Pattern Formation of a Keller-Segel Model with the Source Term 𝑢^𝑝(1−𝑢)

Abstract

Nonlinear dynamics near an unstable constant equilibrium in a Keller-Segel model with the source term 𝑢^𝑝(1−𝑢) is considered. It is proved that nonlinear dynamics of a general perturbation is determined by the finite number of linear growing modes over a time scale of ln(1/𝛿), where 𝛿 is a strength of the initial perturbation.

1. Introduction

Mimura and Tsujikawa in [1] proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis, a dimensionless prototype of which reads 𝑈_𝑡=∇𝐷_𝑢(𝑈),𝑉∇𝑈−𝜒𝑈∇𝑉+𝑔_𝑡=𝐷_𝑣∇²𝑉+𝛼𝑈−𝛽𝑉, (1) where 𝑈(𝑥,𝑡) is the cell density, 𝑉(𝑥,𝑡) is the concentration of chemotactic substance, 𝐷_𝑢>0 is the amoeboid motility, 𝜒>0 is the chemotactic sensitivity, 𝐷_𝑣>0 is the diffusion rate of cyclic adenosine monophosphate (cAMP), 𝛼>0 is the rate of cAMP secretion per unit density of amoebae, and 𝛽>0 is the rate of degradation of cAMP in environment. In [1], the growth term 𝑔(𝑈) is classified into the three cases: (i) 𝑔(0)=0 and 𝑔(𝑢)<0, for any 𝑢>0, (ii) (bistable type) 𝑔(0)=𝑔(𝑎)=𝑔(1)=0, for some 0<𝑎<1, 𝑔(𝑢)<0, for 0<𝑢<𝑎, and 𝑔(𝑢)>0, for 𝑎<𝑢<1, and (iii) (Logistic type) 𝑔(0)=𝑔(1)=0 and 𝑔(𝑢)>0, for 0<𝑢<1.

For model (1), with a Logistic source term 𝑔(𝑢)=𝑢(1−𝑢), Tello and Winkler [2] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al. [3] numerically showed several spatiotemporal patterns in a rectangle. Kuto et al. [4] considered some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints. Banerjee et al. [5] showed that the corresponding dynamics may lead to steady states, to divergences in a finite time as well as to the formation of spatiotemporal irregular patterns. Painter and Hillen [6] demonstrated the capacity of (1) to self-organize into multiple cellular aggregations, which, according to position in parameter space, either form a stationary pattern or undergo a sustained spatiotemporal sequence of merging (two aggregations coalesce) and emerging (a new aggregation appears). Numerical explorations into the latter indicate a positive Lyapunov exponent (sensitive dependence to initial conditions) together with a rich bifurcation structure. They argued that the spatiotemporal irregularity observed here describes a form of spatiotemporal chaos.

For model (1) with a logistic-like growth term 𝑔(𝑢)=𝑢²(1−𝑢), Aida et al. [7] estimated from below the attractor dimension of (1). Efendiev et al. [8] showed that dimension estimates of global attractors for the approximate systems are uniform with respect to the discretization parameter and polynomial order with respect to the chemotactic coefficient in the equation. By using nonnegativity of solutions, Nakaguchi and Efendiev [9] managed significantly to improve dimension estimates with respect to the chemotactic parameter. It is also well-known that the asymptotic behavior of solutions relating to patterns can be described by the dynamical systems of equations and that the degrees of freedom of such processes, which characterize the richness of emerging patterns, correspond to the dimensions of their attractors.

Recently, Guo and Hwang in [10] investigated nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model (i.e., (1) with 𝑔(𝑢)=0, see [11]). Their result can be interpreted as a rigorous mathematical characterization for pattern formation in the Keller-Segel model.

In the present paper, we consider the nonlinear dynamics near an unstable constant equilibrium for the following chemotaxis-diffusion-growth model: 𝑈_𝑡=∇∇𝑈−𝜒𝑈∇𝑉+𝑈^𝑝(1−𝑈),𝑥∈𝕋^𝑑𝑉,𝑡>0,_𝑡=∇²𝑉+𝑈−𝛽𝑉,𝑥∈𝕋^𝑑,𝑡>0, (2) which satisfies the homogeneous Neumann boundary conditions and initial value conditions for 𝑈(𝑥,𝑡) and 𝑉(𝑥,𝑡), that is, 𝜕𝑈𝜕𝑥_𝑖=𝜕𝑉𝜕𝑥_𝑖=0,𝑥_𝑖=0,𝜋,1≤𝑖≤𝑑,𝑡>0,𝑈(𝑥,0)=𝑈₀(𝑥)≥(≢)𝑉()0,𝑥,0=𝑉₀(𝑥)≥(≢)0,𝑥∈𝕋^𝑑, (3)(4) where 𝕋^𝑑=(0,𝜋)^𝑑(𝑑=1,2,3) is a 𝑑-dimensional box, 𝑝≥1. By using the bootstrap technique in [10] and higher-order energy estimates, we prove 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 differently from another, which gives rise to the richness of patterns.

2. Local Stability of Positive Constant Equilibrium Solution

The PDE system (2) without chemotaxis is as follows 𝑈_𝑡=∇²𝑈+𝑈^𝑝(1−𝑈),𝑥∈𝕋^𝑑()𝑉𝑑=1,2,3,𝑡>0,_𝑡=∇²𝑉+𝑈−𝛽𝑉,𝑥∈𝕋^𝑑(𝑑=1,2,3),𝑡>0,𝜕𝑈𝜕𝑥_𝑖=𝜕𝑉𝜕𝑥_𝑖=0,at𝑥_𝑖=0,𝜋,for1≤𝑖≤𝑑, (5)[𝑈,𝑉]=[1,1/𝛽] is the unique positive equilibrium point. We use [⋅,⋅] to denote a column vector. Let 𝐖=[𝑈,𝑉], 𝐖=[𝑈,𝑉], and 𝐆(𝐖)=[𝑈^𝑝(1−𝑈),𝑈−𝛽𝑉]. Then 𝐆_𝐖𝐖=.−101−𝛽 (6)

Theorem 1. The positive equilibrium point [𝑈,𝑉] of (5) is locally asymptotically stable.

Proof. 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 let 𝐸(𝜇_𝑖) be the eigenspace corresponding to 𝜇_𝑖 in 𝐻¹(𝕋^𝑑). Let 𝑋=[𝐻¹(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=⊕^{dim[𝐸(𝜇𝑖)]}_𝑗=1𝑋_𝑖𝑗. (7) Let 𝔇=diag(1,1) and 𝔏=𝔇Δ+𝐆_𝐖(𝐖). The linearization of (5) at [𝑈,𝑉] is 𝐖_𝑡=𝔏𝐖−𝐖. (8) 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+𝐆_𝐖𝐖=−𝜇_𝑖−101−𝜇_𝑖.−𝛽 (9) Then −𝜇_𝑖𝔇+𝐆_𝐖(𝐖) has two negative eigenvalues −𝜇_𝑖−1 and −𝜇_𝑖−𝛽. Hence, [𝑈,𝑉] is locally asymptotically stable (see [12]).

3. Growing Modes in the System (2)

Let 𝑢(𝐱,𝑡)=𝑈(𝐱,𝑡)−𝑈, 𝑣(𝐱,𝑡)=𝑉(𝐱,𝑡)−𝑉. Then 𝑢_𝑡=∇²𝑢−𝜒∇²𝑣−𝜒∇(𝑢∇𝑣)−(𝑢+1)^𝑝𝑣𝑢,_𝑡=∇²𝑣+𝑢−𝛽𝑣. (10) The corresponding linearized system takes the form 𝑢_𝑡=∇²𝑢−𝜒∇²𝑣𝑣−𝑢,_𝑡=∇²𝑣+𝑢−𝛽𝑣. (11)

Let 𝐰(𝐱,𝑡)≡[𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)], 𝐪=(𝑞₁,…,𝑞_𝑑)∈Ω=ℕ^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 satisfy Neumann boundary conditions (3). To find a normal mode to the linear system (11) of the following form 𝐰(𝐱,𝑡)=𝐫_𝐪exp𝜆_𝐪𝑡𝑒_𝐪(𝐱), (12) where 𝐫_𝐪 is a vector depending on 𝐪, we substitute (12) into (11) to get 𝜆_𝐪𝐫_𝐪=−𝑞²−1𝜒𝑞²1−𝑞²𝐫−𝛽^𝐪, (13) where 𝑞²=∑^𝑑_𝑖=1𝑞²_𝑖. This implies that 𝜆det^𝐪+𝑞²+1−𝜒𝑞²−1𝜆_𝐪+𝑞²+𝛽=0,𝜆²_𝐪+2𝑞²+𝛽+1𝜆_𝐪+𝑞²+1𝑞²+𝛽−𝜒𝑞²=0. (14)(15) Thus, if there exists a 𝐪 such that the linear instability criterion holds, that is, 𝑞²+1𝑞²+𝛽−𝜒𝑞²<0, (16) then (15) has at least one positive root 𝜆_𝐪. Obviously, 2𝑞²+𝛽+1²−4𝑞²+1𝑞²+𝛽+4𝜒𝑞²>0. (17) Therefore, we can denote two distinct real roots for all 𝐪 by 𝜆^±_𝐪=−2𝑞²+𝛽+1±𝛽−1²+4𝜒𝑞²2. (18) The corresponding (linearly independent) eigenvectors 𝐫₋(𝐪) and 𝐫₊(𝐪) are given by 𝐫_±(𝐪)=𝜆^±_𝐪+𝑞²+𝛽,1. (19) It is easy to see from (16) that there exist only finitely many 𝑞, such that 𝜆⁺_𝐪>0. We therefore denote the largest eigenvalue by 𝜆_max>0 and define Ω_max≡𝐪∈Ω∶𝜆⁺_𝐪=𝜆_max. (20) Moreover, there is one 𝑞² (possibly two) having 𝜆⁺(𝑞²)=𝜆_max, when we regard 𝜆⁺_𝐪 as a function of 𝑞². We also denote 𝜈>0 to be the gap between the 𝜆_max and the rest, that is, 𝜈=min_{𝐪∈Ω⧵Ωmax}𝜆_max−𝜆_𝐪. (21) Given any initial perturbation 𝐰(𝐱,0), that is, 𝐰(𝐱,0)=_𝐪∈Ω𝐰_𝐪𝑒_𝐪(𝐱)=_𝐪∈Ω𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱), (22) where 𝐰_𝐪=𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪), (23) we know that the unique solution 𝐰(𝐱,𝑡)=[𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] of (11) is given by 𝐰(𝐱,𝑡)=_𝐪∈Ω𝑤⁻_𝐪𝐫₋(𝐪)exp𝜆⁻_𝐪𝑡+𝑤⁺_𝐪𝐫₊(𝐪)exp𝜆⁺_𝐪𝑡×𝑒_𝐪(𝐱)≡𝑒^𝔏𝑡𝐰(𝐱,0). (24) For any 𝐠(⋅,𝑡)∈[𝐿²(𝕋^𝑑)]², we denote ‖𝐠(⋅,𝑡)‖≡‖𝐠(⋅,𝑡)‖_𝐿² and ⟨⋅,⋅⟩ and (⋅,⋅) are the inner product of [𝐿²(𝕋^𝑑)]² and the scaler product of ℝ², respectively. Our main result of this section is the following lemma.

Lemma 2. Assume that the instability criterion (16) holds. Let 𝐰(𝐱,𝑡)=[𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)]≡𝑒^𝔏𝑡𝐰(𝐱,0) (25) be a solution to the linearized system (11) with initial condition 𝐰(𝐱,0). Then there exists a constant 𝐶₁≥1 depending on 𝜒 and 𝛽, such that ‖𝐰(⋅,𝑡)‖≤𝐶₁exp𝜆_max𝑡‖𝐰(⋅,0)‖,∀𝑡>0. (26)

Proof. We prove the Lemma in the following two cases.
(1) 𝑡≥1. By analyzing (15), we have lim_𝐪→∞𝜆^±_𝐪𝑞²=−1. (27) It follows from (18) that 𝜆⁺_𝐪−𝜆⁻_𝐪𝑞²≥2√𝜒𝑞. (28) By solving (23), we have 𝑤^±_𝐪≤𝐫_±(𝐪)×𝐰_𝐪det𝐫₋(𝐪),𝐫₊(𝐪.) (29) Clearly, 𝐫_±(𝐪)=𝜆^±_𝐪𝑞²⋅𝑞²+𝑞²+𝛽²+1. (30)
Later on, we will always denote universal constants by 𝐸_𝑖(𝑖=1,2,…). It is not hard to verify that there exist positive constants 𝐸₁ and 𝐸₂, such that 𝜆^±_𝐪≤𝐸₁𝑞²,𝐫_±(𝐪)≤2𝐸₂𝑞², (31) where 𝐸₂=max{𝐸₁+1,𝛽+1}. It follows from (19) and (28) that 1det𝐫₋(𝐪),𝐫₊(𝐪≤1)2𝑞1𝜒. (32) Combining (29)–(32), we can obtain 𝑤^±_𝐪≤𝐸₃𝑞𝐰_𝐪, (33) where 𝐸₃=𝐸₂1/𝜒. From (31) and (33), we have 𝑤^±_𝐪𝐫_±(𝐪)exp𝜆^±_𝐪𝑡≤2𝐸₂𝐸₃𝑞³𝐰_𝐪exp𝜆^±_𝐪𝑡. (34)
For 𝑡≥1, it follows from (18) that exp𝜆^±_𝐪𝑡−𝑞≤exp²𝑞²+𝛽+1−𝜒+𝛽2𝑞²−𝑞+𝛽+1≤exp²+𝛽+1−𝜒.𝛽+3 (35) Therefore, for all 𝑞>0, there exists a constant 𝐸₄>0, such that 𝑞³exp𝜆^±_𝐪𝑡≤𝐸₄. (36) From (34) and (36), we obtain 𝑤^±_𝐪𝐫_±(𝐪)exp𝜆^±_𝐪𝑡≤𝐸₅𝐰_𝐪, (37) where 𝐸₅=2𝐸₂𝐸₃𝐸₄. By (23), we have 𝐰_𝐪²=𝑤⁻_𝐪²𝐫₋(𝐪)²+2𝑤⁻_𝐪𝑤⁺_𝐪𝐫₋(𝐪),𝐫₊(𝐪)+𝑤⁺_𝐪²𝐫₊(𝐪)². (38) Notice that ‖𝐰(𝐱,0)‖²=𝜋2^𝑑_𝐪∈Ω𝐰_𝐪². (39) It follows from (37) and (39) that ‖𝐰(𝐱,𝑡)‖≤_𝐪∈Ω𝑤⁻_𝐪𝐫₋(𝐪)exp𝜆⁻_𝐪𝑡𝑒_𝐪(𝐱)+_𝐪∈Ω𝑤⁺_𝐪𝐫₊(𝐪)exp𝜆⁺_𝐪𝑡𝑒_𝐪(𝐱)≤𝜋2^𝑑_𝐪∈Ω𝐸²₅𝐰_𝐪^21/2+𝜋2^𝑑_𝐪∈Ω𝐸²₅𝐰_𝐪^21/2=2𝐸₅‖𝐰(𝐱,0)‖. (40) Therefore, ‖𝐰(𝐱,𝑡)‖≤2𝐸₅exp𝜆_max𝑡‖𝐰(𝐱,0)‖,∀𝑡≥1. (41)
(2) 𝑡≤1. By (11), we have 12𝑑𝑑𝑡_𝕋^𝑑|𝑢|²+𝐴|𝑣|²+𝐝𝐱_𝕋𝑑|∇𝑢|²+𝐴|∇𝑣|²−𝜒∇𝑢∇𝑣𝐝𝐱+_𝕋𝑑𝑢²𝐝𝐱+𝐴𝛽_𝕋𝑑𝑣²𝐝𝐱=𝐴_𝕋𝑑𝑢𝑣𝐝𝐱. (42) If 𝐴=𝜒², then the integrand of the second integral can be chosen nonnegative as follows: |∇𝑢|²+𝐴|∇𝑣|²1−𝜒∇𝑢∇𝑣≥2|∇𝑢|²+𝜒²2|∇𝑣|²≥0. (43)
Combining (42) and Young inequality together, we derive 12𝑑𝑑𝑡_𝕋^𝑑|𝑢|²+𝐴|𝑣|²≤𝐴𝐝𝐱2_𝕋^𝑑|𝑢|²+|𝑣|²𝐝𝐱. (44)
By Grownwall inequality, we can obtain ‖𝐰(𝐱,𝑡)‖≤𝐶₁exp𝜆_max𝑡‖𝐰(𝐱,0)‖, (45) where 𝐶₁=max{2𝐸₅,((exp𝐴)/𝐴)^1/2}≥1, if 0<𝐴<1; 𝐶₁=max{2𝐸₅,(𝐴exp𝐴)^1/2}≥1, if 𝐴>1.

4. Bootstrap Lemma

By a standard PDE theory [13], one can establish the existence of local solutions for (10).

Lemma 3 (local existence). For 𝑠≥1(𝑑=1) and 𝑠≥2(𝑑=2,3), there exists a 𝑇₀>0, such that (10) 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, (46) where 𝐶 is a positive constant depending on 𝜒 and 𝛽.

Lemma 4. Let [𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] be a solution of (10). Then 12𝑑𝑑𝑡_{|𝛼|=22𝕋}^𝑑𝐷^𝛼𝑢²+𝜒²𝐷^𝛼𝑣²𝐝𝐱+_{|𝛼|=22𝕋𝑑}14∇𝐷^𝛼𝑢²+𝜒²2∇𝐷^𝛼𝑣²+𝐝𝐱𝐴𝛽2_{|𝛼|=22𝕋}^𝑑𝐷^𝛼𝑣²𝐝𝐱≤𝐶𝜒+2^𝑝−1𝑝+1²×‖𝐰‖_𝐻²+‖𝐰‖^𝑝_𝐻2∇³𝐰²+𝐶^′‖𝑢‖², (47) where 𝐶 is the universal constant and 𝐶^′=4{𝐴/2𝛽−1}³.

Proof. Notice that if 𝐰(𝑥,𝑡) is a solution of (10) on (0,𝜋), then 𝐰(−𝑥_𝑖,𝑡) is a solution of (10) on (−𝜋,0). Let [𝑢(𝐱,𝑡),𝑣(̃𝐱,𝑡)] be a solution of (10) on (−𝜋,𝜋)^𝑑, which is an even extension of [𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] on (−𝜋,0)^𝑑. Then one may assume periodicity at the boundary of the extended 2𝕋^𝑑≡(−𝜋,𝜋)^𝑑.
By (10), one has 12𝑑𝑑𝑡_2𝕋^𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²𝐝𝐱+𝐴𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²𝐝𝐱=−_2𝕋𝑑∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝐴∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²−𝜒∇𝜕_{𝑥𝑖𝑥𝑗}𝑢∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃−𝐝𝐱_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²𝐝𝐱+𝐴_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢𝜕_{𝑥𝑖𝑥𝑗}𝑣̃𝐝𝐱−𝐴𝛽_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²𝐝𝐱+𝜒_2𝕋𝑑∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅𝜕_{𝑥𝑖𝑥𝑗}{𝑣}̃+𝑢∇𝐝𝐱_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²−𝜕_{𝑥𝑖𝑥𝑗}𝑢𝜕_{𝑥𝑖𝑥𝑗}𝑣𝑢(𝑢+1)^𝑝𝐝𝐱=∶𝐼₁+𝐼₂, (48) where 𝐴=𝜒², and 𝐼₁=−_2𝕋𝑑∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝐴∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²−𝜒∇𝜕_{𝑥𝑖𝑥𝑗}𝑢∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃𝐝𝐱−𝐴𝛽_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²𝐝𝐱+𝐴_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢𝜕_{𝑥𝑖𝑥𝑗}̃−𝑣𝐝𝐱_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²𝐝𝐱. (49) Clearly, ∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝐴∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²−𝜒∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃≥12∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝜒²2∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃². (50) Thus, 𝐼₁≤−_2𝕋𝑑12∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝜒²2∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²−𝐝𝐱𝐴𝛽2_2𝕋^𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²+𝐴𝐝𝐱2𝛽−1_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²𝐝𝐱. (51) By interpolation inequalities, one knows that 𝜕_{𝑥𝑖𝑥𝑗}𝑢²≤𝑎∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+14𝑎²‖𝑢‖², (52) for any 𝑎>0. If 𝐴≤2𝛽, then one can delete the last term in the above inequality. If 𝐴>2𝛽, one can choose positive constant 𝑎 such that {𝐴/2𝛽−1}𝑎=1/4. Then 𝐼₁≤−_2𝕋𝑑14∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝜒²2∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²−𝐝𝐱𝐴𝛽2_2𝕋^𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑣̃²𝐝𝐱+𝐶^′‖𝑢‖², (53) where 𝐶^′=4{𝐴/2𝛽−1}³.
One now estimates the second nonlinear term 𝐼₂: 𝐼₂=𝜒_2𝕋𝑑∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅𝜕_{𝑥𝑖𝑥𝑗}{𝑣}̃+𝑢∇𝐝𝐱_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²−𝜕_{𝑥𝑖𝑥𝑗}𝑢𝜕_{𝑥𝑖𝑥𝑗}×𝑢(𝑢+1)^𝑝𝐝𝐱=∶𝐼^′₂+𝐼₂^′′. (54) Clearly, 𝐼^′₂≤𝜒‖∇𝑣‖̃_𝐿^∞∇𝜕_{𝑥𝑖𝑥𝑗}𝑢𝜕_{𝑥𝑖𝑥𝑗}𝑢+2^𝑑_𝑖=1‖∇𝑢‖_𝐿^∞𝜕_{𝑥𝑖𝑥𝑗}𝑣̃∇𝜕_{𝑥𝑖𝑥𝑗}𝑢+‖𝑢‖_𝐿^∞∇𝜕_{𝑥𝑖𝑥𝑗}𝑢∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃. (55) Notice that 𝑔_{𝐿∞(2𝕋𝑑)}≤𝐸₀𝑔_{𝐻2(2𝕋𝑑)},for𝑑≤3,_2𝕋𝑑∇𝑢𝐝𝐱=_2𝕋𝑑∇𝑣̃𝐝𝐱=0,_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢𝐝𝐱=_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑣̃𝐝𝐱=0,𝑔≤𝑔_{𝐿6(2𝕋𝑑)}≤𝐸₁𝑔_𝐻1≤𝐸₁∇𝑔if𝑑≤3,∇𝑔_𝐻2≤𝐸_2|𝛼|=2∇𝐷^𝛼𝑔^21/2. (56)(57)(58)(59)
From (56)–(59), one can obtain 𝐼^′₂≤𝜒𝐸₀‖𝑢‖_𝐻²‖∇𝑣‖̃_𝐻²∇𝜕_{𝑥𝑖𝑥𝑗}𝑢+2𝑑𝐸₀‖𝑣‖̃_𝐻²‖∇𝑢‖_𝐻²∇𝜕_{𝑥𝑖𝑥𝑗}𝑢+𝐸₀‖𝑢‖_𝐻²∇𝜕_{𝑥𝑖𝑥𝑗}𝑢∇𝜕_{𝑥𝑖𝑥𝑗}𝑣̃≤𝜒𝐸₀𝐸₂+2𝑑𝐸₀𝐸₂+𝐸₀×‖𝐰‖_𝐻²_|𝛼|=2∇𝐷^𝛼𝐰²≤𝜒𝐸₃‖𝐰‖_𝐻²∇³𝐰². (60) Similarly, one has 𝐼₂^′′≤_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²−(𝑢+1)^𝑝𝜕_{𝑥𝑖𝑥𝑗}𝑢²𝐝𝐱+𝑝𝑝−1_2𝕋𝑑(𝑢+1)^𝑝−2𝑢𝜕_𝑥𝑖𝑢𝜕_𝑥𝑗𝑢𝜕_{𝑥𝑖𝑥𝑗}𝑢𝐝𝐱+𝑝_2𝕋𝑑(𝑢+1)^𝑝−1𝑢𝜕_{𝑥𝑖𝑥𝑗}𝑢²𝐝𝐱+2𝑝_2𝕋𝑑(𝑢+1)^𝑝−1𝜕_𝑥𝑖𝑢𝜕_𝑥𝑗𝑢𝜕_{𝑥𝑖𝑥𝑗}𝑢𝐝𝐱≤2^𝑝−1𝑀₂𝐸^𝑝₀𝑝𝑝+1𝐸²₂+𝑝+𝑀₁𝐸²₁‖𝐰‖^𝑝_𝐻2∇³𝐰². (61) Here and later, 𝑀₁,𝑀₂,… are universal constants. It follows from (60) and (61) that 𝐼₂≤𝜒𝐸₃‖𝐰‖_𝐻²∇³𝐰²+2^𝑝−1𝑀₂𝐸^𝑝₀𝑝𝑝+1𝐸²₂+𝑝+𝑀₁𝐸²₁×‖𝐰‖^𝑝_𝐻2∇³𝐰²≤𝐶𝜒+2^𝑝−1𝑝+1²×‖𝐰‖_𝐻²+‖𝐰‖^𝑝_𝐻2∇³𝐰², (62) where 𝐶=max{𝐸₃,𝑀₂𝐸^𝑝₀𝐸²₂,𝑀₂𝐸^𝑝₀𝐸²₁,𝑀₁𝑀₂𝐸^𝑝₀𝐸²₁}.
Recall that [𝑢,𝑣]̃ is the even extension of [𝑢,𝑣]. Combining (49), (53), and (62), one has 12𝑑𝑑𝑡_{|𝛼|=22𝕋}^𝑑𝐷^𝛼𝑢²+𝜒²𝐷^𝛼𝑣²𝐝𝐱+_{|𝛼|=22𝕋𝑑}14∇𝐷^𝛼𝑢²+𝜒²2∇𝐷^𝛼𝑣²+𝐝𝐱𝐴𝛽2_{|𝛼|=22𝕋}^𝑑𝐷^𝛼𝑣²𝐝𝐱≤𝐶𝜒+2^𝑝−1𝑝+1²×‖𝐰‖_𝐻²+‖𝐰‖^𝑝_𝐻2∇³𝐰²+𝐶^′‖𝑢‖², (63) where 𝐶=max{𝐸^𝑝₀(𝐸²₁+𝐸²₂)+𝐸²₁,𝐸₃} and 𝐶^′=4{𝐴/2𝛽−1}³.

Lemma 5. Let 𝐰(𝐱,𝑡) be a solution of (10), such that for 0≤𝑡≤𝑇<𝑇₀, ‖𝐰(⋅,𝑡)‖_𝐻²+‖𝐰(⋅,𝑡)‖^𝑝_𝐻2≤1𝐶1min4𝜒+2^𝑝−1𝑝+1²,𝜒²2𝜒+2^𝑝−1𝑝+1²,‖𝐰(⋅,𝑡)‖≤2𝐶₁exp𝜆_max𝑡‖𝐰(⋅,0)‖. (64)(65) Then ‖𝐰(⋅,𝑡)‖^𝑝_𝐻2≤𝐶₃‖𝐰(⋅,0)‖²_𝐻2+exp2𝜆_max𝑡‖𝐰(⋅,0)‖^2𝑝/2,0≤𝑡≤𝑇, (66) where 𝐶₃=max{((𝐸²₁+1)𝜒²)^𝑝/2,{4𝐶²₁[1+(𝐸²₁+1)𝐶^′/𝜆_max]}^𝑝/2}, if 𝜒²≥1 and 𝐶₃=max{((𝐸²₁+1)/𝜒²)^𝑝/2,{4𝐶²₁[1+(𝐸²₁+1)𝐶^′/𝜆_max𝜒²]}^𝑝/2}, if 𝜒²<1.

Proof. By (58), we have ‖∇𝐰(⋅,𝑡)‖²≤𝐸²_1|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)². (67) It follows from (67) that ‖𝐰(⋅,𝑡)‖^𝑝_𝐻2≤‖𝐰(⋅,𝑡)‖²+𝐸²₁+1_|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)^2𝑝/2. (68)
Now, we estimate the second-order derivatives of 𝐰(⋅,𝑡). By Lemma 4, we have 12𝑑𝑑𝑡_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑢²+𝜒²𝐷^𝛼𝑣²𝐝𝐱≤𝐶^′‖𝑢‖²≤𝐶^′‖𝐰(⋅,𝑡)‖². (69) Integrating on both sides of (69) from 0 to 𝑡, we get from (65) the following: _{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢(⋅,𝑡)²+𝜒²𝐷^𝛼𝑣(𝐱,𝑡)²𝐝𝐱≤_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢(⋅,0)²+𝜒²𝐷^𝛼𝑣(⋅,0)²+𝐝𝐱4𝐶²₁𝐶^′𝜆_max‖𝐰(⋅,0)‖²exp2𝜆_max𝑡. (70)
We will proceed in the following two cases: 𝜒²≥1 and 𝜒²<1.
(1) If 𝜒²≥1, we have _|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)²≤_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢(⋅,0)²+𝜒²𝐷^𝛼𝑣(⋅,0)²+𝐝𝐱4𝐶²₁𝐶^′𝜆_max‖𝐰(⋅,0)‖²exp2𝜆_max𝑡≤𝜒²_|𝛼|=2𝐷^𝛼𝐰(⋅,0)²+4𝐶²₁𝐶^′𝜆_max‖𝐰(⋅,0)‖²exp2𝜆_max𝑡. (71) By (68) and (71), one can obtain ‖𝐰(⋅,𝑡)‖^𝑝_𝐻2≤𝐸²₁+1𝜒²_|𝛼|=2𝐷^𝛼𝐰(⋅,0)²+4𝐶²₁+𝐸²₁+14𝐶²₁𝐶^′𝜆_max×‖𝐰(⋅,0)‖²exp2𝜆_max𝑡^𝑝/2≤𝐶₃‖𝐰(⋅,0)‖²_𝐻2+‖𝐰(⋅,0)‖²exp2𝜆_max𝑡^𝑝/2, (72) where 𝐶₃=max{[(𝐸²₁+1)𝜒²]^𝑝/2, {4𝐶²₁[1+(𝐸²₁+1)𝐶^′/𝜆_max]^𝑝/2}.
(2) If 𝜒²<1, from (70), we have 𝜒²_|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)²≤_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢(⋅,0)²+𝜒²𝐷^𝛼𝑣(⋅,0)²+𝐝𝐱4𝐶²₁𝐶^′𝜆_max‖𝐰(⋅,0)‖²exp2𝜆_max𝑡≤_|𝛼|=2𝐷^𝛼𝐰(⋅,0)²+4𝐶²₁𝐶^′𝜆_max‖𝐰(⋅,0)‖²exp2𝜆_max𝑡. (73) Therefore, _|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)²≤1𝜒²_|𝛼|=2𝐷^𝛼𝐰(⋅,0)²+4𝐶²₁𝐶^′𝜒²𝜆_max‖𝐰(⋅,0)‖²exp2𝜆_max𝑡. (74) Combining (68) and (74), we have ‖𝐰(⋅,𝑡)‖^𝑝_𝐻2≤𝐸²₁+1𝜒²_|𝛼|=2𝐷^𝛼𝐰(⋅,0)²+4𝐶²₁𝐸1+²¹+1𝐶^′𝜆_max𝜒²×‖𝐰(⋅,0)‖²exp2𝜆_max𝑡^𝑝/2≤𝐶₃‖𝐰(⋅,0)‖²_𝐻2+‖𝐰(⋅,0)‖²exp2𝜆_max𝑡^𝑝/2, (75) where 𝐶₃=max{[(𝐸²₁+1)/𝜒²]^𝑝/2, {4𝐶²₁[1+(𝐸²₁+1)𝐶^′/𝜆_max𝜒²]}^𝑝/2}.

5. Main Result

Let 𝜃 be a small fixed constant, and let 𝜆_max be the dominant eigenvalue, which is the maximal growth rate. We also denote the gap between the largest growth rate 𝜆_max and the rest by 𝜈>0. Then for 𝛿>0 arbitrary small, we define the escape time 𝑇^𝛿 by 𝜃=𝛿exp𝜆_max𝑇^𝛿, (76) or equivalently 𝑇^𝛿=1𝜆_max𝜃ln𝛿. (77)

Our main result of this paper is the following theorem.

Theorem 6. Assume that the set of 𝑞²=∑^𝑑_𝑖=1𝑞²_𝑖 satisfying instability criterion (16) is not empty for given parameters 𝜒 and 𝛽. Let 𝐰₀(𝐱)=_𝐪∈Ω𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)∈𝐻², (78) such that ‖𝐰₀‖=1. Then there exist constants 𝛿₀>0, 𝐶>0, and 𝜃>0, depending on 𝜒 and 𝛽, 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^−𝜈𝑡+𝛿𝐰₀²_𝐻2+𝛿^𝑝𝐰_0𝐻^𝑝+1₂+𝛿𝑒^𝜆max𝑡+𝛿^𝑝𝑒^{𝑝𝜆max𝑡}𝛿𝑒^𝜆max𝑡, (79) for 0≤𝑡≤𝑇^𝛿, and 𝜈>0 is the gap between 𝜆_max and the rest of 𝜆_𝐪 in (15).

Proof. Let 𝐰^𝛿(𝐱,𝑡) be the solutions of (10) with initial data 𝐰^𝛿(⋅,0)=𝛿𝐰₀. We define 𝑇^∗=sup𝑡∣𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≤𝐶₁2𝛿exp𝜆_max𝑡,𝑇^∗∗=sup𝑡∣𝐰^𝛿(⋅,𝑡)_𝐻2+𝐰^𝛿(⋅,𝑡)^𝑝_𝐻2≤1𝐶^∗₀1min4𝜒+2^𝑝−1𝑝+1²,𝜒²2𝜒+2^𝑝−1𝑝+1². (80)(81) Choose 𝜃, such that 𝐶^∗₀𝐶₃4𝜃+^𝑝𝑝𝐶₃^(𝑝−1)/2𝜃^𝑝𝜆<min^max4,1𝜒+18𝜒+2^𝑝−1𝑝+1²,𝜒²4𝜒+2^𝑝−1𝑝+1². (82)
Now, we establish estimates for 𝐻² norm of 𝐰^𝛿(𝐱,𝑡) for 𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}. For 0<𝑡≤𝑇^∗, by Lemma 2, we have 𝐰^𝛿(⋅,𝑡)≤𝛿𝑒^𝔏𝑡𝐰₀+𝐶₁2𝛿exp𝜆_max𝑡≤𝐶₁𝛿exp𝜆_max𝑡‖𝐰(+𝐶⋅,0)‖₁2𝛿exp𝜆_max𝑡=32𝐶₁𝛿exp𝜆_max𝑡. (83) Applying Lemma 5 and the bootstrap argument yields 𝐰^𝛿(⋅,𝑡)_𝐻2≤𝐶₃𝛿𝐰_0𝐻2+𝛿exp𝜆_max𝑡. (84) From this and (𝑎+𝑏)^𝑝≤2^𝑝−1(𝑎^𝑝+𝑏^𝑝)(𝑎≥0,𝑏≥0,𝑝≥1), we can obtain 𝐰^𝛿(⋅,𝑡)_𝐻^𝑝+1₂≤2^𝑝𝐶₃^(𝑝+1)/2𝛿^𝑝+1𝐰_0𝐻^𝑝+1₂+𝛿^𝑝+1exp𝑝+1𝜆_max𝑡. (85)
Applying Duhamel’s principle, we know that the solution of (10) is 𝐰^𝛿(⋅,𝑡)=𝛿𝑒^𝔏𝑡𝐰₀−^𝑡₀𝑒^{𝔏(𝑡−𝜏)}𝜒∇𝑢^𝛿∇𝑣^𝛿+𝑢^𝛿𝑢^𝛿+1^𝑝−𝑢^𝛿,0𝑑𝜏. (86) Using Lemma 2, (56), (58), and Lemma 5 yields, for 0≤𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}, 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≤𝐶₁^𝑡₀𝑒^{𝜆max(𝑡−𝜏)}𝑢^𝛿(𝜏)−𝑢^𝛿(𝜏)𝑢^𝛿(𝜏)+1^𝑝−𝜒∇𝑢^𝛿(𝜏)∇𝑣^𝛿(𝜏)𝑑𝜏≤𝐶₁𝑀₃^𝑡₀𝑒^{𝜆max(𝑡−𝜏)}𝐰^𝛿(𝜏)_𝐻^𝑝+1₂𝑑𝜏+𝐶₁𝐸²₁𝜒+𝐸₀𝜒^𝑡₀𝑒^{𝜆max(𝑡−𝜏)}𝐰^𝛿(𝜏)²_𝐻2𝑑𝜏≤𝐶₁𝐶^∗₀𝜒+1^𝑡₀𝑒^{𝜆max(𝑡−𝜏)}𝐰^𝛿(𝜏)²_𝐻2+𝐰^𝛿(𝜏)_𝐻^𝑝+1₂𝑑𝜏, (87) where 𝐶^∗₀=max{𝑀₃,𝐸²₁+𝐸₀}. Notice that 𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}. Thus 𝐰^𝛿(𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≤𝐶₁𝐶^∗₀𝐶₃×𝜒+1_𝑡0𝑒^{𝜆max(𝑡−𝜏)}𝛿²𝐰₀²_𝐻2+𝛿²𝑒^2𝜆max𝜏𝑑𝜏+2^𝑝𝐶₁𝐶^∗₀𝐶₃^(𝑝+1)/2×𝜒+1_𝑡0𝑒^{𝜆max(𝑡−𝜏)}𝛿^𝑝+1𝐰_0𝐻^𝑝+1₂+𝛿^𝑝+1𝑒^{(𝑝+1)𝜆max𝜏}𝑑𝜏≤𝐶₁𝐶^∗₀𝐶₃×𝛿𝐰𝜒+1_02𝐻2+2^𝑝𝐶₃^(𝑝−1)/2𝛿^𝑝𝐰_0𝐻^𝑝+1₂𝜆_max+𝛿𝑒^𝜆max𝑡+2^𝑝/𝑝𝐶₃^(𝑝−1)/2𝛿^𝑝𝑒^{𝑝𝜆max𝑡}𝜆_max𝛿𝑒^𝜆max𝑡. (88)
For 𝛿>0 sufficiently small, we claim that 𝑇^𝛿=min𝑇^𝛿,𝑇^∗,𝑇^∗∗. (89) If 𝑇^∗∗ is the smallest, we can let 𝑡=𝑇^∗∗≤𝑇^𝛿 in (84) and (85) to obtain 𝐰^𝛿⋅,𝑇^∗∗_𝐻2+𝐰⋅,𝑇^∗∗𝑝_𝐻2≤2^𝑝−1𝐶₃^𝑝/2𝛿^𝑝𝐰₀^𝑝_𝐻2+𝐶₃𝛿𝐰_0𝐻2+𝐶₃^1/2𝜃+2^𝑝−1𝐶₃^(𝑝−1)/2𝜃^𝑝<1𝐶^∗₀1min4𝜒+2^𝑝−1𝑝+1²,𝜒²2𝜒+2^𝑝−1𝑝+1², (90) for 2^𝑝−1𝐶₃^𝑝/2𝛿^𝑝‖𝐰₀‖^𝑝_𝐻2+𝐶₃𝛿‖𝐰₀‖_𝐻2≤(1/2𝐶^∗₀)min{1/4(𝜒+2^𝑝−1(𝑝+1)²),𝜒²/2(𝜒+2^𝑝−1(𝑝+1)²)} for 𝛿 small, by our choice of 𝜃 in (82) with 𝐶₃≥1. This is a contradiction to the definition of 𝑇^∗∗. On the other hand, if 𝑇^∗ is the smallest, we let 𝑡=𝑇^∗ in (88) to get 𝐰^𝛿⋅,𝑇^∗−𝛿𝑒^𝔏𝑇∗𝐰₀≤𝐶₁𝐶^∗₀𝐶₃×𝛿𝐰𝜒+1_02𝐻2+2^𝑝𝐶₃^(𝑝−1)/2𝛿^𝑝𝐰_0𝐻^𝑝+1₂𝜆_max+𝜃+2^𝑝/𝑝𝐶₃^(𝑝−1)/2𝜃^𝑝𝜆_max𝛿𝑒^{𝜆max𝑇∗}<𝐶₁2𝛿𝑒^{𝜆max𝑇∗}, (91) for 𝐶^∗₀𝐶₃(𝜒+1)(𝛿‖𝐰₀‖²_𝐻2+2^𝑝𝐶₃^(𝑝−1)/2𝛿^𝑝‖𝐰₀‖_𝐻^𝑝+1₂)/𝜆_max<1/4 for 𝛿 small, by our choice of 𝜃 in (82). This again contradicts the definition of 𝑇^∗. Hence, the desired assertion follows.
From (24), we obtain 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≥𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Ωmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)−𝛿_𝐪∈Ωmax𝑤⁻_𝐪𝐫₋(𝐪)exp𝜆⁻_𝐪𝑡𝑒_𝐪(𝐱)−𝛿_{𝐪∈Ω⧵Ωmax}𝑤⁻_𝐪𝐫₋(𝐪)exp𝜆⁻_𝐪𝑡+𝑤⁺_𝐪𝐫₊(𝐪)exp𝜆⁺_𝐪𝑡𝑒_𝐪(𝐱)=𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Ωmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)−𝐼₁−𝐼₂, (92) that is, 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Ωmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)≤𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀+𝐼₁+𝐼₂. (93) Using (34), we deduce that 𝐼²₁=𝛿_𝐪∈Ωmax𝑤⁻_𝐪𝐫₋(𝐪)exp𝜆⁻_𝐪𝑡𝑒_𝐪(𝐱)²≤4𝛿²𝑒^{2(𝜆max−𝜈)𝑡}𝜋2^𝑑_𝐪∈Ωmax𝐸²₂𝐸²₃𝑞⁶𝐰_𝑞². (94) From (18) 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 and if there are 𝑞²₁ and 𝑞²₂ satisfying 𝜆⁺(𝑞²)=𝜆_max, we can let 𝑞²_max=max{𝑞²₁,𝑞²₂}. From (94), we obtain 𝐼²₁≤4𝛿²𝑒^{2(𝜆max−𝜈)𝑡}𝐸²₂𝐸²₃𝑞⁶_max𝜋2^𝑑_𝐪∈Ωmax𝐰_𝐪²≤4𝐸²₂𝐸²₃𝑞⁶_max𝛿²𝑒^{2(𝜆max−𝜈)𝑡}𝐰₀², (95) that is, 𝐼₁≤2𝐸₂𝐸₃𝑞³_max𝛿𝑒^{(𝜆max−𝜈)𝑡}=𝐶^∗𝛿𝑒^{(𝜆max−𝜈)𝑡}, (96) where 𝐶^∗=2𝐸₂𝐸₃𝑞³_max. Now, we consider 𝐼₂𝐼²₂=𝛿²𝜋2^𝑑_{𝐪∈Ω⧵Ωmax}exp2𝜆⁺_𝐪𝑡𝑤⁺_𝐪²𝐫₊(𝐪)²+2exp𝜆⁺_𝐪+𝜆⁻_𝐪𝑡×𝑤⁺_𝐪𝑤⁻_𝐪𝐫₊(𝐪),𝐫₋(𝐪)+exp2𝜆⁻_𝐪𝑡𝑤⁻_𝐪²𝐫₋(𝐪)². (97) From (39) and (41), we get 𝐼²₂≤𝛿²𝑒^{2(𝜆max−𝜈)𝑡}𝜋2^𝑑×_{𝐪∈Ω⧵Ωmax}𝑤⁺_𝐪²𝐫₊(𝐪)²+2𝑤⁺_𝐪𝑤⁻_𝐪𝐫₊(𝐪),𝐫₋(𝐪)+𝑤⁻_𝐪²𝐫₋(𝐪)²≤𝛿²𝑒^{2(𝜆max−𝜈)𝑡}𝜋2^𝑑_𝐪∈Ω𝐰_𝐪²=𝛿²𝑒^{2(𝜆max−𝜈)𝑡}, (98) that is, 𝐼₂≤𝛿𝑒^{(𝜆max−𝜈)𝑡}. (99) From (88), (96), and (99), one can obtain 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Ωmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)≤𝐶₁𝐶^∗₀𝐶₃×𝛿𝐰𝜒+1_02𝐻2+2^𝑝𝐶₃^(𝑝−1)/2𝛿^𝑝𝐰_0𝐻^𝑝+1₂𝜆_max+𝛿𝑒^𝜆max𝑡+2^𝑝/𝑝𝐶₃^(𝑝−1)/2𝛿^𝑝𝑒^{𝑝𝜆max𝑡}𝜆_max𝛿𝑒^𝜆max𝑡+𝐶^∗𝛿𝑒^{(𝜆max−𝜈)𝑡}+𝛿𝑒^{(𝜆max−𝜈)𝑡}≤𝐶^∗+1𝑒^−𝜈𝑡+4^𝑝/𝑝𝐶₁𝐶^∗₀𝐶₃^𝑝+1𝜆_max×𝛿𝐰𝜒+1_02𝐻2+𝛿^𝑝𝐰_0𝐻^𝑝+1₂+𝛿𝑒^𝜆max𝑡+𝛿^𝑝𝑒^{𝑝𝜆max𝑡}𝛿𝑒^𝜆max𝑡≤𝐶𝑒^−𝜈𝑡+𝛿𝐰₀²_𝐻2+𝛿^𝑝𝐰_0𝐻^𝑝+1₂+𝛿𝑒^𝜆max𝑡+𝛿^𝑝𝑒^{𝑝𝜆max𝑡}𝛿𝑒^𝜆max𝑡, (100) where 𝐶=max{𝐶^∗+1,((4^𝑝/𝑝)𝐶₁𝐶^∗₀𝐶₃^𝑝+1/𝜆_max)(𝜒+1)}. This completes the proof.

6. Conclusion

Notice that for 0≤𝑡≤𝑇^𝛿, 𝛿𝑒^𝜆max𝑡≤𝜃 is sufficiently small. As long as 𝑤⁺_𝐪0≠0 for at least one 𝐪₀∈Ω_max, which is generic for perturbations, the corresponding fastest growing modes 𝛿𝑒^𝜆max𝑡_𝐪∈Ωmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)=𝛿𝑒^𝜆max𝑡𝜋2^𝑑/2×_𝐪∈Ωmax𝑤⁺_𝐪²𝐫₊(𝐪)^21/2≥𝛿𝑒^𝜆max𝑡𝑤⁺_𝐪0𝐫₊𝐪₀, (101) have the dominant leading order of 𝛿𝑒^𝜆max𝑡. Theorem 6 implies that the dynamics of a general perturbation is characterized by such linear dynamics over a long time period of 𝜀𝑇^𝛿≤𝑡≤𝑇^𝛿, for any 𝜀>0 (see [10]). In particular, choose a fixed 𝐪₀=(𝑞₀₁,𝑞₀₂,…,𝑞_0𝑑)∈Ω_max and let 𝐰₀(𝐱)=𝐫₊𝐪₀𝐫₊𝐪₀𝑒_𝐪0(𝐱). (102) Then 𝐰₀(𝐱)_𝐻2=𝜋2^𝑑1+𝐪₀²+𝐪₀^41/2. (103) Therefore, if 𝑡≤𝑇^𝛿, we can obtain from (100) 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^{𝜆max𝑇𝛿}𝐫₊𝐪₀𝐫₊𝐪₀𝑒_𝐪0(𝐱𝑒)≤𝐶^{−𝜈𝑇𝛿}+𝛿𝐰₀²_𝐻2+𝛿^𝑝𝐰_0𝐻^𝑝+1₂+𝛿𝑒^{𝜆max𝑇𝛿}+𝛿^𝑝𝑒^{𝑝𝜆max𝑇𝛿}𝛿𝑒^{𝜆max𝑇𝛿}≤𝐶𝜃exp−𝜈𝜆_max𝜃ln𝛿𝜋+𝜃𝛿2^𝑑1+𝐪₀²+𝐪₀⁴+𝜃𝛿^𝑝𝜋2^{(𝑝+1)𝑑/2}×1+|𝐪₀|²+|𝐪₀|^4(𝑝+1)/2+𝜃²+𝜃^𝑝+1. (104) Note that 𝛿≤𝜃, 𝜃, and 𝜈 are fixed constants, and 𝐪₀ is a fixed vector. Hence, we have 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^{𝜆max𝑇𝛿}𝐫₊𝐪₀𝐫₊𝐪₀𝑒_𝐪0(𝐱)≤𝐶𝐶₄𝛿^𝜈/𝜆max+𝜃²+𝜃^𝑝+1, (105) where 𝐶₄=max{𝜃^{1−𝜈/𝜆max},(𝜋/2)^{(𝑝+1)𝑑/2}(1+|𝐪₀|²+|𝐪₀|⁴)^(𝑝+1)/2+1}. Thus, 𝐰^𝛿(⋅,𝑡)≥𝛿𝑒^{𝜆max𝑇𝛿}𝐫₊𝐪₀𝐫₊𝐪₀𝑒_𝐪0(𝐱)−𝐶𝐶₄𝛿^𝜈/𝜆max+𝜃²+𝜃^𝑝+1≥𝜃−𝐶^∗∗𝛿^𝜈/𝜆max+𝜃²+𝜃^𝑝+1. (106)

Let 𝜃 be sufficiently small so that 𝜃/2𝐶^∗∗−𝜃²−𝜃^𝑝+1>0 holds. Then 𝛿₁=∶(𝜃/2𝐶^∗∗−𝜃²−𝜃^𝑝+1)^𝜆max/𝜈, and 𝐰^𝛿(⋅,𝑡)≥𝜃−𝐶^∗∗𝛿^𝜈/𝜆max+𝜃²+𝜃^𝑝+1≥𝜃2>0,0≤𝛿≤𝛿₁. (107) This implies nonlinear instability as 𝛿→0. Then instability occurs before the possible blow-up time.

Recall that the unique positive equilibrium point of (2) without chemotaxis is locally asymptotically stable. This means that the instability in (2) is called chemotaxis-driven instability since the presence of chemotaxis is essential for the instability mechanism leading to nonuniform patterns in the system (2).

Acknowledgments

The authors would like to thank the referee for their helpful comments. This work is supported by the China National Natural Science Foundation (no. 11061031; 11261053) and the Fundamental Research Funds for the Gansu University.