≥1, then it follows from (57) that |𝑢_𝕋𝑁(𝑥,𝑡)|²+|𝑣(𝑥,𝑡)|²𝑑𝑥≤𝐴_0𝕋𝑁|𝑢(𝑥,0)|²+|𝑣(𝑥,0)|²+𝐴𝑑𝑥₀𝜇²+𝛽𝑢^𝑡_0𝕋𝑁𝑢²+𝑣²𝑑𝑥𝑑𝑡; (58) thus, the Gronwall inequality implies ‖𝐰(⋅,𝑡)‖_𝐿²_(𝕋^𝑁₎≤𝐴₀‖𝐰(⋅,0)‖_𝐿²_(𝕋^𝑁₎⋅𝑒^{((𝐴0𝜇2+𝛽}𝑢_)/2)𝑡. (59) Consequently, there exists a positive constant 𝐶₈, such that ‖𝐰(⋅,𝑡)‖_𝐿²_(𝕋^𝑁₎≤𝐶₈𝐴₀‖𝐰(⋅,0)‖_𝐿²_(𝕋^𝑁₎𝑒^𝜆max𝑡 (60) for all 𝑡∈[0,1] due to the boundedness of 𝑒^{((𝐴0𝜇2+𝛽}𝑢_{)/2−𝜆max)𝑡}. (61)
If 0<𝐴₀<1, in the same way as above, there exists a positive constant 𝐶₉, such that ‖𝐰(⋅,𝑡)‖_𝐿²_(𝕋^𝑁₎≤𝐶₉‖𝐰(⋅,0)‖_𝐿²_(𝕋^𝑁₎𝑒^𝜆max𝑡. (62) The proof is completed by taking 𝐶₁=max{𝐶₇,𝐶₈𝐴₀,𝐶₉}.

3. The Estimates for the Solutions of the Full System (5)

The general theory in [21] guarantees that (5) has a unique nonnegative local solution. The results can be summarized as follows.

Lemma 2. Suppose that 𝐰(𝑥,𝑡)=[𝑢,𝑣] is a solution of the full system (5). For 𝑠≥1(𝑁=1) and 𝑠≥2(𝑁=2,3), there exist a 𝑇>0 and a constant 𝐶, such that ‖𝐰(⋅,𝑡)‖_𝐻^𝑠_(𝕋^𝑁₎≤𝐶‖𝐰(⋅,0)‖_𝐻^𝑠_(𝕋^𝑁₎,0≤𝑡<𝑇 (63) if 𝑢₀(𝑥),𝑣₀(𝑥)∈𝐻^𝑠(𝕋^𝑁).

Denote 𝜕_𝑖𝑗𝜕𝑢∶=²𝑢𝜕𝑥_𝑖𝜕𝑥_𝑗,𝜕_𝑖𝑢∶=𝜕𝑢𝜕𝑥_𝑖,𝑖,𝑗=1,…,𝑁. (64) In order to derive the 𝐻² estimate for the solution of (5), we first prove the following energy estimates.

Lemma 3. Suppose that 𝐰(𝑥,𝑡)=[𝑢,𝑣] is a solution of the full system (5). Then, 12𝑑𝑑𝑡^𝑁_{𝑖,𝑗=1𝕋}^𝑁𝜕_𝑖𝑗𝑢²+𝜕_𝑖𝑗𝑣²+𝑑𝑥_{𝑁𝑖,𝑗=1𝕋𝑁}𝑑₁4∇𝜕_𝑖𝑗𝑢²𝐴𝑑𝑥+⁰2𝑑₂1+𝑑₃𝑢∇𝜕_𝑖𝑗𝑣²+𝑢𝑑𝑥_{𝑁𝑖,𝑗=1𝕋𝑁}𝜕_𝑖𝑗𝑢²𝐴𝑑𝑥+⁰𝜇2^𝑁_{𝑖,𝑗=1𝕋}^𝑁𝜕_𝑖𝑗𝑣²≤𝐶𝑑𝑥₂1+𝜂²‖𝐰(⋅,𝑡)‖_𝐻²_(𝕋^𝑁₎∇³𝐰(⋅,𝑡)²_{𝐿2𝕋𝑁}+𝐶₂‖𝑢(⋅,𝑡)‖²_{𝐿2𝕋𝑁} (65) for ‖𝐰‖_𝐻²_(𝕋^𝑁₎≤𝜂.

Proof. We first notice that system (5) preserves the evenness of the solution; that is, if 𝐰(𝑥₁,𝑥₂,𝑥₃,𝑡) is a solution to (5), then 𝐰(−𝑥₁,𝑥₂,𝑥₃,𝑡), 𝐰(𝑥₁,−𝑥₂,𝑥₃,𝑡), and 𝐰(𝑥₁,𝑥₂,−𝑥₃,𝑡) are also solutions of (5). We can regard system (5) as a special case with the evenness of the periodic problem by a reflective and an even extension. For this reason, we may assume periodicity at the boundary of the extended 2𝕋^𝑁=(−𝜋,𝜋)^𝑁. Taking the second order partial derivative of the first equation of (5), multiplying 𝜕_𝑖𝑗𝑢, and integrating over the domain 2𝕋^𝑁 to get 12𝑑𝑑𝑡_2𝕋^𝑁𝜕_𝑖𝑗𝑢²=𝑑𝑥_2𝕋𝑁𝜕_𝑖𝑗𝑑₁Δ𝑢+𝑔₁(𝑢+𝑢,𝑣+𝑣)̃𝜕_𝑖𝑗𝑢𝑑𝑥=_2𝕋𝑁𝜕_𝑖𝑗𝐿₁(𝑢,𝑣)+𝑁₁(𝑢,𝑣)𝜕_𝑖𝑗𝑢𝑑𝑥, (66) where 𝐿₁(𝑢,𝑣)=𝑑₁𝑁Δ𝑢−𝑢𝑢−𝛽𝑢𝑣,₁(𝑢,𝑣)=𝑔₁(𝑢+𝑢,𝑣+𝑣)̃+𝑢𝑢+𝛽𝑢𝑣=−𝑢²−𝛽𝑢𝑣 (67)(68) are the linear and nonlinear terms, respectively, then, we have _2𝕋𝑁𝜕_𝑖𝑗𝐿₁(𝑢,𝑣)𝜕_𝑖𝑗𝑢𝑑𝑥=−𝑑_12𝕋𝑁∇𝜕_𝑖𝑗𝑢²𝑑𝑥−𝑢_2𝕋𝑁𝜕_𝑖𝑗𝑢²𝑑𝑥−𝛽𝑢_2𝕋𝑁𝜕_𝑖𝑗𝑢𝜕_𝑖𝑗𝑣𝑑𝑥,_2𝕋𝑁𝜕_𝑖𝑗𝑁₁(𝑢,𝑣)𝜕_𝑖𝑗𝑢𝑑𝑥=−_2𝕋𝑁𝜕_𝑖𝑗𝑢²+𝛽𝑢𝑣𝜕_𝑖𝑗𝑢𝑑𝑥,=−_2𝕋𝑁2𝜕_𝑖𝑢𝜕_𝑗𝑢+2𝑢𝜕_𝑖𝑗𝑢+𝛽𝑣𝜕_𝑖𝑗𝑢+𝛽𝜕_𝑖𝑢𝜕_𝑗𝑣+𝛽𝜕_𝑗𝑢𝜕_𝑖𝑣+𝛽𝑢𝜕_𝑖𝑗𝑣𝜕_𝑖𝑗𝑢𝑑𝑥=−_2𝕋𝑁2𝑢+𝛽𝑣𝜕_𝑖𝑗𝑢+𝛽𝑢𝜕_𝑖𝑗𝑣+2𝜕_𝑖𝑢𝜕_𝑗𝑢+𝛽𝜕_𝑖𝑢𝜕_𝑗𝑣+𝛽𝜕_𝑗𝑢𝜕_𝑖𝑣𝜕_𝑖𝑗≤‖𝐰‖𝑢𝑑𝑥2+2𝛽_𝐿^∞_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰²_{𝐿22𝕋𝑁}+2+2𝛽‖∇𝐰‖_𝐿^∞_(2𝕋^𝑁₎‖∇𝐰‖_𝐿²_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰_{𝐿2(2𝕋𝑁)}. (69) Similarly, taking the second order partial derivative of the second equation of (5), multiplying 𝜕_𝑖𝑗𝑣, and integrating over the domain 2𝕋^𝑁 to get 12𝑑𝑑𝑡_2𝕋^𝑁𝜕_𝑖𝑗𝑣²=𝑑𝑥_2𝕋𝑁𝜕_𝑖𝑗𝑑₂Δ1+𝑑₃(𝑢+𝑢)(𝑣+𝑣)̃+𝑔₂(𝑢+𝑢,𝑣+𝑣)̃𝜕_𝑖𝑗=𝑣𝑑𝑥_2𝕋𝑁𝜕_𝑖𝑗𝐿₂(𝑢,𝑣)+𝑁₂(𝑢,𝑣)𝜕_𝑖𝑗𝑣𝑑𝑥, (70) where 𝐿₂(𝑢,𝑣)=𝑑₂𝑑₃𝑣Δ̃𝑢+𝑑₂1+𝑑₃𝑢Δ𝑣+𝜇²𝑁𝑢−𝜇𝑣,₂(𝑢,𝑣)=𝑑₂Δ1+𝑑₃(𝑢+𝑢)(𝑣+𝑣)̃+𝑔₂(𝑢+𝑢,𝑣+𝑣)̃−𝐿₂(𝑢,𝑣)=𝑑₂𝑑₃(𝑣Δ𝑢+2∇𝑢∇𝑣+𝑢Δ𝑣)−𝜇𝑢−𝑣²𝑚+𝑢+𝑢∶=𝑁₂⁽¹⁾(𝑢,𝑣)+𝑁₂⁽²⁾(𝑢,𝑣), (71)(72) thus, _2𝕋𝑁𝜕_𝑖𝑗𝐿₂(𝑢,𝑣)𝜕_𝑖𝑗𝑣𝑑𝑥=−𝑑₂𝑑₃𝑣̃_2𝕋𝑁∇𝜕_𝑖𝑗𝑢∇𝜕_𝑖𝑗𝑣𝑑𝑥−𝑑₂1+𝑑₃𝑢_2𝕋𝑁∇𝜕_𝑖𝑗𝑣²𝑑𝑥+𝜇²_2𝕋𝑁𝜕_𝑖𝑗𝑢𝜕_𝑖𝑗𝑣𝑑𝑥−𝜇_2𝕋𝑁|𝜕_𝑖𝑗𝑣|²𝑑𝑥,_2𝕋𝑁𝜕_𝑖𝑗𝑁₂⁽¹⁾(𝑢,𝑣)𝜕_𝑖𝑗𝑣𝑑𝑥=𝑑₂𝑑_32𝕋𝑁𝜕_𝑖𝑗(𝑣Δ𝑢+2∇𝑢∇𝑣+𝑢Δ𝑣)𝜕_𝑖𝑗𝑣𝑑𝑥=𝑑₂𝑑_32𝕋𝑁𝜕_𝑖𝑗𝑣Δ𝑢+𝑣Δ𝜕_𝑖𝑗𝑢+𝜕_𝑖𝑣Δ𝜕_𝑗𝑢+𝜕_𝑗𝑣Δ𝜕_𝑖𝑢𝜕_𝑖𝑗𝑣𝑑𝑥+2𝑑₂𝑑_32𝕋𝑁∇𝜕_𝑖𝑗𝑢∇𝑣+∇𝑢∇𝜕_𝑖𝑗𝑣+∇𝜕_𝑖𝑢∇𝜕_𝑗𝑣+∇𝜕_𝑗𝑢∇𝜕_𝑖𝑣𝜕_𝑖𝑗𝑣𝑑𝑥+𝑑₂𝑑_32𝕋𝑁𝜕_𝑖𝑗𝑢Δ𝑣+𝑢Δ𝜕_𝑖𝑗𝑣+𝜕_𝑖𝑢Δ𝜕_𝑗𝑣+𝜕_𝑗𝑢Δ𝜕_𝑖𝑣𝜕_𝑖𝑗𝑣𝑑𝑥=−𝑑₂𝑑_32𝕋𝑁𝜕_𝑖𝑗𝑣∇𝜕_𝑖𝑗𝑣∇𝑢𝑑𝑥−𝑑₂𝑑_32𝕋𝑁∇𝑣𝜕_𝑖𝑗𝑢∇𝜕_𝑖𝑗𝑣𝑑𝑥−𝑑₂𝑑_32𝕋𝑁𝑣∇𝜕_𝑖𝑗𝑣∇𝜕_𝑖𝑗𝑢𝑑𝑥+𝑑₂𝑑₃×_2𝕋𝑁𝜕_𝑖𝑣Δ𝜕_𝑗𝑢+𝜕_𝑗𝑣Δ𝜕_𝑖𝑢𝜕_𝑖𝑗𝑣𝑑𝑥−2𝑑₂𝑑_32𝕋𝑁Δ𝜕_𝑖𝑢𝜕_𝑖𝑗𝑣𝜕_𝑗𝑣+∇𝜕_𝑖𝑢∇𝜕_𝑖𝑗𝑣𝜕_𝑗𝑣+Δ𝜕_𝑗𝑢𝜕_𝑖𝑗𝑣𝜕_𝑖𝑣+∇𝜕_𝑗𝑢∇𝜕_𝑖𝑗𝑣𝜕_𝑖𝑣𝑑𝑥−𝑑₂𝑑_32𝕋𝑁𝑢|∇𝜕_𝑖𝑗𝑣|²𝑑𝑥+𝑑₂𝑑₃×_2𝕋𝑁𝜕_𝑖𝑢Δ𝜕_𝑗𝑣+𝜕_𝑗𝑢Δ𝜕_𝑖𝑣𝜕_𝑖𝑗𝑣𝑑𝑥≤14𝑑₂𝑑₃‖∇𝐰‖_𝐿^∞_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰_{𝐿2(2𝕋𝑁)}∇³𝐰_{𝐿2(2𝕋𝑁)}+2𝑑₂𝑑₃‖𝐰‖_𝐿^∞_(2𝕋^𝑁₎∇³𝐰²_{𝐿22𝕋𝑁},_2𝕋𝑁𝜕_𝑖𝑗𝑁₂⁽²⁾(𝑢,𝑣)𝜕_𝑖𝑗𝑣𝑑𝑥=−_2𝕋𝑁𝜕_𝑖𝑗𝜇𝑢−𝑣²𝑢𝜕𝑚+𝑢+^𝑖𝑗=𝑣𝑑𝑥_2𝕋𝑁−2𝜇𝑢−𝑣𝜇𝜕_𝑖𝑗𝑢−𝜕_𝑖𝑗𝑣+2𝜇𝜕_𝑖𝑢−𝜕_𝑖𝑣𝜇𝜕_𝑗𝑢−𝜕_𝑗𝑣×(𝑚+𝑢+𝑢)⁻¹−𝜇𝑢−𝑣²2𝜕_𝑖𝑢𝜕_𝑗𝑢(𝑚+𝑢+𝑢)³−𝜕_𝑖𝑗𝑢(𝑚+𝑢+𝑢)²+2𝜇𝑢−𝑣𝜇𝜕_𝑖𝑢−𝜕_𝑖𝑣𝜕_𝑗𝑢+𝜇𝜕_𝑗𝑢−𝜕_𝑗𝑣𝜕_𝑖𝑢×(𝑚+𝑢+𝑢)⁻²𝜕_𝑖𝑗≤𝑣𝑑𝑥2(𝜇+1)²𝑚‖𝐰‖_𝐿^∞_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰²_{𝐿22𝕋𝑁}+2(𝜇+1)²𝑚‖∇𝐰‖_𝐿^∞_(2𝕋^𝑁₎‖∇𝐰‖_𝐿²_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰_{𝐿2(2𝕋𝑁)}+2𝜇+1²𝑚³‖𝐰‖²_{𝐿∞2𝕋𝑁}‖∇𝐰‖_𝐿^∞_(2𝕋^𝑁₎×‖∇𝐰‖_𝐿²_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰_{𝐿2(2𝕋𝑁)}+(𝜇+1)²𝑚²‖𝐰‖²_{𝐿∞2𝕋𝑁}𝜕_𝑖𝑗𝐰²_{𝐿22𝕋𝑁}+4𝜇+1²𝑚²‖𝐰‖_𝐿^∞_(2𝕋^𝑁₎‖∇𝐰‖_𝐿^∞_(2𝕋^𝑁₎×‖∇𝐰‖_𝐿²_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰_{𝐿2(2𝕋𝑁)}≤(𝜇+1)²𝑚‖𝐰‖2+^{𝐿∞(2𝕋𝑁)}𝑚‖𝐰‖_𝐿^∞_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰²_{𝐿22𝕋𝑁}+2𝜇+1²𝑚‖𝐰‖1+^{𝐿∞(2𝕋𝑁)}𝑚²‖∇𝐰‖_𝐿^∞_(2𝕋^𝑁₎×‖∇𝐰‖_𝐿²_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰_{𝐿2(2𝕋𝑁)}. (73) Substituting (70) ×𝐴₀+ (66) into (69) and (73) to get 12𝑑𝑑𝑡_2𝕋^𝑁𝜕_𝑖𝑗𝑢²+𝐴₀𝜕_𝑖𝑗𝑣²+𝑑𝑥_2𝕋𝑁𝑑₁∇𝜕_𝑖𝑗𝑢²+𝐴₀𝑑₂𝑑₃𝑣∇̃𝜕_𝑖𝑗𝑢∇𝜕_𝑖𝑗𝑣+𝐴₀𝑑₂1+𝑑₃𝑢∇𝜕_𝑖𝑗𝑣²+𝑢𝑑𝑥_2𝕋𝑁𝜕_𝑖𝑗𝑢²𝑑𝑥+𝐴₀𝜇_2𝕋𝑁𝜕_𝑖𝑗𝑣²𝑑𝑥≤𝐼₁+𝐼₂, (74) where 𝐼₁=𝐶₁₀1+𝜂‖𝐰‖_𝐿^∞_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰²_{𝐿22𝕋𝑁}+𝐶₁₁1+𝜂²‖∇𝐰‖_𝐿^∞_(2𝕋^𝑁₎‖∇𝐰‖_𝐿²_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰_{𝐿2(2𝕋𝑁)}+𝐶₁₂‖∇𝐰‖_𝐿^∞_(2𝕋^𝑁₎𝜕_𝑖𝑗𝐰_{𝐿2(2𝕋𝑁)}∇³𝐰_{𝐿2(2𝕋𝑁)}+𝐶₁₃‖𝐰‖_𝐿^∞_(2𝕋^𝑁₎∇³𝐰²_{𝐿22𝕋𝑁},𝐼₂=𝐴₀𝜇²−𝛽𝑢_2𝕋𝑁𝜕_𝑖𝑗𝑢𝜕_𝑖𝑗𝑣𝑑𝑥, (75) we apply the Sobolev imbedding to control the 𝐿^∞ norm by 𝑔_{𝐿∞(2𝕋𝑁)}≤𝐶₀𝑔_{𝐻2(2𝕋𝑁)} (76) for 𝑁≤3. From the Hölder inequality, the Poincaré inequality, and the Sobolev imbedding theorem, it follows that 𝑔_{𝐿2(2𝕋𝑁)}≤(2𝜋)^𝑁/4𝑔_{𝐿4(2𝕋𝑁)}≤𝐶₀∇𝑔_{𝐿2(2𝕋𝑁)} (77) for ∫_2𝕋𝑁𝑔𝑑𝑥=0. Recall the even extension of (5), and the solution [𝑢,𝑣] satisfies _2𝕋𝑁∇𝑢𝑑𝑥=_2𝕋𝑁∇𝑣𝑑𝑥=0,_2𝕋𝑁𝜕_𝑖𝑗𝑢𝑑𝑥=_2𝕋𝑁𝜕_𝑖𝑗𝑣𝑑𝑥=0,𝑖,𝑗=1,…,𝑁. (78) By (76) and (77), we find that ‖∇𝐰‖_𝐿^∞_(2𝕋^𝑁₎≤𝐶₀‖∇𝐰‖_𝐻²_(2𝕋^𝑁₎≤𝐶₀∇³𝐰_{𝐿2(2𝕋𝑁)}, (79) where 𝐶₀ is a universal constant. Therefore, when 𝑁≤3, it follows from (76) and (79) that 𝐼₁≤𝐶₁₄(1+𝜂)²‖𝐰‖_𝐻²_(2𝕋^𝑁₎∇³𝐰²_{𝐿22𝕋𝑁}. (80) Applying the Young inequality to get 𝐼₂≤𝐴₀𝜇²+𝛽𝑢²2𝐴₀𝜇_2𝕋^𝑁𝜕_𝑖𝑗𝑢²𝐴𝑑𝑥+⁰𝜇2_2𝕋^𝑁|𝜕_𝑖𝑗𝑣|²𝑑𝑥, (81) which is combined with the interpolation inequality and the 𝜀-Young inequality to imply _2𝕋𝑁|𝜕_𝑖𝑗𝑢|²𝑑𝑥≤𝐶₀𝑎∇𝜕_𝑖𝑗𝑢²_{𝐿22𝕋𝑁}+14𝑎²‖𝑢‖²_{𝐿22𝕋𝑁}, (82) where 𝑎 is a positive constant, in the same way as above, we obtain that the second integral satisfies _2𝕋𝑁𝑑₁∇𝜕_𝑖𝑗𝑢²+𝐴₀𝑑₂𝑑₃𝑣∇̃𝜕_𝑖𝑗𝑢∇𝜕_𝑖𝑗𝑣+𝐴₀𝑑₂1+𝑑₃𝑢∇𝜕_𝑖𝑗𝑣²≥𝑑𝑑𝑥₁2∇𝜕_𝑖𝑗𝑢²+𝐴₀𝑑₂1+𝑑₃𝑢2∇𝜕_𝑖𝑗𝑣²𝑑𝑥. (83) Substituting (80)–(83) into (74), take ((𝐴₀𝜇²+𝛽𝑢)²/2𝐴₀𝜇)𝐶₀𝑎=𝑑₁/4 to get 12𝑑𝑑𝑡_2𝕋^𝑁𝜕_𝑖𝑗𝑢²+𝐴₀𝜕_𝑖𝑗𝑣²+𝑑𝑥_2𝕋𝑁𝑑₁4∇𝜕_𝑖𝑗𝑢²+𝐴₀2𝑑₂1+𝑑₃𝑢∇𝜕_𝑖𝑗𝑣²+𝑢𝑑𝑥_2𝕋𝑁𝜕_𝑖𝑗𝑢²𝐴𝑑𝑥+⁰𝜇2_2𝕋^𝑁𝜕_𝑖𝑗𝑣²𝑑𝑥≤𝐶₁₄1+𝜂²‖𝐰‖_𝐻²_(2𝕋^𝑁₎∇³𝐰²_{𝐿22𝕋𝑁}+𝐶₁₅‖𝑢‖²_{𝐿22𝕋𝑁}. (84)
Similar as the proof of Lemma 1, we proceed in the two cases: 𝐴₀≥1 and 0<𝐴₀<1. Then, we conclude 12𝑑𝑑𝑡_2𝕋^𝑁𝜕_𝑖𝑗𝑢²+𝜕_𝑖𝑗𝑣²+𝑑𝑥_2𝕋𝑁𝑑₁4∇𝜕_𝑖𝑗𝑢²+𝐴₀2𝑑₂1+𝑑₃𝑢∇𝜕_𝑖𝑗𝑣²+𝑢𝑑𝑥_2𝕋𝑁𝜕_𝑖𝑗𝑢²𝐴𝑑𝑥+⁰𝜇2_2𝕋^𝑁𝜕_𝑖𝑗𝑣²𝑑𝑥≤𝐶₁₆1+𝜂²‖𝐰‖_𝐻²_(2𝕋^𝑁₎∇³𝐰²_{𝐿22𝕋𝑁}+𝐶₁₇‖𝑢‖²_{𝐿22𝕋𝑁}. (85) So, the even extension implies 12𝑑𝑑𝑡^𝑁_{𝑖,𝑗=1𝕋}^𝑁𝜕_𝑖𝑗𝑢²+𝜕_𝑖𝑗𝑣²+𝑑𝑥_{𝑁𝑖,𝑗=1𝕋𝑁}𝑑₁4∇𝜕_𝑖𝑗𝑢²+𝐴₀2𝑑₂1+𝑑₃𝑢∇𝜕_𝑖𝑗𝑣²+𝑢𝑑𝑥_{𝑁𝑖,𝑗=1𝕋𝑁}𝜕_𝑖𝑗𝑢²𝐴𝑑𝑥+⁰𝜇2^𝑁_{𝑖,𝑗=1𝕋}^𝑁𝜕_𝑖𝑗𝑣²≤𝐶𝑑𝑥₂(1+𝜂)²‖𝐰‖_𝐻²_(𝕋^𝑁₎∇³𝐰²_{𝐿2𝕋𝑁}+𝐶₂‖𝑢‖²_{𝐿2𝕋𝑁}. (86)

Next, we control the 𝐻² growth of 𝐰(𝑥,𝑡) in terms of its 𝐿² growth.

Lemma 4. Suppose that 𝐰(𝑥,𝑡) is a solution of the full system (5), such that ‖𝐰(⋅,𝑡)‖_𝐻²_(𝕋^𝑁₎𝑑≤min𝜂,¹4𝐶₂1+𝜂²,𝐴₀𝑑₂1+𝑑₃𝑢2𝐶₂1+𝜂²,‖𝐰(⋅,𝑡)‖_𝐿²_(𝕋^𝑁₎≤2𝐶₁𝑒^𝜆max𝑡‖𝐰(⋅,0)‖_𝐿²_(𝕋^𝑁₎. (87)(88) Then, ‖𝐰(⋅,𝑡)‖²_{𝐻2𝕋𝑁}≤𝐶₃‖𝐰(⋅,0)‖²_{𝐻2𝕋𝑁}+𝑒^2𝜆max𝑡‖𝐰(⋅,0)‖²_{𝐿2𝕋𝑁}, (89) where 𝐶₃=2max{4𝐶²₁,4(𝐶₀+1)𝐶²₁𝐶₂/𝜆_max,(𝐶₀+1)/2}.

Proof. We first consider the second-order derivatives of 𝐰(𝑥,𝑡). From Lemmas 3 and 4 and our assumption for 𝐰(𝑥,𝑡), we have 12𝑑𝑑𝑡^𝑁_{𝑖,𝑗=1𝕋}^𝑁𝜕_𝑖𝑗𝑢²+𝜕_𝑖𝑗𝑣²𝑑𝑥≤𝐶₂‖𝑢‖²_{𝐿2𝕋𝑁}≤4𝐶²₁𝐶₂𝑒^2𝜆max𝑡‖𝐰(⋅,0)‖²_{𝐿2𝕋𝑁}. (90) By an integration from 0 to 𝑡, we deduce that ^𝑁_{𝑖,𝑗=1𝕋𝑁}𝜕_𝑖𝑗𝑢(𝑥,𝑡)²+𝜕_𝑖𝑗𝑣(𝑥,𝑡)²≤𝑑𝑥_{𝑁𝑖,𝑗=1𝕋𝑁}𝜕_𝑖𝑗𝑢(𝑥,0)²+𝜕_𝑖𝑗𝑣(𝑥,0)²+4𝐶𝑑𝑥₂₁𝐶₂𝜆_max𝑒^2𝜆max𝑡‖𝐰(⋅,0)‖²_{𝐿2𝕋𝑁}. (91) For the first derivations of 𝐰(𝑥,𝑡), we apply the Poincaré inequality to get ‖∇𝐰‖_𝐿²_(2𝕋^𝑁₎≤𝐶₀∇²𝐰_{𝐿2(2𝕋𝑁)}. (92) Applying the even extension, we have ‖∇𝐰‖_𝐿²_(𝕋^𝑁₎≤𝐶₀∇²𝐰_{𝐿2(𝕋𝑁)}. (93) It follows from (88)–(93) that ‖𝐰(⋅,𝐭)‖²_{𝐻2𝕋𝑁}=‖𝐰(⋅,𝐭)‖²_{𝐿2𝕋𝑁}+‖∇𝐰‖²_{𝐿2𝕋𝑁}+∇²𝐰²_{𝐿2𝕋𝑁}≤‖𝐰(⋅,𝐭)‖²_{𝐿2𝕋𝑁}+𝐶₀+1∇²𝐰²_{𝐿2𝕋𝑁}≤4𝐶²₁𝑒^2𝜆max𝑡‖𝐰(⋅,𝟎)‖²_{𝐿2𝕋𝑁}+𝐶₀+1‖𝐰(⋅,𝟎)‖²_{𝐻2𝕋𝑁}+4𝐶₀+1𝐶²₁𝐶₂𝜆_max𝑒^2𝜆max𝑡‖𝐰(⋅,0)‖²_{𝐿2𝕋𝑁}≤𝐶₃‖𝐰(⋅,𝟎)‖²_{𝐻2𝕋𝑁}+𝑒^2𝜆max𝑡‖𝐰(⋅,𝟎)‖²_{𝐿2𝕋𝑁}, (94) where 𝐶₃=2max{4𝐶²₁,4(𝐶₀+1)𝐶²₁𝐶₂/𝜆_max,(𝐶₀+1)/2}.

4. Nonlinear Instability and Pattern Formation

Let 𝜃 be a fixed constant. For an arbitrary positive number 𝛿, we define the escape time 𝑇^𝛿 by 𝜃=𝛿𝑒^{𝜆max𝑇𝛿} (95) or equivalently 𝑇^𝛿=1𝜆_max𝜃ln𝛿. (96)

Theorem 5. Assume that there exists a 𝐪, such that the instability criterion (13) holds. Let 𝐰₀(𝑥)=_𝐪∈Λ1𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝑥)+_𝐪∈Λ2𝑤_𝐪𝐫(𝐪)+𝑤^′_𝐪𝐫^′(𝐪)𝑒_𝐪(𝑥)+_𝐪∈Λ3𝑤_𝐪^ReRe𝐫(𝐪)+𝑤_𝐪^ImIm𝐫(𝐪)𝑒_𝐪(𝑥), (97) such that 𝐰₀(𝑥)∈𝐻²(𝕋^𝑁), ‖𝐰_𝟎‖_𝐿2(𝕋^𝑁₎=1. Then, there exist 𝜃₀>0, 𝛿₀>0, and 𝐶>0, such that for all 𝜃∈(0,𝜃₀) and 𝛿∈(0,𝛿₀], the nonlinear evolution 𝐰^𝛿(𝑥,𝑡) of (5) with the initial perturbation 𝐰^𝛿(𝑥,0)=𝛿𝐰₀(𝑥) satisfies 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Λmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝑥)_{𝐿2(𝕋𝑁)}≤𝐶𝑒^−𝜎𝑡+𝛿𝐰_𝟎²_{𝐻2𝕋𝑁}+𝛿𝑒^𝜆max𝑡𝛿𝑒^𝜆max𝑡,∀𝑡∈0,𝑇^𝛿, (98) where the constant 𝜎, called the gap between 𝜆_max and the rest of eigenvalues, is positive, and 𝛿₀1=min2𝐶₃𝐰_{𝟎𝐻2(𝑇𝑁)},𝑑min𝜂,¹4𝐶²1+𝜂²,𝐴₀𝑑₂1+𝑑₃𝑢2𝐶₁1+𝜂²,𝜆_max4𝐶₃𝐶₄𝐰_𝟎²_{𝐻2𝑇𝑁},𝜃₀𝜆=min^max4𝐶₃𝐶₄,𝜂2𝐶₃,𝑑₁8𝐶₂𝐶₃1+𝜂²,𝐴₀𝑑₂1+𝑑₃𝑢4𝐶₂𝐶₃1+𝜂². (99)

Remark 6. First, we notice that the part of the fastest growing modes 𝛿𝑒^𝜆max𝑡_𝐪∈Λmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝑥) (100) in (98) is nontrivial. Recall that the initial profile 𝐰₀(𝑥) is any 𝐻² function satisfying ‖𝐰_𝟎‖_𝐿2(𝕋^𝑁₎=1. Thus, we can choose 𝐰₀(𝑥), such that there is at least one 𝐪_𝟎∈Λ_max with 𝑤⁺_𝐪𝟎≠0. Consequently, 𝛿𝑒^𝜆max𝑡_𝐪∈Λmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝑥)_{𝐿2(𝕋𝑁)}≥𝛿𝑒^𝜆max𝑡𝑤⁺_𝐪𝟎𝐫₊𝐪_𝟎>0. (101)

Remark 7. Fix 𝜃 to be a positive small number. If 𝛿 is small sufficiently, then 𝑇^𝛿 is large, and for 0≤𝑡≤𝑇^𝛿, 𝛿𝑒^𝜆max𝑡≤𝜃. Our estimate (98) implies that the dynamics of a general perturbation can be characterized by such linear dynamics over a long time period (0,𝑇^𝛿], when the initial perturbation is small.

Remark 8. In particular, if we take 𝐰₀(𝑥)=𝐫₊𝐪_𝟎𝐫₊𝐪_𝟎𝑒_𝐪𝟎(𝑥) (102) in Remark 6, then, at the time 𝑡=𝑇^𝛿, the estimate (98) gives 𝐰^𝛿⋅,𝑇^𝛿−𝛿𝑒^{𝜆max𝑇𝛿}𝐫₊𝐪_𝟎𝐫₊𝐪_𝟎𝑒_𝐪𝟎(⋅)_{𝐿2(𝕋𝑁)}≤𝐶𝑒^{−𝜎𝑇𝛿}+𝛿𝐰_𝟎²_{𝐻2𝕋𝑁}𝜃+𝜃𝜃,=𝐶^{1−𝜎/𝜆max}𝛿^𝜎/𝜆max+𝛿𝜃𝐰_𝟎²_{𝐻2𝕋𝑁}+𝜃². (103) Therefore, 𝐰^𝛿⋅,𝑇^𝛿≥𝜃−𝐶𝜃^{1−𝜎/𝜆max}𝛿^𝜎/𝜆max+𝛿𝜃𝐰_𝟎²_{𝐻2𝕋𝑁}+𝜃². (104) For 𝜃 sufficiently small, ‖𝐰^𝛿(.,𝑇^𝛿)‖≥𝜃/2 as 𝛿→0, which implies that nonlinear instability occurs.

Remark 9. From a view of pattern formation, Theorem 5 shows that if the unique positive equilibrium (𝑢,𝑣)̃ of (2) is linear unstable, then a general small perturbation near (𝑢,𝑣)̃ can induce pattern formation. Furthermore, the patterns can be characterized by the fastest growing modes in the corresponding linear dynamics over a long time period (0,𝑇^𝛿].

Proof of Theorem 5. Define 𝑇^∗=sup𝑡∣𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝐿𝑡𝐰_{0𝐿2(𝕋𝑁)}≤𝐶₁2𝛿𝑒^𝜆max𝑡,𝑇^∗∗=sup𝑡∣𝐰^𝛿(⋅,𝑡)_{𝐻2(𝕋𝑁)}𝑑≤min𝜂,¹4𝐶₂(1+𝜂)²,𝐴₀𝑑₂1+𝑑₃𝑢2𝐶₂(1+𝜂)². (105)(106) Now, we proceed in the following four steps.
Step  1. We establish 𝐻² estimate for the solution 𝐰^𝛿(𝑥,𝑡) of 0≤𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}.
From Lemma 1, for any 𝑡≥0, we have 𝛿𝑒^𝐿𝑡𝐰_{0𝐿2(𝕋𝑁)}≤𝛿𝐶₁𝑒^𝜆max𝑡. (107) By the definition of 𝑇^∗, for any 0≤𝑡≤𝑇^∗, it follows that 𝐰^𝛿(⋅,𝑡)_{𝐿2(𝕋𝑁)}−𝛿𝑒^𝐿𝑡𝐰_{0𝐿2(𝕋𝑁)}≤𝐶₁2𝛿𝑒^𝜆max𝑡. (108) Substituting (107) into (108), we obtain 𝐰^𝛿(⋅,𝑡)_{𝐿2(𝕋𝑁)}≤3𝐶₁2𝛿𝑒^𝜆max𝑡. (109) Furthermore, it follows from (109) and Lemma 4 that 𝐰^𝛿(⋅,𝑡)_{𝐻2(𝕋𝑁)}≤𝐶₃⋅𝛿²𝐰₀²_{𝐻2𝕋𝑁}+𝛿²𝑒^2𝜆max𝑡≤𝐶₃⋅𝛿𝐰_{0𝐻2(𝕋𝑁)}+𝛿𝑒^𝜆max𝑡 (110)(111) for any 𝑡≤𝑇^∗∗.
Step  2. We establish 𝐿² estimate for 𝐰^𝛿(𝑥,𝑡) of 0≤𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}. Applying Duhamel’s principle, we have 𝐰^𝛿(𝑥,𝑡)=𝛿𝑒^𝐿𝑡𝐰₀−^𝑡₀𝑒^{𝐿(𝑡−𝜏)}𝑁𝐰^𝛿(𝑥,𝜏)𝑑𝜏, (112) where 𝑁[𝐰^𝛿(𝑥,𝜏)]=(𝑁₁(𝐰^𝛿(𝑥,𝜏)),𝑁₂(𝐰^𝛿(𝑥,𝜏))), 𝑁₁(𝐰^𝛿(𝑥,𝜏)), and 𝑁₂(𝐰^𝛿(𝑥,𝜏)) are given by (68) and (72). Using Lemma 1, it follows that 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝐿𝑡𝐰_{0𝐿2(𝕋𝑁)}=^𝑡₀𝑒^{𝐿(𝑡−𝜏)}𝑁𝐰^𝛿(𝑥,𝜏)𝑑𝜏_{𝐿2(𝕋𝑁)}≤^𝑡₀𝑒^{𝐿(𝑡−𝜏)}𝑁𝐰^𝛿(𝑥,𝜏)_{𝐿2(𝕋𝑁)}≤𝐶𝑑𝜏_1𝑡0𝑒^{𝜆max(𝑡−𝜏)}𝑁𝐰^𝛿(𝑥,𝜏)_{𝐿2(𝕋𝑁)}𝑑𝜏,𝑁(𝐰^𝛿(𝑥,𝜏))_{𝐿2(𝕋𝑁)}≤𝑁₁𝐰^𝛿(𝑥,𝜏)_{𝐿2(𝕋𝑁)}+𝑁₂(𝐰^𝛿(𝑥,𝜏))_{𝐿2(𝕋𝑁)}=−(𝑢^𝛿)²−𝛽𝑢^𝛿𝑣^𝛿_{𝐿2(𝕋𝑁)}+𝑑₂𝑑₃𝑣^𝛿Δ𝑢^𝛿+2𝑑₂𝑑₃∇𝑢^𝛿∇𝑣^𝛿+𝑑₂𝑑₃𝑢^𝛿Δ𝑣^𝛿−𝜇𝑢^𝛿−𝑣^𝛿2𝑚+𝑢^𝛿+𝑢_{𝐿2(𝕋𝑁)}≤𝛽+1𝐰^𝛿_{𝐿∞(𝕋𝑁)}𝐰^𝛿_{𝐿2(𝕋𝑁)}+2𝑑₂𝑑₃𝐰^𝛿_{𝐿∞(𝕋𝑁)}×𝐰^𝛿_{𝐻2(𝕋𝑁)}+2𝑑₂𝑑₃∇𝐰^𝛿2_{𝐿4(𝕋𝑁)}+(𝜇+1)²𝑚𝐰^𝛿_{𝐿∞(𝕋𝑁)}𝐰^𝛿_{𝐿2(𝕋𝑁)}≤𝐶₄𝐰^𝛿2_{𝐻2(𝕋𝑁)}, (113) where 𝐶₄=𝛽+1+4𝑑₂𝑑₃+(𝜇+1)²/𝑚. Therefore, 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝐿𝑡𝐰_{0𝐿2(𝕋𝑁)}≤𝐶₁𝐶₄^𝑡₀𝑒^{𝜆max(𝑡−𝜏)}‖𝐰^𝛿(𝜏)‖²_𝐻²_𝕋^𝑁𝑑𝜏. (114) Substituting (110) into (114), we obtain 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝐿𝑡𝐰_{0𝐿2(𝕋𝑁)}≤𝐶₁𝐶₃𝐶₄^𝑡₀𝑒^{𝜆max(𝑡−𝜏)}×𝛿²𝐰₀²_{𝐻2𝕋𝑁}+𝛿²𝑒^2𝜆max𝜏≤𝐶𝑑𝜏₁𝐶₃𝐶₄𝛿𝐰₀²_{𝐻2𝕋𝑁}𝜆_max+𝛿𝑒^𝜆max𝑡𝜆_max𝛿𝑒^𝜆max𝑡. (115)
Step  3. We prove by contradiction that 𝑇^𝛿=min𝑇^𝛿,𝑇^∗,𝑇^∗∗ (116) for 𝛿 sufficiently small and 𝜆𝜃<min^max4𝐶₃𝐶₄,𝜂2𝐶₃,𝑑₁8𝐶₂𝐶₃(1+𝜂)²,𝐴₀𝑑₂1+𝑑₃𝑢4𝐶₂𝐶₃(1+𝜂)². (117)
If 𝑇^∗∗=min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}, we can let 𝑡=𝑇^∗∗≤𝑇^𝛿 in (111) to obtain 𝐰^𝛿⋅,𝑇^∗∗_{𝐻2(𝕋𝑁)}≤𝐶₃𝛿𝐰_{0𝐻2(𝕋𝑁)}+𝛿𝑒^{𝜆max𝑇∗∗}≤𝐶₃𝛿𝐰_{0𝐻2(𝕋𝑁)}+𝜃. (118) By choosing 𝛿 sufficiently small, such that 𝐶₃𝛿𝐰_{0𝐻2(𝕋𝑁)}≤12𝑑min𝜂,¹4𝐶₂1+𝜂²,𝐴₀𝑑₂1+𝑑₃𝑢2𝐶₂1+𝜂², (119) and the choice of 𝜃 in (117), we have 𝐰^𝛿(⋅,𝑇^∗∗)_{𝐻2(𝕋𝑁)}𝑑<min𝜂,¹4𝐶₂1+𝜂²,𝐴₀𝑑₂1+𝑑₃𝑢2𝐶₂(1+𝜂)². (120) This is a contradiction to the definition of 𝑇^∗∗.
On the other hand, if 𝑇^∗=min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}, we can let 𝑡=𝑇^∗ in (115) to get 𝐰^𝛿⋅,𝑇^∗−𝛿𝑒^𝐿𝑇∗𝐰_{0𝐿2(𝕋𝑁)}≤𝐶₁𝐶₃𝐶₄𝛿𝐰₀²_{𝐻2𝕋𝑁}𝜆_max+𝜃𝜆_max𝛿𝑒^{𝜆max𝑇∗}. (121) By choosing 𝛿 sufficiently small, such that 𝐶₃𝐶₄𝛿𝐰₀²_{𝐻2𝕋𝑁}𝜆_max≤14, (122) and the choice of 𝜃 in (117), we have 𝐰^𝛿(⋅,𝑇^∗)−𝛿𝑒^𝐿𝑇∗𝐰_{0𝐿2(𝕋𝑁)}<𝐶₁2𝛿𝑒^{𝜆max𝑇∗}. (123) This again contradicts the definition of 𝑇^∗. From these arguments, (116) holds.
Step  4. Rewriting the left-hand term in (115) as the form of (26), and separating 𝐪∈Λ_max and moving 𝐪∉Λ_max to the right-hand side, it follows that 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Λmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝑥)_{𝐿2(𝕋𝑁)}≤𝛿_𝐪∈Λmax𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}𝑒_𝐪(𝑥)_{𝐿2(𝕋𝑁)}+𝛿_{𝐪∈Λ1⧵Λmax}𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}+𝑤⁺_𝐪𝐫₊(𝐪)𝑒^{𝜆+𝐪𝑡}𝑒_𝐪(𝑥)_{𝐿2(𝕋𝑁)}+𝛿_𝐪∈Λ2𝑤_𝐪𝐫(𝐪)+𝑤^′_𝐪𝐫^′(𝐪)+𝑤^′_𝐪𝐫(𝐪)𝑡𝑒^𝜆𝐪𝑡𝑒_𝐪(𝑥)_{𝐿2(𝕋𝑁)}+𝛿_𝐪∈Λ3𝑤_𝐪^ReRe𝐫(𝐪)cosIm𝜆_𝐪𝑡−Im𝐫(𝐪)sinIm𝜆_𝐪𝑡+𝑤_𝐪^ImRe𝐫(𝐪)sinIm𝜆_𝐪𝑡+Im𝐫(𝐪)cosIm𝜆_𝐪𝑡×𝑒^{(Re𝜆𝐪)𝑡}𝑒_𝐪(𝑥)_{𝐿2(𝕋𝑁)}+𝐶₁𝐶₃𝐶₄𝛿𝐰₀²_{𝐻2𝕋𝑁}𝜆_max+𝛿𝑒^𝜆max𝑡𝜆_max𝛿𝑒^𝜆max𝑡. (124)
Next, we process the first term on the right side of (124) to get 𝛿_𝐪∈Λmax𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}𝑒_𝐪(𝑥)_{𝐿2(𝕋𝑁)}=𝛿_𝐪∈Λmax(𝑤⁻_𝐪)²𝐫₋(𝐪)²𝑒^{2𝜆−𝐪𝑡}𝑑𝑥^1/2. (125) Recall that there are only finite elements in Λ_max and for any 𝐪∈Λ_max, there exists a constant 𝑀₀, such that 0<|𝐪|<𝑀₀. Therefore, 𝛿_𝐪∈Λmax𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}𝑒_𝐪(𝑥)_{𝐿2(𝕋𝑁)}≤𝐶𝛿𝑒^{(𝜆max−𝜎)𝑡}. (126) Similar to the arguments in the proof of Lemma 1, we can treat the second, third, and fourth terms to obtain some similar estimates as (126), and our theorem follows.

Acknowledgments

This work was supported by the National Natural Science Foundation of China as received Grant nos. 10871160 and 11361055, the Fundamental Research Funds for the Gansu University, and NWNU-LKQN-11-21 Foundations.