Nonlinear Instability for a Volume-Filling Chemotaxis Model with Logistic Growth

Abstract

This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a 𝑑-dimensional box 𝕋^𝑑=(0,𝜋)^𝑑(𝑑=1,2,3). It is proved that given any general perturbation of magnitude 𝛿, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order ln(1/𝛿). Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.

1. Introduction

An important variant of the chemotaxis model was initially proposed by Painter and Hillen in [1] to model the volume-filling effect. In the volume-filling effect, it is assumed that cells have a certain finite (nonzero) volume and that the occupation of an area limits other cells from penetrating it. A simple version of the volume-filling chemotaxis model (VF) is the following: 𝑈_𝑡=∇𝑑₁(),𝑉∇𝑈−𝜒𝑈1−𝑈∇𝑉_𝑡=𝑑₂∇²𝑉+𝛼𝑈−𝛽𝑉, (1) where 𝑑₁, 𝑑₂, 𝜒, 𝛼, and 𝛽 are given positive constants. 𝑈(𝑥,𝑡) is the cell density and 𝑉(𝑥,𝑡) denotes the density of the external chemical substance which is secreted by the cells themselves. 𝑑₁ and 𝑑₂ denote the cell and chemical diffusion coefficients, respectively. 𝜒 is called chemosensitivity. The term 𝑈(1−𝑈)∇𝑉 denotes the chemotactic flux under a volume constraint 1 (called crowding capacity), meaning that the chemotactic movement will be inhibited at the aggregation location where the cell density reaches 1.

In recent years, the chemotaxis models with volume-filling effect have been studied extensively. Hillen and Painter [2] firstly proved the global existence of solutions. Numerical simulations in one and two dimensions show interesting phenomena of pattern formation and formation of stable aggregates. Wrzosek [3] showed the existence of a compact global attractor in the space 𝑊^1,𝑝(Ω,ℝ²), 𝑝>𝑛, Ω⊂ℝ^𝑛 for some cases. In [4], the structure of the attractor can be understood using Lyapunov functions. Stationary solutions which are inhomogeneous in space were investigated for a given range of parameters. In [1], a numerical exploration was conducted to determine the longtime patterning behaviour, revealing formation of multiple plateau type patterns which undergo a coarsening process with increasingly long transient times. Potapov and Hillen [5] investigated the metastability of steady states. The underlying bifurcation diagram was identified, revealing that the unstable eigenvalues are exponentially small. The plateau interactions were studied using asymptotic methods. In [6], it was obtained that a priori estimates for the classical chemotaxis model of Patlak, Keller, and Segel when a nonlinear diffusion or a nonlinear chemosensitivity was considered accounting for the finite size of the cells and how entropy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solutions were showed. Burger et al. [7] discussed the effects of linear and nonlinear diffusion in the large time asymptotic behavior of the Keller-Segel model of chemotaxis with volume-filling effect. Moreover, the global existence of solutions and nontrivial steady states were also studied. Wang and Hillen [8] established the global existence of classical solutions to a generalized chemotaxis model, which includes the volume-filling effect expressed through a nonlinear squeezing probability. Necessary and sufficient conditions for spatial pattern formation were given and the underlying bifurcations were analyzed. In [9], the stationary solutions of the volume-filling chemotaxis model without a growth term were obtained by Jiang and Zhang.

Moreover, Wrzosek [10] considered various assumptions on nonlinear diffusion and chemotactic sensitivity function which lead to the existence of global in time solutions, thus preventing blow-up. In [11], Winkler proved that if certain conditions were fulfilled, then there were solutions that blow up in either finite or infinite time. In particular, in the framework of chemotaxis models incorporating a volume-filling effect in the sense of Painter and Hillen [1], his results indicated how strongly the cellular movement must be inhibited at large cell densities in order to rule out chemotactic collapse. Winkler and Djie [12] discussed boundedness and finite-time collapse for a chemotaxis system with volume-filling effect. Wang et al. [13] proved that for a wide range of nonlinear diffusion operators, including singular and degenerate ones, if the taxis force was strong enough with respect to diffusion and the initial data were chosen properly, then there was a classical solution which reaches the threshold at the maximal time of its existence; no matter whether the latter was finite or infinite. Zhang and Zheng [14] obtained the crucial uniform boundedness of the solution for a quasilinear nonuniform parabolic system modelling chemotaxis with volume-filling effect and the results on convergence to equilibrium and the decay rate using a suitable nonsmooth Simon-Łojasiewicz approach. In [15], the uniform boundedness, global in time existence and uniqueness of classical solution, were proved. With the help of a suitable nonsmooth Simon-Łojasiewicz approach, the results on convergence of the solution to equilibrium and the convergence rate were obtained. Li and Zhang [16] classified the existence or nonexistence of steady state solutions of a 1-D chemotaxis model with volume-filling effect. Their results provided insights on how the biological parameters affect pattern formation.

The chemotaxis models with logistic growth but without a volume-filling effect were studied (see [17–22]). The global attractor and traveling wave solutions of a volume-filling chemotaxis model with logistic growth were obtained in [3] and [23], respectively. Ma et al. [24] studied the existence of stationary solutions of a volume-filling chemotaxis model with logistic cell growth. Moreover, based on an explicit formula for the stationary solutions, which is derived by asymptotic bifurcation analysis, the stability criteria were established and a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue was found. Quite recently, Ma et al. in [25] studied the nonexistence of nonconstant steady state (i.e., stationary pattern) for a chemotaxis model with the volume-filling effect and logistic cell growth and established the critical value of the chemotactic coefficient between the existence and the nonexistence of stationary pattern.

Guo and Hwang [26] investigated nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Their results can be interpreted as a rigorous mathematical characterization for the early-stage pattern formation in the Keller-Segel model. Very recently, Fu and Liu in [22] and [27] studied instability in the Keller-Segel model with a logistic source and cubic source term, respectively. Their results indicated that chemotaxis-driven nonlinear instability occurs in these models.

In this paper, we mainly consider the nonlinear instability for the following chemotaxis model: 𝑈_𝑡=∇𝑑₁()𝑈∇𝑈−𝜒𝑈1−𝑈∇𝑉+𝜇𝑈1−𝑈_𝑐,𝑥∈𝕋^𝑑𝑉,𝑡>0,_𝑡=𝑑₂∇²𝑉+𝛼𝑈−𝛽𝑉,𝑥∈𝕋^𝑑,𝑡>0 (2) which is subject to the Neumann boundary conditions 𝜕𝑈𝜕𝑥_𝑖=𝜕𝑉𝜕𝑥_𝑖=0,at𝑥_𝑖=0,𝜋,1<𝑖<𝑑 (3) and the nonnegative initial data 𝑈(𝑥,0)=𝑈₀(𝑥)≥0,𝑉(𝑥,0)=𝑉₀(𝑥)≥0,𝑥∈𝕋^𝑑, (4) where 𝕋^𝑑=(0,𝜋)^𝑑(𝑑=1,2,3) is a 𝑑-dimensional box. The term 𝜇𝑈(1−𝑈/𝑈_𝑐) describes the logistic growth of cells with growth rate 𝜇>0 and carrying capacity 𝑈_𝑐 fulfilling 0<𝑈_𝑐≤1. Our main result (see Theorem 6) indicates that the nonlinear dynamics near an unstable constant equilibrium points in the classical Keller-Segel model, the Keller-Segel model with a logistic source and cubic source term, respectively, and a volume-filling chemotaxis model with logistic source term are almost similar.

The organization of this paper is as follows: in Section 2, we show that the unique positive equilibrium point of (2) without chemotaxis is globally asymptotically stable and cross diffusion cannot induce the instability of the positive equilibrium. In Section 3, we consider the growing modes of (2). In Section 4, we present and prove the Bootstrap lemma, which was first introduced in [28]. In Section 5, for any given general perturbation of magnitude 𝛿, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of the order ln(1/𝛿). Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.

2. Stability of Positive Equilibrium Point of (2) without Chemotaxis

We first discuss the following corresponding kinetic equations of (2): 𝑈_𝑡𝑈=𝜇𝑈1−𝑈_𝑐𝑉,𝑡>0,_𝑡=𝛼𝑈−𝛽𝑉,𝑡>0. (5) We use [⋅,⋅] to denote a column vector. Evidently, (5) has the unique positive equilibrium point 𝐖=[𝑈,𝑉]=[𝑈_𝑐,(𝛼/𝛽)𝑈_𝑐]. For simplicity, we denote 𝐅(𝐖)=[𝜇𝑈(1−𝑈/𝑈_𝑐),𝛼𝑈−𝛽𝑉], and a direct calculation yields 𝐅_𝐖𝐖=.−𝜇0𝛼−𝛽 (6)

The characteristic polynomial of 𝐅_𝐖(𝐖) is 𝜑(𝜆)=(𝜆+𝜇)(𝜆+𝛽); −𝜇 and −𝛽 are the two roots of 𝜑(𝜆)=0. Hence [𝑈_𝑐,(𝛼/𝛽)𝑈_𝑐] is locally stable. Define 𝐸(𝑡)=𝑝(𝑈−𝑈_𝑐−𝑈_𝑐ln(𝑈/𝑈_𝑐))+(𝑉−(𝛼/𝛽)𝑈_𝑐)², where 𝑝=2𝛼²𝑈_𝑐/𝛽𝜇. By the Lyapunov-LaSalle invariance principle [29], [𝑈_𝑐,(𝛼/𝛽)𝑈_𝑐] is globally asymptotically stable.

We now consider system (2) without chemotaxis in the following form: 𝑈_𝑡=𝑑₁𝑈Δ𝑈+𝜇𝑈1−𝑈_𝑐,𝑥∈𝕋^𝑑(),𝑉𝑑=1,2,3𝑡>0,_𝑡=𝑑₂Δ𝑉+𝛼𝑈−𝛽𝑉,𝑥∈𝕋^𝑑(𝑑=1,2,3),𝑡>0,𝜕𝑈𝜕𝑥_𝑖=𝜕𝑉𝜕𝑥_𝑖=0,at𝑥_𝑖=0,𝜋,for1≤𝑖≤𝑑. (7)

Let 0=𝜇₁<𝜇₂<𝜇₃<⋯ be the eigenvalues of the operator −Δ on 𝕋^𝑑(𝑑=1,2,3) with the homogeneous Neumann boundary condition and 𝐸(𝜇_𝑖) the eigenspace corresponding to 𝜇_𝑖 in 𝐿²(𝕋^𝑑). Let 𝑋=[𝐿²(𝕋^𝑑)]², {𝜙_𝑖𝑗∶𝑗=1,…,dim𝐸(𝜇_𝑖)} be an orthonormal basis of 𝐸(𝜇_𝑖), and 𝑋_𝑖𝑗={𝐜⋅𝜙_𝑖𝑗∣𝐜∈ℝ²}. Then 𝑋=⊕^∞_𝑖=1𝑋_𝑖, 𝑋_𝑖=⊕^{dim𝐸(𝜇𝑖)}_𝑗=1𝑋_𝑖𝑗.

Let 𝔇=diag(𝑑₁,𝑑₂) and 𝔏=𝔇Δ+𝐅_𝐖(𝐖). The linearization of (7) at [𝑈,𝑉] can be expressed by 𝐖_𝑡=𝔏(𝐖−𝐖). For each 𝑖≥1, 𝑋_𝑖 is invariant under the operator 𝔏, and 𝜆 is an eigenvalue of  𝔏 on 𝑋_𝑖 if and only if it is an eigenvalue of the matrix −𝜇_𝑖𝔇+𝐅_𝐖𝐖=−𝜇_𝑖𝑑₁−𝜇0𝛼−𝜇_𝑖𝑑₂.−𝛽 (8) Thus, −𝜇_𝑖𝔇+𝐅_𝐖(𝐖) has two negative eigenvalues −𝜇_𝑖𝑑₁−𝜇 and −𝜇_𝑖𝑑₂−𝛽. It follows from Theorem 5.1.1 in [30] that [𝑈_𝑐,(𝛼/𝛽)𝑈_𝑐] is locally asymptotically stable.

Let 𝐖=[𝑈,𝑉] be the unique nonnegative global solution. The maximum principle gives 0≤𝑈(𝑥,𝑡)≤max𝑈_𝑐,sup_𝕋𝑑𝑈₀(𝑥,()𝛼)0≤𝑉𝑥,𝑡≤max𝛽𝑈_𝑐,𝛼𝛽sup_𝕋𝑑𝑈₀(𝑥),sup_𝕋𝑑𝑉₀(𝑥). (9) Moreover, by the strong maximum principle [31], we know that if 𝑈₀, 𝑉₀≥(≢)0, then 𝑈(𝑥,𝑡)>0, 𝑉(𝑥,𝑡)>0 on 𝕋^𝑑 for all 𝑡>0.

We define the Lyapunov function 𝐸(𝑡)=_𝕋𝑑𝑝𝑈−𝑈_𝑐−𝑈_𝑐𝑈ln𝑈_𝑐+𝛼𝑉−𝛽𝑈_𝑐²𝑑𝑥, (10) where 𝑝=2𝛼²𝑈_𝑐/𝛽𝜇. Then 𝐸(𝑡)≥0 for all 𝑡≥0. Applying (7) and integration by parts, we have 𝐸^′(𝑡)=−_𝕋𝑑𝑝𝑑₁𝑈_𝑐𝑈²|∇𝑈|²+2𝑑₂|∇𝑉|²−𝑑𝑥_𝕋𝑑𝑝𝜇𝑈_𝑐𝑈−𝑈_𝑐²−2𝛼𝑈−𝑈_𝑐𝛼𝑉−𝛽𝑈_𝑐𝛼+2𝛽𝑉−𝛽𝑈_𝑐²𝛼𝑑𝑥.≤−²𝛽_𝕋^𝑑𝑈−𝑈_𝑐²𝑑𝑥−𝛽_𝕋𝑑𝛼𝑉−𝛽𝑈_𝑐²𝑑𝑥. (11) By (9), (11), the basic estimates for parabolic equations [31], and Lemma 2.1 in [22] (which is given in [32] in Chinese), we can conclude that lim_𝑡→∞𝑈(⋅,𝑡)−𝑈_{𝑐𝐿2(𝕋𝑑)}=0,lim_𝑡→∞𝛼𝑉(⋅,𝑡)−𝛽𝑈_{𝑐𝐿2(𝕋𝑑)}=0. (12) The global asymptotic stability of [𝑈_𝑐,(𝛼/𝛽)𝑈_𝑐] follows from (12) together with the local stability of [𝑈_𝑐,(𝛼/𝛽)𝑈_𝑐].

Next, we consider that the cross diffusion model 𝑈_𝑡=𝑑₁Δ𝑈+𝑑₃𝑈𝑈𝑉+𝜇𝑈1−𝑈_𝑐,𝑥∈𝕋^𝑑𝑉,𝑡>0,_𝑡=𝑑₂Δ𝑉+𝛼𝑈−𝛽𝑉,𝑥∈𝕋^𝑑,𝑡>0,𝜕𝑈𝜕𝑥_𝑖=𝜕𝑉𝜕𝑥_𝑖=0,at𝑥_𝑖𝑈()=0,𝜋,for1≤𝑖≤𝑑,𝑥,0=𝑈₀(𝑥)≥0,𝑉(𝑥,0)=𝑉₀(𝑥)≥0,𝑥∈𝕋^𝑑. (13)

Let Φ(𝐖)=[𝑑₁(𝑈+𝑑₃𝑈𝑉),𝑑₂𝑉]. Then the linearizing system (13) at 𝐖 can be written as 𝐖_𝑡=Φ_𝐖Δ+𝐅_𝐖𝐖𝐖, (14) where Φ_𝐖𝐖=Φ_𝐖=𝑑₁1+𝑑₃𝑉𝑑₁𝑑₃𝑈0𝑑₂. (15) Then, for each 𝑖∈{1,2,…}, 𝑋_𝑖 is invariant under the operator Φ_𝐖Δ+𝐅_𝐖(𝐖), and 𝜉 is an eigenvalue of Φ_𝐖Δ+𝐅_𝐖(𝐖) on 𝑋_𝑖 if and only if 𝜉 is an eigenvalue of the matrix 𝒜_𝑖=−𝑑₁1+𝑑₃𝑉𝜇_𝑖−𝜇−𝑑₁𝑑₃𝑈𝜇_𝑖𝛼−𝑑₂𝜇_𝑖.−𝛽 (16) Notice that det𝒜_𝑖=𝑑₁𝑑₂1+𝑑₃𝑉𝜇²_𝑖+𝑑₂𝜇+𝛽𝑑₁1+𝑑₃𝑉+𝑑₁𝑑₃𝛼𝑈𝜇_𝑖+𝜇𝛽>0,Tr𝒜_𝑖=−𝑑₁+𝑑₂+𝑑₁𝑑₃𝑉𝜇_𝑖+𝜇+𝛽<0. (17) Thus, the two eigenvalues 𝜉⁺_𝑖 and 𝜉⁻_𝑖 of 𝒜_𝑖 have negative real parts.

From this, we see that adding the cross diffusion to the system (7), the positive constant solution is also locally stable, which means that Turing instability does not occur. The information above indicates that the aggregation of individuals does not occur in the absence of chemotactic effect. It is the purpose of the present paper to clarify the effect of the chemotaxis and nonlinear patterns created by chemotaxis for a volume-filling chemotaxis model with logistic growth.

3. Growing Modes in the System (2)

The nonlinear evolution of a perturbation 𝑢(𝐱,𝑡)=𝑈(𝐱,𝑡)−𝑈, 𝑣(𝐱,𝑡)=𝑉(𝐱,𝑡)−𝑉 around [𝑈,𝑉] satisfies 𝑢_𝑡=𝑑₁∇²𝑢−𝜒𝑈_𝑐1−𝑈_𝑐∇²𝑢𝑣−𝜇𝑢−𝜒∇1−𝑢−2𝑈_𝑐∇𝑣−𝜇𝑢²𝑈_𝑐,𝑣_𝑡=𝑑₂∇²𝑣+𝛼𝑢−𝛽𝑣. (18) The corresponding linearized system is as follows: 𝑢_𝑡=𝑑₁∇²𝑢−𝜒𝑈_𝑐1−𝑈_𝑐∇²𝑣𝑣−𝜇𝑢,_𝑡=𝑑₂∇²𝑣+𝛼𝑢−𝛽𝑣. (19) Let 𝐰(𝐱,𝑡)≡[𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)], 𝐪=(𝑞₁,…,𝑞_𝑑)∈Ω=(ℕ∪{0})^𝑑, and 𝑒_𝐪(𝐱)=∏^𝑑_𝑖=1cos(𝑞_𝑖𝑥_𝑖). Then {𝑒_𝐪(𝐱)}_𝐪∈Ω forms a basis of the space of functions in 𝕋^𝑑 that satisfies Neumann boundary conditions. We try to find a normal mode to the linear system (19) of the following form: 𝐰(𝐱,𝑡)=𝐫_𝐪𝑒^𝜆𝐪𝑡𝑒_𝐪(𝐱), (20) where 𝐫_𝐪 is a vector depending on 𝐪. Substituting (20) into (19) yields 𝜆_𝐪𝐫_𝐪=−𝑑₁𝑞²−𝜇𝜒𝑈_𝑐1−𝑈_𝑐𝑞²𝛼−𝑑₂𝑞²𝐫−𝛽^𝐪, (21) where 𝑞²=∑^𝑑_𝑖=1𝑞²_𝑖. We can obtain the following dispersion formula for 𝜆_𝐪: 𝜆²_𝐪+𝑑₁+𝑑₂𝑞²+𝛽+𝜇𝜆_𝐪+𝑞²𝑑₁𝑑₂𝑞²+𝑑₁𝛽+𝑑₂𝜇−𝛼𝜒𝑈_𝑐1−𝑈_𝑐+𝜇𝛽=0. (22) Thus, we deduce the following linear instability criterion by requiring that there exist a 𝐪 such that 𝑞²𝑑₁𝑑₂𝑞²+𝑑₁𝛽+𝑑₂𝜇−𝛼𝜒𝑈_𝑐1−𝑈_𝑐+𝜇𝛽<0 (23) to ensure that (22) has at least one positive root 𝜆_𝐪. This means that 𝑞²𝑑₁−𝑑₂−𝛽+𝜇²+4𝛼𝜒𝑈_𝑐1−𝑈_𝑐𝑞²>0. (24) There exist two distinct real roots: 𝜆^±_𝐪=−𝑞²𝑑₁+𝑑₂±+𝛽+𝜇𝑞²𝑑₁−𝑑₂−𝛽+𝜇²+4𝛼𝜒𝑈_𝑐1−𝑈_𝑐𝑞²×(2)⁻¹ (25) for all 𝐪≠𝟎 to the quadratic equation (22), which are denoted by 𝜆⁻_𝐪<𝜆⁺_𝐪. We denote the corresponding (linearly independent) eigenvectors by 𝐫₋(𝐪) and 𝐫₊(𝐪), such that 𝐫_±(𝐪)=𝜆^±_𝐪+𝑑₂𝑞²+𝛽𝛼,1. (26) Clearly, for 𝑞 large, 𝑞²𝑑₁𝑑₂𝑞²+𝑑₁𝛽+𝑑₂𝜇−𝛼𝜒𝑈_𝑐1−𝑈_𝑐+𝜇𝛽>0. (27) Thus, there are only finitely many 𝐪 such that 𝜆⁺_𝐪>0. We denote the largest eigenvalue by 𝜆_max>0 and define Ω_max≡{𝐪∈Ω∣𝜆⁺_𝐪=𝜆_max}. From (25) we can regard 𝜆⁺_𝐪 as a function of 𝑞². Therefore, there is one 𝑞² (possibly two) having 𝜆⁺(𝑞²)=𝜆_max. We also denote 𝜌>0 to be the gap between the 𝜆_max and the rest; that is, 𝜌=min_{𝐪∈Ω⧵Ωmax}|𝜆_max−𝜆_𝐪|.

Given any initial perturbation 𝐰(𝐱,0), we can expand it as 𝐰(𝐱,0)=_𝐪∈Ω𝐰_𝐪𝑒_𝐪(𝐱)=_𝐪∈Ω𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱),𝐰_𝐪=𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪). (28)(29)

In the sequel, denote by ⟨⋅,⋅⟩ and (⋅,⋅) the inner product of [𝐿²(𝕋^𝑑)]² and the scalar product of ℝ², respectively. For any 𝐠(⋅,𝑡)∈[𝐿²(𝕋^𝑑)]², we denote ‖𝐠(⋅,𝑡)‖≡‖𝐠(⋅,𝑡)‖_𝐿². Throughout this paper, we always denote universal constants by 𝐶_𝑖(𝑖=1,2,…).

Clearly, ‖𝐰(𝐱,0)‖²=𝜋2^𝑑_𝐪∈Ω𝐰_𝐪². (30)

The unique solution 𝐰(𝐱,𝑡)=[𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] of (19) is given by 𝐰(𝐱,𝑡)=_𝐪∈Ω𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}+𝑤⁺_𝐪𝐫₊(𝐪)𝑒^{𝜆+𝐪𝑡}𝑒_𝐪(𝐱)≡𝑒^𝔏𝑡𝐰(𝐱,0). (31)

Our main result in this section is the following lemma.

Lemma 1. Suppose that the instability criterion (23) holds. Let 𝐰(𝐱,𝑡)≡𝑒^𝔏𝑡𝐰(𝐱,0) be a solution to the linearized system (19) with initial condition 𝐰(𝐱,0). Then there exists a constant 𝐶₁≥1 depending on 𝑑₁, 𝑑₂, 𝑈_𝑐, 𝜒, 𝜇, 𝛼, and 𝛽, such that ‖𝐰(⋅,𝑡)‖≤𝐶₁𝑒^𝜆max𝑡‖𝐰(⋅,0)‖,∀𝑡≥0. (32)

Proof. We divide the proof into the following two cases.(1)𝑡≥1. It follows from (25) that lim_𝐪→∞𝜆^±_𝐪𝑞²=−𝑑₁,−𝑑₂, (33)respectively. Thus, 𝜆^±_𝐪≤−min{𝑑₁,𝑑₂}𝑞² and there exists a positive constant 𝐶₁ for all 𝐪>0, such that 𝜆^±_𝐪𝑞²≤𝐶₁. (34) By the quadratic formula of (22), one can obtain 𝜆⁺_𝐪−𝜆⁻_𝐪≥2𝑞𝛼𝜒𝑈_𝑐1−𝑈_𝑐. (35) It follows from (29) that 𝑤^±_𝐪≤𝐫_±(𝐪)×𝐰_𝐪det𝐫₋(𝐪),𝐫₊(𝐪.) (36)
From (26) and (34), we can conclude that there exists a positive constant 𝐶₂ such that 𝐫_±(𝐪)=𝜆^±_𝐪𝛼𝑞²+𝑑₂𝛼𝑞²+𝛽𝛼²+1≤𝐶₂𝑞² (37) for all 𝑞>0, where 𝐶₂=2max{𝐶₁+𝑑₂/𝛼,1+𝛽/𝛼}. By (34), one can deduce 1det𝐫₋(𝐪),𝐫₊(𝐪≤1)2𝑞𝛼𝜒𝑈_𝑐1−𝑈_𝑐. (38) Combining (37) and (38), we find that 𝑤^±_𝐪≤𝐶₃𝑞𝐰_𝐪, (39) where 𝐶₃=(𝐶₂/2)𝛼/𝜒𝑈_𝑐(1−𝑈_𝑐). For 𝑡≥1 and 𝑞 large, it is not difficult to verify by (33), (37), and (39) that 𝑤^±_𝐪𝐫_±(𝐪)𝑒^{𝜆±𝐪𝑡}≤𝐶₂𝐶₃𝑞³𝐰_𝐪𝑒^{𝜆±𝐪𝑡}≤𝐶₄𝐰_𝐪𝑞³expmin𝑑₁,𝑑₂𝑞²≤𝐶₅𝐰_𝐪. (40) In view of (30) and (40), we observe that ‖𝐰(𝐱,𝑡)‖≤2𝐶₅𝑒^𝜆max𝑡‖𝐰(𝐱,0)‖,for𝑡≥1. (41)(2)𝑡≤1. Multiplying the first equation of (19) by 𝑢 and the second by 𝐾𝑣, adding them together, and integrating the result in 𝕋^𝑑, we have 12𝑑𝑑𝑡_𝕋^𝑑|𝑢|²+𝐾|𝑣|²+𝐝𝐱_𝕋𝑑𝑑₁|∇𝑢|²+𝐾𝑑₂|∇𝑣|²−𝜒𝑈_𝑐1−𝑈_𝑐∇𝑢∇𝑣𝐝𝐱=−𝜇_𝕋𝑑𝑢²𝐝𝐱−𝐾𝛽_𝕋𝑑𝑣²𝐝𝐱+𝛼𝐾_𝕋𝑑𝑢𝑣𝐝𝐱. (42)Let 𝐾=𝜒𝑈_𝑐1−𝑈_𝑐²𝑑₁𝑑₂. (43) Then the integrand of the second integral can be estimated as follows: 𝑑₁|∇𝑢|²+𝐾𝑑₂|∇𝑣|²−𝜒𝑈_𝑐1−𝑈_𝑐≥𝑑∇𝑢∇𝑣₁2|∇𝑢|²+𝜒𝑈_𝑐1−𝑈_𝑐²2𝑑₁|∇𝑣|²≥0. (44) Using Young inequality, we deduce that 12𝑑𝑑𝑡_𝕋^𝑑|𝑢|²+𝐾𝑣²𝛼√𝐝𝐱≤𝐾2_𝕋^𝑑|𝑢|²+𝐾|𝑣|²𝐝𝐱. (45) It follows from Gronwall inequality that |𝑢_𝕋𝑑(𝐱,𝑡)|²+𝐾|𝑣(𝐱,𝑡)|²𝐝𝐱≤𝑒^{𝛼√𝐾𝑡}_𝕋𝑑|𝑢(𝐱,𝑡)|²+𝐾|𝑣(𝐱,𝑡)|²𝐝𝐱. (46)
If 𝐾≥1 and 𝑡≤1, then it is clear from (46) that ‖𝐰(𝐱,𝑡)‖≤𝐾𝑒^𝛼√𝐾𝑒^𝜆max𝑡‖𝐰(𝐱,0)‖. (47) Similarly, if 0<𝐾<1, 𝑡≤1, then from (46), one has ‖𝐰(𝐱,𝑡)‖≤𝑒^𝛼/𝐾𝑒^𝜆max𝑡‖𝐰(𝐱,0)‖. (48) Let 𝐶₁=max{2𝐶₅,𝑒^𝛼/𝐾}≥1 if 0<𝐾<1, and let 𝐶₁=max{2𝐶₅,𝐾𝑒^𝛼√𝐾}≥1 if 𝐾≥1. Then ‖𝐰(𝐱,𝑡)‖≤𝐶₁𝑒^𝜆max𝑡‖𝐰(𝐱,0)‖. (49) This completes the proof of Lemma 1.

4. Bootstrap Lemma

By a standard PDE theory [31], we can establish the existence of local solutions for (18).

Lemma 2 (Local existence). For 𝑠≥1(𝑑=1) and 𝑠≥2(𝑑=2,3), there exists a 𝑇₀>0 such that (18) with 𝑢(⋅,0), 𝑣(⋅,0)∈𝐻^𝑠 has a unique solution 𝐰(⋅,𝑡) on (0,𝑇₀) which satisfies ‖𝐰(𝑡)‖_𝐻^𝑠≤𝐶‖𝐰(0)‖_𝐻^𝑠,0<𝑡<𝑇₀, (50) where 𝐶 is a positive constant depending on 𝑑₁, 𝑑₂, 𝑈_𝑐, 𝛼, 𝛽, and 𝜒.

It is not difficult to verify the following result.

Lemma 3. Let 𝐰(𝐱,𝑡) be a solution of (18). Then the even extension of 𝐰(𝐱,𝑡) on 2𝕋^𝑑=(−𝜋,𝜋)^𝑑(𝑑=1,2,3) is also the solution of (18) which satisfies homogeneous Neumann boundary conditions and periodical boundary conditions on 2𝕋^𝑑=(−𝜋,𝜋)^𝑑(𝑑=1,2,3).

Lemma 4. Let [𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] be a solution of (18). Then 12𝑑𝑑𝑡_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑢²+𝜒𝑈_𝑐1−𝑈_𝑐²𝑑₁𝑑₂𝐷^𝛼𝑣²𝐝𝐱+_{|𝛼|=2𝕋𝑑}𝑑₁4∇𝐷^𝛼𝑢²+𝜒𝑈_𝑐1−𝑈_𝑐²2𝑑₁∇𝐷^𝛼𝑣²𝐝𝐱+𝜇_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢²𝛽𝐝𝐱+𝜒𝑈_𝑐1−𝑈_𝑐²2𝑑₁𝑑₂×_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑣²≤𝐶𝐝𝐱₂‖𝐰‖_𝐻²+‖𝐰‖²_𝐻2∇³𝐰²+𝐶₃‖𝑢‖², (51) where 𝐶₃=𝐶³₀𝛼³[𝜒𝑈_𝑐(1−𝑈_𝑐)]⁶/2𝛽³𝑑⁵₁𝑑³₂.

Proof. It is known by Lemma 3 that 𝑢_𝑡=𝑑₁∇²𝑢−𝜒𝑈_𝑐1−𝑈_𝑐∇²̃𝑢𝑢𝑣−𝜇−𝜒∇1−𝑢−2𝑈_𝑐∇𝑣̃−𝜇𝑢²𝑈_𝑐,𝑥∈2𝕋^𝑑𝑣̃,𝑡>0,_𝑡=𝑑₂∇²𝑣̃+𝛼𝑢−𝛽𝑣̃,𝑥∈2𝕋^𝑑,𝑡>0,𝜕𝑢𝜕𝑥_𝑖=𝜕𝑣̃𝜕𝑥_𝑖=0,at𝑥_𝑖=−𝜋,0,𝜋,for1≤𝑖≤𝑑, (52)(53) where [𝑢(𝐱,𝑡),𝑣̃(𝐱,𝑡)] is the even extension of [𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] on 2𝕋^𝑑. Taking the second-order derivative of (52) for 𝑥_𝑖, 𝑥_𝑗 and making inner product with 𝜕_{𝑥𝑖𝑥𝑗}𝑢 and 𝐾𝜕_{𝑥𝑖𝑥𝑗}𝑣̃, respectively, and adding them together, we deduce that 12𝑑𝑑𝑡_2𝕋^𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝐾𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²+𝐝𝐱_2𝕋𝑑𝑑₁∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝐾𝑑₂∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²−𝜒𝑈_𝑐1−𝑈_𝑐∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃𝐝𝐱+𝜇_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²𝐝𝐱+𝛽𝐾_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²𝐝𝐱=𝜒1−2𝑈_{𝑐2𝕋𝑑}∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅𝜕_{𝑥𝑖𝑥𝑗}(𝑢⋅∇𝑣̃)𝐝𝐱−𝜒_2𝕋𝑑∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅𝜕_{𝑥𝑖𝑥𝑗}𝑢²∇𝑣̃𝐝𝐱+𝐾𝛼_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅𝜕_{𝑥𝑖𝑥𝑗}̃−𝑣𝐝𝐱2𝜇𝑈_𝑐2𝕋^𝑑𝑢𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝜕_𝑥𝑖𝑢⋅𝜕_𝑥𝑗𝑢⋅𝜕_{𝑥𝑖𝑥𝑗}𝑢𝐝𝐱≡𝐼₁+𝐼₂+𝐼₃+𝐼₄. (54)
We can apply Young inequality and (43) to get 𝑑₁∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝐾𝑑₂∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²−𝜒𝑈_𝑐1−𝑈_𝑐∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃≥𝑑₁2∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝜒𝑈_𝑐1−𝑈_𝑐²2𝑑₁∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃². (55)
Now we estimate each term on the right-hand sides of (54). By using Hölder inequality, 𝐼₁≤𝜒1−2𝑈_𝑐×‖∇𝑣̃‖_𝐿^∞∇𝜕_{𝑥𝑖𝑥𝑗}𝑢𝜕_{𝑥𝑖𝑥𝑗}𝑢+2^𝑑_𝑖=1‖∇𝑢‖_𝐿^∞𝜕_{𝑥𝑖𝑥𝑗}𝑣̃∇𝜕_{𝑥𝑖𝑥𝑗}𝑢+‖𝑢‖_𝐿^∞∇𝜕_{𝑥𝑖𝑥𝑗}𝑢∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃. (56) Notice that 𝑔_{𝐿∞(2𝕋𝑑)}≤𝐶₆𝑔_{𝐻2(2𝕋𝑑)},𝑔_{𝐿4(2𝕋𝑑)}≤𝐶₇𝑔_{𝐻2(2𝕋𝑑)},𝑔_{𝐿6(2𝕋𝑑)}≤𝐶₈𝑔_{𝐻2(2𝕋𝑑)}, (57) for 𝑑≤3. Clearly, _2𝕋𝑑∇𝑢𝑑𝐱=_2𝕋𝑑∇𝑣̃𝑑𝐱=0,_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢𝑑𝐱=_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑣̃𝑑𝐱=0. (58) Using the Poincaré inequality, we have 𝑔≤𝐶₉𝑔_{𝐿4(2𝕋𝑑)}≤𝐶₁₀𝑔_{𝐿6(2𝕋𝑑)}≤𝐶₁₁𝑔_𝐻1≤𝐶₁₂∇𝑔,𝑑≤3. (59) It follows from (58) and (59) that 𝜕_𝑥𝑖𝑔≤𝐶₁₂∇𝜕_𝑥𝑖𝑔,𝜕_{𝑥𝑖𝑥𝑗}𝑔≤𝐶₁₂∇𝜕_{𝑥𝑖𝑥𝑗}𝑔,∇𝑔≤𝐶_12|𝛼|=2𝐷^𝛼𝑔^21/2≤𝐶²_12|𝛼|=2∇𝐷^𝛼𝑔^21/2. (60)(61) Furthermore, ∇𝑔_𝐻2≤𝐶_13|𝛼|=2∇𝐷^𝛼𝑔^21/2,𝐶₁₃=𝐶⁴₁₂+𝐶²₁₂+1^1/2. (62) Combining (56), (57), and (62), we observe that _|𝛼|=2𝐼₁≤𝜒1−2𝑈_𝑐𝐶₁₄‖𝐰‖_𝐻²∇³𝐰², (63) where 𝐶₁₄=𝐶₆[(1+2𝑑)𝐶₁₃+1].
By Hölder inequality, it follows from (57) and (62) that _|𝛼|=2𝐼₂≤𝜒𝐶₁₅‖𝐰‖²_𝐻2∇³𝐰², (64) where 𝐶₁₅=𝐶₆(2𝐶₉⁻²𝐶²₁₂𝐶₁₃+6𝐶₆𝐶₁₃+𝐶₆).
Now we estimate 𝐼₃. By interpolation for all 𝑎>0, it can be proved that 𝜕_{𝑥𝑖𝑥𝑗}𝑢²≤𝐶₀𝑎∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+‖𝑢‖²4𝑎². (65) Then it is easy to see that _|𝛼|=2𝐼₃≤𝐾𝛽2_{|𝛼|=22𝕋}^𝑑𝐷^𝛼𝑣̃²+𝑑𝐝𝐱₁4_{|𝛼|=22𝕋}^𝑑∇𝐷^𝛼𝑢²𝐝𝐱+𝐶₃‖𝑢‖², (66) where 𝐶₃=𝐶³₀𝛼³[𝜒𝑈_𝑐(1−𝑈_𝑐)]⁶/2𝛽³𝑑⁵₁𝑑³₂ and 𝑎=𝑑²₁𝑑₂𝛽/2[𝜒𝑈_𝑐(1−𝑈_𝑐)]²𝐶₀𝛼.
Finally, from (57) and (61), 𝐼₄ is bounded by _|𝛼|=2𝐼₄≤4𝜇𝑈_𝑐𝐶₁₆‖𝐰‖_𝐻²∇³𝐰². (67)
Combining (54), (55), and (63)–(67), one can obtain 12𝑑𝑑𝑡_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑢²+𝐾𝐷^𝛼𝑣²𝐝𝐱+_{|𝛼|=2𝕋𝑑}𝑑₁4∇𝐷^𝛼𝑢²+𝐾𝑑₂2∇𝐷^𝛼𝑣²𝐝𝐱+𝜇_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢²𝐝𝐱+𝛽𝐾2_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑣²≤𝐶𝐝𝐱₂‖𝐰‖_𝐻²+‖𝐰‖²_𝐻2∇³𝐰²+𝐶₃‖𝑢‖², (68) where 𝐶₂=(𝜒|1−2𝑈_𝑐|+𝜒+(4𝜇/𝑈_𝑐))max{𝐶₁₄,𝐶₁₅,𝐶₁₆} and the proof is completed.

Lemma 5. Let 𝐰(𝐱,𝑡) be a solution of (18) such that for 0≤𝑡≤𝑇<𝑇₀, ‖𝐰(⋅,𝑡)‖_𝐻²+‖𝐰(⋅,𝑡)‖²_𝐻2≤1𝐶₂𝑑min¹4,𝜒𝑈_𝑐1−𝑈_𝑐²2𝑑₁,‖𝐰(⋅,𝑡)‖≤2𝐶₁𝑒^𝜆max𝑡‖𝐰(⋅,0)‖. (69)(70) Then ‖𝐰(⋅,𝑡)‖²_𝐻2≤𝐶₄‖𝐰(⋅,0)‖²_𝐻2+𝑒^2𝜆max𝑡‖𝐰(⋅,0)‖²,0≤𝑡≤𝑇, (71) where 𝐶₄=max{(1+𝐶²₁₂)[𝜒𝑈_𝑐(1−𝑈_𝑐)]²/𝑑₁𝑑₂,4𝐶²₁[1+𝐶₃(1+𝐶²₁₂)/𝜆_max]} if [𝜒𝑈_𝑐(1−𝑈_𝑐)]²/𝑑₁𝑑₂≥1 and 𝐶₄=max{(1+𝐶²₁₂)𝑑₁𝑑₂/[𝜒𝑈_𝑐(1−𝑈_𝑐)]²,4𝐶²₁[1+𝐶₃(1+𝐶²₁₂)𝑑₁𝑑₂/𝜆_max[𝜒𝑈_𝑐(1−𝑈_𝑐)]²]} if [𝜒𝑈_𝑐(1−𝑈_𝑐)]²/𝑑₁𝑑₂<1.

Proof. It follows from (61) that ‖∇𝐰(⋅,𝑡)‖²≤𝐶²_12|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)². (72) Thus, ‖𝐰(⋅,𝑡)‖²_𝐻2≤‖𝐰(⋅,𝑡)‖²+𝐶²₁₂+1_|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)². (73)
Now we estimate the second-order derivatives of 𝐰(⋅,𝑡). From Lemma 4 and (69), we can obtain 12𝑑𝑑𝑡_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑢²+𝜒𝑈_𝑐1−𝑈_𝑐²𝑑₁𝑑₂𝐷^𝛼𝑣²≤𝐶𝐝𝐱₃‖𝑢‖²≤𝐶₃‖𝐰(⋅,𝑡)‖². (74) Integrating this from 0 to 𝑡, we deduce from (70) that _{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢(⋅,𝑡)²+𝜒𝑈_𝑐1−𝑈_𝑐²𝑑₁𝑑₂𝐷^𝛼𝑣(⋅,𝑡)²𝐝𝐱≤_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢(⋅,0)²+𝜒𝑈_𝑐1−𝑈_𝑐²𝑑₁𝑑₂𝐷^𝛼𝑣(⋅,0)²+4𝐶𝐝𝐱₂₁𝐶₃𝜆_max𝑒^2𝜆max𝑡‖𝐰(⋅,0)‖². (75)
We will proceed in the following two cases: [𝜒𝑈_𝑐(1−𝑈_𝑐)]²/𝑑₁𝑑₂≥1, [𝜒𝑈_𝑐(1−𝑈_𝑐)]²/𝑑₁𝑑₂<1.(1)If [𝜒𝑈_𝑐(1−𝑈_𝑐)]²/𝑑₁𝑑₂≥1, then it is clear from (75) that _|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)²≤𝜒𝑈_𝑐1−𝑈_𝑐²𝑑₁𝑑_2|𝛼|=2𝐷^𝛼𝐰(⋅,0)²+4𝐶²₁𝐶₃𝜆_max𝑒^2𝜆max𝑡‖𝐰(⋅,0)‖². (76)Using (73) and (76), we know that ‖𝐰(⋅,𝑡)‖²_𝐻2≤𝐶₄‖𝐰(⋅,0)‖²_𝐻2+‖𝐰(⋅,0)‖²𝑒^2𝜆max𝑡, (77) where 𝐶₄=max{(1+𝐶²₁₂)[𝜒𝑈_𝑐(1−𝑈_𝑐)]²/𝑑₁𝑑₂,4𝐶²₁[1+𝐶₃(1+𝐶²₁₂)/𝜆_max]}.(2)If [𝜒𝑈_𝑐(1−𝑈_𝑐)]²/𝑑₁𝑑₂<1, then it is not hard to verify by (70), (73), and (75) that ‖𝐰(⋅,𝑡)‖²_𝐻2≤𝐶₄‖𝐰(⋅,0)‖²_𝐻2+‖𝐰(⋅,0)‖²𝑒^2𝜆max𝑡, (78)where 𝐶₄=max{(1+𝐶²₁₂)(𝑑₁𝑑₂/[𝜒𝑈_𝑐(1−𝑈_𝑐)]²),4𝐶²₁[1+𝐶₃(1+𝐶²₁₂)𝑑₁𝑑₂/𝜆_max[𝜒𝑈_𝑐(1−𝑈_𝑐)]²]} and thereby completing the proof.

5. Main Result

Let 𝜃>0 be a small fixed constant, and 𝜆_max the dominant eigenvalue which is the maximal growth rate. For 𝛿>0 arbitrary small we define the escape time 𝑇^𝛿 by 𝜃=𝛿𝑒^{𝜆max𝑇𝛿} (79) or equivalently 𝑇^𝛿=1𝜆_max𝜃ln𝛿. (80)

Our main result in this paper reads as follows.

Theorem 6. Assume that the set of 𝑞²=∑^𝑑_𝑖=1𝑞²_𝑖 satisfying instability criterion (23) is not empty for given parameters 𝑑₁, 𝑑₂, 𝛼, 𝛽, 𝜒, 𝜇, and 𝑈_𝑐. Let 𝐰₀(𝐱)=_𝐪∈Ω𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)∈𝐻², (81) such that ‖𝐰₀‖=1. Then there exist constants 𝛿₀>0, 𝐶>0, and 𝜃>0, depending on 𝑑₁, 𝑑₂, 𝛼, 𝛽, 𝜒, 𝜇, and 𝑈_𝑐, such that for all 0<𝛿≤𝛿₀, if the initial perturbation of the steady state [𝑈,𝑉] is 𝐰^𝛿(⋅,0)=𝛿𝐰₀, then its nonlinear evolution 𝐰^𝛿(⋅,𝑡) satisfies 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Ωmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)≤𝐶𝑒^−𝜌𝑡+𝛿𝐰₀²_𝐻2+𝛿²𝐰₀³_𝐻2+𝛿𝑒^𝜆max𝑡+𝛿²𝑒^2𝜆max𝑡𝛿𝑒^𝜆max𝑡 (82) for 0≤𝑡≤𝑇^𝛿, and 𝜌>0 is the gap between 𝜆_max and the rest of 𝜆_𝐪 in (82).

Proof. Let 𝐰^𝛿(𝐱,𝑡) be the solutions of (18) with initial data 𝐰^𝛿(⋅,0)=𝛿𝐰₀. Define 𝑇^∗=sup𝑡∣𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≤𝐶₁2𝛿𝑒^𝜆max𝑡.𝑇^∗∗=sup𝑡∣𝐰^𝛿(⋅,𝑡)_𝐻2+𝐰^𝛿(⋅,𝑡)²_𝐻2≤1𝐶^∗₂𝑑min¹4,𝜒𝑈_𝑐1−𝑈_𝑐²2𝑑₁. (83) We recall (80) and choose 𝜃 such that 𝐶^∗₂𝐶₄𝜃1+2𝐶₄^1/2𝜃𝜆<min^max4,𝑑₁8,𝜒𝑈_𝑐1−𝑈_𝑐²4𝑑₁. (84)
We first estimate 𝐻² norm of 𝐰^𝛿(𝐱,𝑡) for 0≤𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}. By the definition of 𝑇^∗ and Lemma 1, for 0<𝑡≤𝑇^∗, we have 𝐰^𝛿(≤3⋅,𝑡)2𝐶₁𝛿𝑒^𝜆max𝑡. (85) From Lemma 5, direct computation gives 𝐰^𝛿(⋅,𝑡)_𝐻2≤𝐶₄𝛿𝐰_0𝐻2+𝛿𝑒^𝜆max𝑡. (86) It follows from (86) and (𝑎+𝑏)^𝑝≤2^𝑝−1(𝑎^𝑝+𝑏^𝑝)(𝑎,𝑏≥0,𝑝≥1) that 𝐰^𝛿(⋅,𝑡)³_𝐻2≤4𝐶₄^3/2𝛿³𝐰₀³_𝐻2+𝛿³𝑒^3𝜆max𝑡. (87)
Secondly, we establish a sharper 𝐿² estimate for 𝐰^𝛿(𝐱,𝑡) for 0≤𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}. Applying Duhamel’s principle, we get 𝐰^𝛿(⋅,𝑡)=𝛿𝑒^𝔏𝑡𝐰₀−^𝑡₀𝑒^{𝔏(𝑡−𝜏)}𝜒1−2𝑈_𝑐∇𝑢^𝛿(𝜏)∇𝑣^𝛿(𝜏)−2𝜒𝑢^𝛿(𝜏)∇𝑢^𝛿(𝜏)∇𝑣^𝛿(𝜏)+𝜒1−2𝑈_𝑐𝑢^𝛿(𝜏)∇²𝑣^𝛿(𝜏)−𝜒𝑢^𝛿(𝜏)²∇²𝑣^𝛿(𝜏)+𝜇𝑈_𝑐𝑢^𝛿(𝜏)²,0𝑑𝜏. (88) By Lemma 1, (57), (59), and Lemma 5, for 0≤𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}, we know 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≤𝐶₁𝐶₅^𝑡₀𝑒^{𝜆max(𝑡−𝜏)}𝐰^𝛿(𝜏)²_𝐻2+𝐰^𝛿(𝜏)³_𝐻2𝑑𝜏, (89) where 𝐶₅=𝜒|1−2𝑈_𝑐|(𝐶²₁₂/𝐶²₉+𝐶₆)+𝜇𝐶₇/𝜒|1−2𝑈_𝑐|𝑈_𝑐+𝜒𝐶₆(2𝐶²₁₂/𝐶²₉+1). By our choice of 𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}, it is further bounded by 𝐰^𝛿(𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≤𝐶₁𝐶₄𝐶₅𝛿𝐰₀²_𝐻2+4𝐶₄𝛿²𝐰₀³_𝐻2𝜆_max+𝛿𝑒^𝜆max𝑡+2𝐶₄𝛿²𝑒^2𝜆max𝑡𝜆_max𝛿𝑒^𝜆max𝑡. (90)
Next, We prove by contradiction that for 𝛿 sufficiently small, 𝑇^𝛿=min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}. If 𝑇^∗∗ is the smallest, we can let 𝑡=𝑇^∗∗≤𝑇^𝛿 in (86) and (87) to get 𝐰^𝛿𝑇^∗∗_𝐻2+𝐰^𝛿𝑇^∗∗2_𝐻2≤𝐶₄𝛿𝐰_0𝐻2+𝐶₄𝛿²𝐰₀²_𝐻2+𝐶₄𝜃1+𝐶₄𝜃<1𝐶^∗₂𝑑min¹4,𝜒𝑈_𝑐1−𝑈_𝑐²2𝑑₁, (91) where 𝜃 satisfies (84) with 𝐶₄≥1 and 𝛿 is sufficiently small such that 𝐶₄𝛿‖𝐰₀‖_𝐻2+𝐶₄𝛿²‖𝐰₀‖²_𝐻2≤(1/2𝐶^∗₂)min{𝑑₁/4,[𝜒𝑈_𝑐(1−𝑈_𝑐)]²/2𝑑₁}. This is a contradiction to the definition of 𝑇^∗∗.
If 𝑇^∗ is the smallest, let 𝑡=𝑇^∗≤𝑇^𝛿 in (88), we see that 𝐰^𝛿⋅,𝑇^∗−𝛿𝑒^𝔏𝑇∗𝐰₀≤𝐶₁𝐶₄𝐶₅𝛿𝐰₀²_𝐻2+4𝐶₄𝛿²𝐰₀³_𝐻2𝜆_max+𝜃+2𝐶₄𝜃²𝜆_max𝛿𝑒^{𝜆max𝑇∗}<𝐶₁2𝛿𝑒^{𝜆max𝑇∗}, (92) for 𝛿 is small such that 𝐶₄𝐶₅((𝛿‖𝐰₀‖²_𝐻2+4𝐶₄𝛿²‖𝐰₀‖³_𝐻2)/𝜆_max)<1/4, by our choice of 𝜃 in (84) and let 𝐶₅/𝐶^∗₂≤1. This again contradicts the definition of 𝑇^∗. Hence, 𝑇^𝛿 is the smallest.
Finally, we can obtain from (31) that 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Ωmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)≤𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀+𝛿_𝐪∈Ωmax𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}𝑒_𝐪(𝐱)+𝛿_{𝐪∈Ω⧵Ωmax}𝑤⁻_𝐪𝐫₋(𝐪)𝑒^{𝜆−𝐪𝑡}+𝑤⁺_𝐪𝐫₊(𝐪)𝑒^{𝜆+𝐪𝑡}𝑒_𝐪(𝐱)=𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀+𝐼₁+𝐼₂. (93) By (30), (37), (39), and the definition of 𝜌, we get 𝐼²₁≤𝛿²𝑒^{2(𝜆max−𝜌)𝑡}𝜋2^𝑑_𝐪∈Ωmax𝑤⁻_𝐪²𝐫₋(𝐪)²≤𝛿²𝑒^{2(𝜆max−𝜌)𝑡}𝐶²₂𝐶²₄𝜋2^𝑑_𝐪∈Ωmax𝑞⁶𝐰_𝑞². (94) From (25) we know that there is one (or two) 𝑞² satisfying 𝜆⁺(𝑞²)=𝜆_max. If there is only one 𝑞² satisfying 𝜆⁺(𝑞²)=𝜆_max, we denote it by 𝑞²_max; if there are 𝑞²₁ and 𝑞²₂ satisfying 𝜆⁺(𝑞²_𝑖)=𝜆_max(𝑖=1,2), we can let 𝑞²_max=max{𝑞²₁,𝑞²₂}. It follows from (30) and (94) that 𝐼₁≤𝐶₆𝛿𝑒^{(𝜆max−𝜌)𝑡}, (95) where