A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term

Abstract

This paper deals with a Neumann boundary value problem for a Keller-Segel model with a cubic source term in a d-dimensional box (𝑑=1,2,3), which describes the movement of cells in response to the presence of a chemical signal substance. 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 differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model.

1. Introduction

Keller and Segel in their pioneering work [1] proposed the following model 𝑈_𝑡=∇𝐷_𝑢,𝑉∇𝑈−𝜒𝑈∇𝑉_𝑡=𝐷_𝑣∇²𝑉+𝛼𝑈−𝛽𝑉, (1) where 𝑈(𝑥,𝑡) is cell density, 𝑉(𝑥,𝑡) is chemoattractant concentration, 𝐷_𝑢 is the amoeboid motility, 𝜒 is the chemotactic sensitivity, 𝐷_𝑣 is the diffusion rate of cyclic adenosine monophosphate (cAMP), 𝛼 is the rate of cAMP secretion per unit density of amoebae, and 𝛽 is the rate of degradation of cAMP in environment. Keller and Segel wanted to model the chemotactic movement of the cellular slime mold Dictyostelium discoideum during its aggregation phase, where population growth does not occur. Therefore, they considered a population in the absence of “death” and “birth.” For some main results on the Keller-Segel model, please see [2–4] and references therein.

Recently, Guo and Hwang in [5] investigated the nonlinear dynamics near an unstable constant equilibrium of the Keller-Segel model satisfying the Neumann boundary conditions for 𝑈(𝑥,𝑡) and 𝑉(𝑥,𝑡) on a 𝑑-dimensional box 𝕋^𝑑=(0,𝜋)^𝑑(𝑑=1,2,3); that is, 𝜕𝑈𝜕𝑥_𝑖=𝜕𝑉𝜕𝑥_𝑖=0,at𝑥_𝑖=0,𝜋,for1≤𝑖≤𝑑. (2) Let [𝑈,𝑉] be the uniform constant solution of the Keller-Segel model, and 𝑢(𝑥,𝑡)=𝑈(𝑥,𝑡)−𝑈,𝑣(𝑥,𝑡)=𝑉(𝑥,𝑡)−𝑉. Then [𝑢,𝑣] satisfies the equivalent Keller-Segel system below: 𝑢_𝑡=𝐷_𝑢∇²𝑢−𝜒𝑈∇²(),𝑣𝑣−𝜒∇𝑢∇𝑣_𝑡=𝐷_𝑣∇²𝑣+𝛼𝑢−𝛽𝑣. (3) Guo and Hwang proved that linear fastest growing modes determine unstable patterns for the above system. Their result can be interpreted as a rigorous mathematical characterization for early pattern formation in the Keller-Segel model.

In recent years, more and more attention has been given to the Keller-Segel model with the reaction terms, that is, the following chemotaxis-diffusion-growth model: 𝑈_𝑡=∇𝐷_𝑢(𝑈),𝑉∇𝑈−𝜒𝑈∇𝑉+𝑓_𝑡=𝐷_𝑣∇²𝑉+𝛼𝑈−𝛽𝑉. (4)

For 𝑓(𝑈)=𝑟𝑈(1−𝑈/𝐾), Painter and Hillen [6] demonstrated the capacity of the above model to self-organize into multiple cellular aggregations which, according to its 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). This spatiotemporal patterning can be further subdivided into either a time-periodic or time-irregular fashion. Numerical explorations into the latter indicate a positive Lyapunov exponent (sensitive dependence to initial conditions) together with a rich bifurcation structure. In particular, they found stationary patterns that bifurcate onto a path of periodic patterns which, prior to the onset of spatiotemporal irregularity, undergo a “periodic-doubling” sequence. Based on these results and comparisons with other systems, they argued that the spatiotemporal irregularity observed here describes a form of spatiotemporal chaos.

For 𝑓(𝑈)=𝑟𝑈(1−𝑈), Banerjee et al. [7] showed that the dynamics of the chemotaxis-diffusion-growth model may lead to steady states, to divergencies in a finite time, and to the formation of spatiotemporal irregular patterns. The latter, in particular, appears to be chaotic in part of the range of bounded solutions, as demonstrated by the analysis of wavelet power spectra. Steady states are achieved with sufficiently large values of the chemotactic coefficient 𝜒 and/or with growth rate 𝑟 below a critical value 𝑟_𝑐. For 𝑟>𝑟_𝑐, the solutions of the differential equations of the model diverge in a finite time. They also reported on the pattern formation regime, for different value of 𝜒, 𝑟 and of the diffusion coefficient 𝐷_𝑢. For the same 𝑓(𝑈), Kuto et al. [8] considered some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints. They studied the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity and construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle. For this case, the directions of the branches near the bifurcation points are also obtained. Finally, they exhibited several numerical results for the stationary and oscillating patterns.

In [9], Okuda and Osaki studied the chemotaxis-diffusion-growth model with 𝑓=𝑝𝑈(1−𝑈)+𝑞𝑈(1−𝑈)(𝑈−𝜈) in a rectangular domain by applying the center manifold theory, where constant 𝜈∈(0,1) and either 𝑝>0,𝑞=0, or 𝑝=0,𝑞>0. It is observed that the trivial solutions are destabilized due to the chemotaxis term. They obtained the normal form on the center manifold, and it is proved that the locally asymptotically stable hexagonal patterns exist.

Another extended formation of logistic source term is the cubic source term 𝑓=𝑈(𝑏₁+𝑏₂𝑈−𝑏₃𝑈²), where 𝑏₁≥0 is the intrinsic growth rate, the sign of 𝑏₂ is undetermined, 𝑏₃>0 is a positive constant, and 𝑏₂𝑈−𝑏₃𝑈² is the density restriction term (see [10, 11] for more information and references). Recently, Cao and Fu in [11] studied global existence and convergence of solutions to a cross-diffusion cubic predator-prey system with stage structure for the prey. In this paper, we investigate dynamics of the chemotaxis-diffusion-growth model with the source term 𝑓=(𝑏₁+𝑏₂𝑈−𝑏₃𝑈²)𝑈; that is, 𝑈_𝑡=∇𝐷_𝑢+𝑏∇𝑈−𝜒𝑈∇𝑉₁+𝑏₂𝑈−𝑏₃𝑈²𝑉𝑈,_𝑡=𝐷_𝑣∇²𝑉+𝛼𝑈−𝛽𝑉, (5) where 𝐷_𝑢, 𝜒, 𝐷_𝑣, 𝛼, and 𝛽 are positive constants and 𝑈(𝑥,𝑡), 𝑉(𝑥,𝑡) satisfies the Neumann boundary conditions. We will prove that given any general perturbation of magnitude 𝛿, its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of ln(1/𝛿). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model (5).

The organization of this paper is as follows: In Section 2, we prove that the positive constant equilibrium solution of (5) without chemotaxis is globally asymptotically stable if 𝑏₁>0. In Section 3, we investigate the growing modes of (5). In Section 4, we present and prove the Bootstrap lemma. In Section 5, given any 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 ln(1/𝛿).

2. Stability of Positive Equilibrium Point of (5) without Chemotaxis

The corresponding semilinear system of (5) without chemotaxis is as follows: 𝑈_𝑡=𝐷_𝑢∇²𝑈+𝑏₁+𝑏₂𝑈−𝑏₃𝑈²𝑈,𝑥∈𝕋^𝑑𝑉,𝑡>0,_𝑡=𝐷_𝑣∇²𝑉+𝛼𝑈−𝛽𝑉,𝑥∈𝕋^𝑑,𝑡>0,𝜕𝑈𝜕𝑥_𝑖=𝜕𝑉𝜕𝑥_𝑖=0,𝑥_𝑖𝑈()=0,𝜋,1≤𝑖≤𝑑,𝑥,0=𝑈₀(𝑥)𝑉()≥0,𝑥,0=𝑉₀(𝑥)≥0,𝑥∈𝕋^𝑑(𝑑=1,2,3). (6)

Obviously, [𝑈,𝑉]≡[(𝑏₂+𝑏²₂+4𝑏₁𝑏₃)/2𝑏₃,𝛼(𝑏₂+𝑏²₂+4𝑏₁𝑏₃)/2𝛽𝑏₃] is a positive equilibrium point of (6) if and only if either of the following two cases happens: (i)𝑏₁>0,𝑏₂∈ℝ, (ii)𝑏₁=0,𝑏₂>0. In the following we will discuss the stability of [𝑈,𝑉] in (6).

Let 𝑊=[𝑈,𝑉],𝑊=[𝑈,𝑉], and 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 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=⊕^{dim[𝐸(𝜇𝑖)]}_𝑗=1𝑋_𝑖𝑗. (7)

Let 𝔇= diag(𝐷_𝑢,𝐷_𝑣) and 𝔏=𝔇Δ+𝐺_𝑊(𝑊), where 𝐺_𝑊𝑊=−3𝑏₃𝑈²+2𝑏₂𝑈+𝑏₁0𝛼−𝛽. (8) Then the linearization of (6) at [𝑈,𝑉] is 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+𝐺_𝑊𝑊=−𝜇_𝑖𝐷_𝑢−3𝑏₃𝑈²+2𝑏₂𝑈+𝑏₁0𝛼−𝜇_𝑖𝐷_𝑣.−𝛽 (9) Notice that (𝑏₂+𝑏²₂+3𝑏₁𝑏₃)/3𝑏₃ is the positive root of −3𝑏₃𝑥²+2𝑏₂𝑥+𝑏₁=0 and 𝑏𝑈>²+𝑏²₂+3𝑏₁𝑏₃3𝑏₃. (10) Thus, −𝜇_𝑖𝔇+𝐺_𝑊(𝑊) has two negative eigenvalues −𝜇_𝑖𝐷_𝑢−3𝑏₃𝑈²+2𝑏₂𝑈+𝑏₁ and −𝜇_𝑖𝐷_𝑣−𝛽. It follows from [12, Theorem 5.1.1] that [𝑈,𝑉] is locally asymptotically stable.

Let [𝑈,𝑉] be a unique nonnegative global solution of (6). It is not hard to verify by the maximum principle that 0≤𝑈≤max𝑈,𝑈₀(𝑥)_∞,0≤𝑉≤max𝑉,𝑉₀(𝑥)_∞,∀𝑡≥0. (11) Moreover, if 𝑈₀,𝑉₀≥(≢)0, then 𝑈(𝑥,𝑡)>0, 𝑉(𝑥,𝑡)>0, for all 𝑡>0.

According to the main result in [13], we have ‖𝑈(⋅,𝑡)‖_{𝐶2,𝛼(𝕋}^𝑑₎‖𝑉(≤𝐶,⋅,𝑡)‖_{𝐶2,𝛼(𝕋}^𝑑₎≤𝐶,∀𝑡≥1. (12)

(i) If 𝑏₁>0, 𝑏₂∈ℝ, then 𝑏₃𝑈−𝑏₂>0. We define the Lyapunov function 𝐸(𝑡)=_𝕋𝑑𝑝𝑈−𝑈−𝑈𝑈ln𝑈+𝑉−𝑉²𝑑𝑥, (13) where 𝑝=2𝛼²/(𝛽(𝑏₃𝑈−𝑏₂)).

Calculating the derivative of 𝐸(𝑡) along positive solution of (6) by integration by parts and the Cauchy inequality, we have 𝑑𝐸(𝑢,𝑣)𝑑𝑡≤−_𝕋𝑑𝑝𝐷_𝑢𝑈𝑈²|∇𝑈|²+2𝐷_𝑣|∇𝑉|²−𝑑𝑥_𝕋𝑑𝑝𝑏₃𝑈+𝑏₃𝑈−𝑏₂𝑈−𝑈²−2𝛼𝑈−𝑈𝑉−𝑉+2𝛽𝑉−𝑉²𝑑𝑥≤−_𝕋𝑑𝑝𝑏₃𝑈−𝑏₂𝑈−𝑈²−2𝛼𝑈−𝑈𝑉−𝑉+2𝛽𝑉−𝑉²𝛼𝑑𝑥≤−²𝛽_𝕋^𝑑𝑈−𝑈²𝑑𝑥−𝛽_𝕋𝑑𝑉−𝑉²𝑑𝑥. (14)

(ii) If 𝑏₁=0, 𝑏₂>0, then 𝑏₂=𝑏₃𝑈. We define the Lyapunov function 𝐸(𝑡)=_𝕋𝑑𝑞𝑈𝑈−𝑈²𝑉−𝑉²𝑑𝑥, (15) where 𝑞=2𝛼²/𝛽𝑏₃𝑈.

Calculating in the same way as (14), we have 𝑑𝐸(𝑢,𝑣)𝑑𝑡≤−_𝕋𝑑2𝑞𝐷_𝑢𝑈²𝑢³|∇𝑈|²+𝐷_𝑣|∇𝑉|²−𝑑𝑥_𝕋𝑑𝑞𝑏₃𝑈+𝑈𝑈−𝑈²−2𝛼𝑈−𝑈𝑉−𝑉+2𝛽𝑉−𝑉²𝑑𝑥≤−_𝕋𝑑𝑞𝑏₃𝑈𝑈−𝑈²−2𝛼𝑈−𝑈𝑉−𝑉+2𝛽𝑉−𝑉²𝛼𝑑𝑥≤−²𝛽_𝕋^𝑑𝑈−𝑈²𝑑𝑥−𝛽_𝕋𝑑𝑉−𝑉²𝑑𝑥. (16)

Combining (12)–(16) and Lemma  3.2 in [11], we conclude that lim_𝑡→∞𝑈(⋅,𝑡)−𝑈_{𝐿2(𝕋𝑑)}=0,lim_𝑡→∞𝑉(⋅,𝑡)−𝑉_{𝐿2(𝕋𝑑)}=0. (17)

The global asymptotic stability of [𝑈,𝑉] follows from (17) and the local stability of [𝑈,𝑉].

Theorem 1. The positive equilibrium point [𝑈,𝑉] of (6) is locally asymptotically stable. If either 𝑏₁>0, 𝑏₂∈ℝ or 𝑏₁=0, 𝑏₂>0 holds, then [𝑈,𝑉] is globally asymptotically stable.

3. Growing Modes in the System (5)

Let 𝑢(𝐱,𝑡)=𝑈(𝐱,𝑡)−𝑈,𝑣(𝐱,𝑡)=𝑉(𝐱,𝑡)−𝑉. Then 𝑢_𝑡=𝐷_𝑢∇²𝑢−𝜒𝑈∇²𝑣−3𝑏²₃𝑈²−2𝑏₂𝑈−𝑏₁𝑢−𝜒∇(𝑢∇𝑣)+𝑏₂−3𝑏₃𝑈𝑢²−𝑏₃𝑢³,𝑣_𝑡=𝐷_𝑣∇²𝑣+𝛼𝑢−𝛽𝑣. (18) The corresponding linearized system takes the form 𝑢_𝑡=𝐷_𝑢∇²𝑢−𝜒𝑈∇²𝑣−3𝑏²₃𝑈²−2𝑏₂𝑈−𝑏₁𝑣𝑢,_𝑡=𝐷_𝑣∇²𝑣+𝛼𝑢−𝛽𝑣. (19) 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. We look for a normal mode to the linearized system (19) of the following form 𝐰(𝐱,𝑡)=𝐫_𝐪exp𝜆_𝐪𝑡𝑒_𝐪(𝐱), (20) where 𝐫_𝐪 is a vector depending on 𝐪. Plugging (20) into (19) we have the following dispersion formula for 𝜆_𝐪𝜆²_𝐪+𝑞²𝐷_𝑢+𝐷_𝑣+𝛽+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁𝜆_𝐪+𝑞²𝐷_𝑢𝐷_𝑣𝑞²+𝛽𝐷_𝑢+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁𝐷_𝑣−𝛼𝜒𝑈+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁𝛽=0. (21) Thus we deduce the following well-known aggregation (i.e., linear instability) criterion by requiring there exists a 𝐪 such that 𝑞²𝐷_𝑢𝐷_𝑣𝑞²+𝛽𝐷_𝑢+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁𝐷_𝑣−𝛼𝜒𝑈+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁𝛽<0 (22) to ensure that (21) has at least one positive root 𝜆_𝐪. This implies that for 𝐪, 𝑞²𝐷_𝑢+𝐷_𝑣+𝛽+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁²−4𝑞²𝐷_𝑢𝐷_𝑣𝑞²+𝛽𝐷_𝑢+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁𝐷_𝑣−𝛼𝜒𝑈+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁𝛽=𝑞²𝐷_𝑢−𝐷_𝑣−𝛽−3𝑏₃𝑈²+2𝑏₂𝑈+𝑏₁²+4𝛼𝜒𝑈𝑞²≥0. (23) If 𝐪=𝟎, then (21) has two negative roots −𝛽 and −3𝑏₃𝑈²+2𝑏₂𝑈+𝑏₁. Therefore, the positive equilibrium point of (18) is locally asymptotically stable.

Now we investigate nonlinear dynamics near the unstable constant equilibrium solution of (18) in the case 𝐪≠𝟎. If 𝐪≠𝟎, the right side of (23) is positive. Therefore, there exist two distinct real roots 𝜆^±_𝐪 for all 𝐪≠𝟎 to the quadratic equation (21). We denote the corresponding (linearly independent) eigenvectors by 𝐫₋(𝐪) and 𝐫₊(𝐪), such that 𝐫_±(𝐪)=𝜆^±_𝐪+𝐷_𝑣𝑞²+𝛽𝛼,1. (24) Clearly, for 𝑞 large 𝑞²𝐷_𝑢𝐷_𝑣𝑞²+𝛽𝐷_𝑢+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁𝐷_𝑣−𝛼𝜒𝑈+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁𝛽>0. (25) Hence, 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}. It is easy to see that there is one 𝑞² (possibly two) having 𝜆⁺(𝑞²)=𝜆_max if 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. 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), (26) so that 𝐰_𝐪=𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪). (27) The unique solution 𝐰(𝐱,𝑡)=[𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] to (19) is given by 𝐰(𝐱,𝑡)=_𝐪∈Ω𝑤⁻_𝐪𝐫₋(𝐪)exp𝜆⁻_𝐪𝑡+𝑤⁺_𝐪𝐫₊(𝐪)exp𝜆⁺_𝐪𝑡𝑒_𝐪(𝐱)≡𝑒^𝔏𝑡𝐰(𝑥,0). (28) For any 𝐠(⋅,𝑡)∈[𝐿²(𝕋^𝑑)]², we denote ‖𝐠(⋅,𝑡)‖≡‖𝐠(⋅,𝑡)‖_𝐿². Our main result of this section is the following lemma.

Lemma 2. Suppose that the instability criterion (22) 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 𝐷_𝑢, 𝜒, 𝑈, 𝑏₁, 𝑏₂, 𝑏₃, 𝐷_𝑣, 𝛼, and 𝛽, such that ‖𝐰(⋅,𝑡)‖≤𝐶₁exp𝜆_max𝑡‖𝐰(⋅,0)‖,∀𝑡≥0. (29)

Proof. We first consider the case for 𝑡≥1. By analyzing (21), for 𝑞 large, we have lim_𝑞→∞𝜆^±_𝐪𝑞²=−𝐷_𝑢,−𝐷_𝑣, (30) respectively. From the quadratic formula of (21), we can see that 𝜆⁺_𝐪−𝜆⁻_𝐪𝑞²≥2𝛼𝜒𝑈𝑞. (31) It follows from (27) that 𝑤^±_𝐪≤𝐫_±(𝐪)×𝐰_𝐪det𝐫₋(𝐪),𝐫₊(𝐪.) (32) Later on we will always denote universal constants by 𝐸_𝑖(𝑖=1,2,…). Note that 𝐪∈ℕ^𝑑 and 𝐪≠𝟎. From (24) and (30), for all 𝑞>0, there exists a positive constant 𝐸₁ and 𝐸₂, such that |𝜆^±_𝐪/𝛼𝑞²|≤𝐸₁, and 𝐫_±(𝐪)≤𝐸₂𝑞². (33) By (24), (31), (32), and (33), we deduce that 𝑤^±_𝐪≤𝐸₃𝑞𝐰_𝐪. (34) Thus, it is clear from (33) and (34) that 𝑤^±_𝐪𝐫_±(𝐪)exp𝜆^±_𝐪𝑡≤𝐸₂𝐸₃𝑞³𝐰_𝐪exp𝜆^±_𝐪𝑡. (35) For 𝑡≥1, it is not hard to verify that there exists a constant 𝐸₄>0, such that 𝑞³exp𝜆^±_𝐪𝑡≤𝐸₄. (36) It follows from (35) and (36) that 𝑤^±_𝐪𝐫_±(𝐪)exp𝜆^±_𝐪𝑡≤𝐸₅𝐰_𝐪. (37) Denote by ⟨⋅,⋅⟩ and (⋅,⋅) the inner product of [𝐿²(𝕋^𝑑)]² and the scaler product of ℝ², respectively. A simple computation shows that ‖𝐰(𝐱,0)‖²=𝜋2^𝑑_𝐪∈Ω𝐰_𝐪². (38) From (28), (37) and (38), we have ‖𝐰(𝐱,𝑡)‖≤2𝐸₅exp𝜆_max𝑡‖𝐰(𝐱,0)‖,for𝑡≥1. (39) On the other hand, for 𝑡<1, it is sufficient to derive the standard energy estimate in 𝐿². By (19), we have 12𝑑𝑑𝑡_𝕋^𝑑|𝑢|²+𝐴|𝑣|²+𝐝𝐱_𝕋𝑑𝐷_𝑢|∇𝑢|²+𝐴𝐷_𝑣|∇𝑣|²−𝜒+𝑈∇𝑢∇𝑣𝐝𝐱3𝑏₃𝑈²−2𝑏₂𝑈−𝑏_1𝕋𝑑𝑢²𝐝𝐱+𝐴𝛽_𝕋𝑑𝑣²𝐝𝐱=𝐴𝛼_𝕋𝑑𝑢𝑣𝐝𝐱. (40) Let 𝜒𝐴=𝑈²𝐷_𝑢𝐷_𝑣. (41) Then the integrand of the second term on the left side of (40) satisfies 𝐷_𝑢|∇𝑢|²+𝐴𝐷_𝑣|∇𝑣|²−𝜒≥𝐷𝑈∇𝑢∇𝑣_𝑢2|∇𝑢|²+𝜒𝑈²2𝐷_𝑢|∇𝑣|²≥0. (42) By (40), Young's inequality, and 𝐴≥1, we deduce that 12𝑑𝑑𝑡_𝕋^𝑑|𝑢|²+𝐴|𝑣|²𝐝𝐱≤𝐴𝛼2_𝕋^𝑑|𝑢|²+𝐴|𝑣|²𝐝𝐱. (43) Using Grownwall's inequality and noticing 𝐴≥1 and 𝑡<1, we can obtain ‖𝐰(𝐱,𝑡)‖≤𝐴exp(𝐴𝛼)^1/2exp𝜆_max𝑡‖𝐰(𝐱,0)‖. (44) If 0<𝐴<1, by (43), 𝑡<1, and Grownwall inequality, we have ‖𝐰(𝐱,𝑡)‖≤exp𝛼𝐴^1/2exp𝜆_max𝑡‖𝐰(𝐱,0)‖. (45) Let 𝐶₁=max{2𝐸₅,(𝐴exp(𝐴𝛼))^1/2}≥1 if 0<𝐴<1. Then ‖𝐰(𝐱,𝑡)‖≤𝐶₁exp(𝜆_max𝑡)‖𝐰(𝐱,0)‖ if 𝐴>1 and 𝐶₁=max{2𝐸₅,(exp𝛼/𝐴)^1/2}≥1.

4. Bootstrap Lemma

By a standard PDE theory [14], we can establish the existence of local solutions for (18).

Lemma 3 (local existence). For 𝑠≥1(𝑑=1) and 𝑠≥2(𝑑=2,3), there exists 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, (46) where 𝐶 is a positive constant depending on 𝐷_𝑢, 𝜒, 𝑈, 𝑏₁, 𝑏₂, 𝑏₃, 𝐷_𝑣, 𝛼, and 𝛽.

It is not hard to verify the following result.

Lemma 4. 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 5. Let [𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] be a solution of (18). Then 12𝑑𝑑𝑡_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑢²+𝜒𝑈²𝐷_𝑢𝐷_𝑣𝐷^𝛼𝑣²𝐝𝐱+_{|𝛼|=2𝕋𝑑}𝐷_𝑢4∇𝐷^𝛼𝑢²+𝜒𝑈²2𝐷_𝑢∇𝐷^𝛼𝑣²+𝐝𝐱𝐴𝛽2_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑣²𝐝𝐱+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁2×_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢²𝐝𝐱≤𝐶₀𝜒2+𝑏₂−3𝑏₃𝑈+32𝑏₃×‖𝐰‖_𝐻²+‖𝐰‖²_𝐻2∇³𝐰²+𝐶₂‖𝑢‖², (47) where 𝐶₂=𝐸₁₅𝜒¹²𝑈¹²𝛼¹²/2𝐷⁸_𝑢𝐷⁶_𝑣𝛽⁶(3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁)³.

Proof. It is known by Lemma 4 that 𝑢_𝑡=𝐷_𝑢∇²𝑢−𝜒𝑈∇²𝑣−3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁(𝑣)+𝑏𝑢−𝜒∇𝑢∇₂−3𝑏₃𝑈𝑢²−𝑏₃𝑢³,𝑣_𝑡=𝐷_𝑣∇²𝜕𝑢𝑣+𝛼𝑢−𝛽𝑣,𝜕𝑥_𝑖=𝜕𝑣𝜕𝑥_𝑖=0,at𝑥_𝑖=−𝜋,0,𝜋,for1≤𝑖≤𝑑, (48) where [𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] is the even extension of [𝑢(𝐱,𝑡),𝑣(𝐱,𝑡)] on (−𝜋,0)^𝑑. Taking the second-order derivative of (48) for 𝑥_𝑖,𝑥_𝑗 and making inner product with 𝜕_{𝑥𝑖𝑥𝑗}𝑢 and 𝐴𝜕_{𝑥𝑖𝑥𝑗}𝑣, respectively, on both sides then adding the two equations together, we have 12𝑑𝑑𝑡_2𝕋^𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝐴𝜕_{𝑥𝑖𝑥𝑗}𝑣²+𝐝𝐱_2𝕋𝑑𝐷_𝑢∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝐴𝐷_𝑣∇𝜕_{𝑥𝑖𝑥𝑗}𝑣²−𝜒𝑈∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅∇𝜕_{𝑥𝑖𝑥𝑗}𝑣𝐝𝐱+𝐴𝛽_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑣²+𝐝𝐱3𝑏₃𝑈³−2𝑏₂𝑈−𝑏_12𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢²𝐝𝐱=𝜒_2𝕋𝑑∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅𝜕_{𝑥𝑖𝑥𝑗}(𝑢⋅∇𝑣)𝐝𝐱+𝐴𝛼_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅𝜕_{𝑥𝑖𝑥𝑗}𝑣𝐝𝐱+2𝑏₂−3𝑏₃𝑈×_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢𝜕_𝑥𝑖𝑢⋅𝜕_𝑥𝑗𝑢+𝑢⋅𝜕_{𝑥𝑖𝑥𝑗}𝑢𝐝𝐱−3𝑏_32𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢2𝑢⋅𝜕_𝑥𝑖𝑢⋅𝜕_𝑥𝑗𝑢+𝑢²⋅𝜕_{𝑥𝑖𝑥𝑗}𝑢𝐝𝐱≡𝐼₁+𝐼₂+𝐼₃+𝐼₄. (49) Clearly 𝐷_𝑢∇𝜕_{𝑥𝑖,𝑥𝑗}𝑢²+𝐴𝐷_𝑣∇𝜕_{𝑥𝑖,𝑥𝑗}𝑣²−𝜒𝑈∇𝜕_{𝑥𝑖𝑥𝑗}𝑢⋅∇𝜕_{𝑥𝑖𝑥𝑗}𝑣≥𝐷_𝑢2∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+𝜒𝑈²2𝐷_𝑢∇𝜕_{𝑥𝑖𝑥𝑗}𝑣². (50) The nonlinear term 𝐼₁ is bounded by 𝐼₁≤𝜒‖∇𝑣‖_𝐿^∞∇𝜕_{𝑥𝑖𝑥𝑗}𝑢𝜕_{𝑥𝑖𝑥𝑗}𝑢+2^𝑑_𝑖=1‖∇𝑢‖_𝐿^∞𝜕_{𝑥𝑖𝑥𝑗}𝑣×∇𝜕_{𝑥𝑖𝑥𝑗}𝑢+‖𝑢‖_𝐿^∞∇𝜕_{𝑥𝑖𝑥𝑗}𝑢∇𝜕_{𝑥𝑖𝑥𝑗}𝑣. (51) We know that 𝑔_{𝐿∞(2𝕋𝑑)}≤𝐸₆𝑔_{𝐻2(2𝕋𝑑)},𝑔_{𝐿4(2𝕋𝑑)}≤𝐸₇𝑔_{𝐻2(2𝕋𝑑)}.𝑔_{𝐿6(2𝕋𝑑)}≤𝐸₈𝑔_{𝐻2(2𝕋𝑑)}, (52) for 𝑑≤3, and _2𝕋𝑑∇𝑢𝐝𝐱=_2𝕋𝑑∇𝑣𝐝𝐱=0,_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑢𝐝𝐱=_2𝕋𝑑𝜕_{𝑥𝑖𝑥𝑗}𝑣𝐝𝐱=0. (53) Applying the Poincaré inequality, we have 𝑔≤𝐸₉𝑔_{𝐿4(2𝕋𝑑)}≤𝐸₁₀𝑔_𝐻1≤𝐸₁₁∇𝑔,𝑑≤3. (54) It follows from (53) and (54) that ‖𝜕_𝑥𝑖𝑔‖≤𝐸₁₁‖∇𝜕_𝑥𝑖𝑔‖. Thus ∇𝑔≤𝐸₁₁^𝑑_𝑖,𝑗=1𝜕_{𝑥𝑖𝑥𝑗}𝑔^21/2. (55) Furthermore, 1𝑑^2𝑑_𝑖,𝑗=1𝜕_{𝑥𝑖𝑥𝑗}𝑔²≤_|𝛼|=2𝐷^𝛼𝑔²≤^𝑑_𝑖,𝑗=1𝜕_{𝑥𝑖𝑥𝑗}𝑔². (56) This implies that (∑^𝑑_𝑖,𝑗=1‖𝜕_{𝑥𝑖𝑥𝑗}𝑔‖²)^1/2 is equivalent to (∑_|𝛼|=2‖𝐷^𝛼𝑔‖²)^1/2. From (53)–(56), we have 𝜕_{𝑥𝑖𝑥𝑗}𝑔≤𝐸₁₁∇𝜕_{𝑥𝑖𝑥𝑗}𝑔,∇𝑔≤𝐸₁₁𝑑_|𝛼|=2𝐷^𝛼𝑔^21/2≤𝐸²₁₁𝑑_|𝛼|=2∇𝐷^𝛼𝑔^21/2. (57) It follows from (56) and (57) that ∇𝑔_𝐻2≤𝐸_12|𝛼|=2∇𝐷^𝛼𝑔^21/2,𝐸₁₂=𝐸⁴₁₁𝑑²+𝐸²₁₁𝑑²+1^1/2. (58) By (51), (52), (57), and (58), we have _|𝛼|=2𝐼₁≤𝜒𝐸₁₃‖𝐰‖_𝐻²∇³𝐰². (59) Now we consider 𝐼₂. From Gagliardo-Nirenberg inequality and Young inequality, we obtain 𝜕_{𝑥𝑖𝑥𝑗}𝑢²≤8𝐸²₁₄9𝑎∇𝜕_{𝑥𝑖𝑥𝑗}𝑢²+‖𝑢‖²4𝑎². (60) Let 𝑎=9𝐷³_𝑢𝐷²_𝑣𝛽²(3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁)/4𝐸²₁₄𝜒⁴𝑈⁴𝛼⁴. Then _|𝛼|=2𝐼₂≤3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁2_|𝛼|=2𝐷^𝛼𝑢²+𝐴𝛽2_|𝛼|=2𝐷^𝛼𝑣²+𝐷_𝑢4_|𝛼|=2∇𝐷^𝛼𝑢²+𝐸₁₅𝜒¹²𝑈¹²𝛼¹²𝐷⁸_𝑢𝐷⁶_𝑣𝛽⁶3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁³‖𝑢‖². (61) It follows from (52), (57), and (58) that _|𝛼|=2𝐼₃≤2𝑏₂−3𝑏₃𝑈𝐸₁₆‖𝐰‖_𝐻²∇³𝐰²,_|𝛼|=2𝐼₄≤3𝑏₃𝐸₁₇‖𝐰‖²∇³𝐰. (62) Combining (49), (50), (59), (61), and (62), we have 12𝑑𝑑𝑡_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑢²+𝜒𝑈²𝐷_𝑢𝐷_𝑣𝐷^𝛼𝑣²𝐝𝐱+_{|𝛼|=2𝕋𝑑}𝐷_𝑢4∇𝐷^𝛼𝑢²+𝜒𝑈²2𝐷_𝑢∇𝐷^𝛼𝑣²+𝐝𝐱𝐴𝛽2_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑣²𝐝𝐱+3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁2×_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢²𝐝𝐱≤𝐶₀𝜒+2𝑏₂−3𝑏₃𝑈+3𝑏₃2×‖𝐰‖_𝐻²+‖𝐰‖²_𝐻2∇³𝐰²+𝐶₂‖𝑢‖², (63) where 𝐶₀=max{𝐸₁₃,𝐸₁₆,𝐸₁₇}, 𝐶₂=𝐸₁₅𝜒¹²𝑈¹²𝛼¹²/2𝐷⁸_𝑢𝐷⁶_𝑣𝛽⁶(3𝑏₃𝑈²−2𝑏₂𝑈−𝑏₁)³.

Lemma 6. Let 𝐰(𝐱,𝑡) be a solution of (18) such that for 0≤𝑡≤𝑇<𝑇₀‖𝐰(⋅,𝑡)‖_𝐻²+‖𝐰(⋅,𝑡)‖²_𝐻2≤1𝐶₀𝐷min^𝑢2𝜒+2𝑏₂−3𝑏₃𝑈+3𝑏₃,𝜒𝑈²𝐷_𝑢𝜒+2𝑏₂−3𝑏₃𝑈+3𝑏₃,‖𝐰(⋅,𝑡)‖≤2𝐶₁exp𝜆_max𝑡‖𝐰(⋅,0)‖. (64)(65) Then ‖𝐰(⋅,𝑡)‖²_𝐻2≤𝐶₃‖𝐰(⋅,0)‖²_𝐻2+exp2𝜆_max𝑡‖𝐰(⋅,0)‖²,0≤𝑡≤𝑇, (66) where 𝐶₃=max{(𝐸²₁₁𝑑²+1)((𝜒𝑈)²/𝐷_𝑢𝐷_𝑣),4𝐶²₁{1+(𝐸²₁₁𝑑²+1)(𝐶₂/𝜆_max)}} if (𝜒𝑈)²/𝐷_𝑢𝐷_𝑣≥1,𝐶₃=max{(𝐸²₁₁𝑑²+1)(𝐷_𝑢𝐷_𝑣/(𝜒𝑈)²),4𝐶²₁{1+(𝐸²₁₁𝑑²+1)(𝐶₂𝐷_𝑢𝐷_𝑣/𝜆_max(𝜒𝑈)²)}} if (𝜒𝑈)²/𝐷_𝑢𝐷_𝑣<1.

Proof. It is clear from (57) that ‖∇𝐰(⋅,𝑡)‖²≤𝐸²₁₁𝑑²_|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)². (67) It follows from (67) that ‖𝐰(⋅,𝑡)‖²_𝐻2≤‖𝐰(⋅,𝑡)‖²+𝐸²₁₁𝑑²+1_|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)². (68) Now we estimate the second-order derivatives of 𝐰(⋅,𝑡). By (65) and Lemma 5, we immediately see that 12𝑑𝑑𝑡_|𝛼|=2𝕋^𝑑𝐷^𝛼𝑢²+𝜒𝑈²𝐷_𝑢𝐷_𝑣𝐷^𝛼𝑣²𝐝𝐱≤𝐶₂‖𝑢‖²≤𝐶₂‖𝐰(⋅,𝑡)‖². (69) Integrating on both sides of (69) from 0 to 𝑡 and from (65), we have _{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢(⋅,𝑡)²+𝜒𝑈²𝐷_𝑢𝐷_𝑣𝐷^𝛼𝑣(𝐱,𝑡)²𝐝𝐱≤_{|𝛼|=2𝕋𝑑}𝐷^𝛼𝑢(⋅,0)²+𝜒𝑈²𝐷_𝑢𝐷_𝑣𝐷^𝛼𝑣(⋅,0)²+𝐝𝐱4𝐶²₁𝐶₂𝜆_max‖𝐰(⋅,0)‖²exp2𝜆_max𝑡. (70) We will proceed in the following two cases: (𝜒𝑈)²/𝐷_𝑢𝐷_𝑣≥1, (𝜒𝑈)²/𝐷_𝑢𝐷_𝑣<1.
(1) If (𝜒𝑈)²/𝐷_𝑢𝐷_𝑣≥1, it follows from (70) that _|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)²≤𝜒𝑈²𝐷_𝑢𝐷_𝑣|𝛼|=2𝐷^𝛼𝐰(⋅,0)²+4𝐶²₁𝐶₂𝜆_max‖𝐰(⋅,0)‖²exp2𝜆_max𝑡. (71) By (68) and (71), we have ‖𝐰(⋅,𝑡)‖²_𝐻2≤𝐶₃‖𝐰(⋅,0)‖²_𝐻2+‖𝐰(⋅,0)‖²exp2𝜆_max𝑡, (72) where 𝐶₃=max{(𝐸²₁₁𝑑²+1)((𝜒𝑈)²/𝐷_𝑢𝐷_𝑣),4𝐶²₁{1+(𝐸²₁₁𝑑²+1)(𝐶₂/𝜆_max)}}.
(2) If (𝜒𝑈)²/𝐷_𝑢𝐷_𝑣<1, it follows from (71) that _|𝛼|=2𝐷^𝛼𝐰(⋅,𝑡)²≤𝐷_𝑢𝐷_𝑣𝜒𝑈²_|𝛼|=2𝐷^𝛼𝐰(⋅,0)²+𝐷_𝑢𝐷_𝑣𝜒𝑈²⋅4𝐶²₁𝐶₂𝜆_max‖𝐰(⋅,0)‖²exp2𝜆_max𝑡. (73)
By (68) and (73), we have ‖𝐰(⋅,𝑡)‖²_𝐻2≤𝐶₃‖𝐰(⋅,0)‖²_𝐻2+‖𝐰(⋅,0)‖²exp2𝜆_max𝑡, (74) where 𝐶₃=max{(𝐸²₁₁𝑑²+1)(𝐷_𝑢𝐷_𝑣/(𝜒𝑈)²),4𝐶²₁{1+(𝐸²₁₁𝑑²+1)(𝐶₂𝐷_𝑢𝐷_𝑣/𝜆_max(𝜒𝑈)²)}}.

5. Main Result

Let 𝜃 be a small fixed constant, and 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, (75) or equivalently 𝑇^𝛿=1𝜆_max𝜃ln𝛿. (76) Our main result is as follows.

Theorem 7. Assume that the set of 𝑞²=∑^𝑑_𝑖=1𝑞²_𝑖 satisfying instability criterion (22) is not empty for given parameters 𝐷_𝑢, 𝜒, 𝑈, 𝑏₁, 𝑏₂, 𝑏₃, 𝐷_𝑣, 𝛼, 𝛽. Let 𝐰₀(𝐱)=_𝐪∈Ω𝑤⁻_𝐪𝐫₋(𝐪)+𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)∈𝐻², (77) such that ‖𝐰₀‖=1. Then there exist constants 𝛿₀>0,𝐶>0 and 𝜃>0 depending on 𝐷_𝑢, 𝜒, 𝑈, 𝑏₁, 𝑏₂, 𝑏₃, 𝐷_𝑣, 𝛼, 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+𝛿²𝐰₀³_𝐻2+𝛿𝑒^𝜆max𝑡+𝛿²𝑒^2𝜆max𝑡𝛿𝑒^𝜆max𝑡 (78) for 0≤𝑡≤𝑇^𝛿, and 𝜈>0 is the gap between 𝜆_max and the rest of 𝜆_𝐪 in (21).

Proof. Let 𝐰^𝛿(𝐱,𝑡) be the solutions to (18) with initial data 𝐰^𝛿(⋅,0)=𝛿𝐰₀. We define 𝑇^∗=sup𝑡∣𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≤𝐶₁2𝛿exp𝜆_max𝑡. (79)
We also define 𝑇^∗∗×=sup_{𝐰𝑡∣𝛿(⋅,𝑡)𝐻2+𝐰𝛿(⋅,𝑡)2𝐻2≤1𝐶∗0𝐷min𝑢2𝑏𝜒+22−3𝑏3𝑈+3𝑏3,𝜒𝑈2𝐷𝑢𝑏𝜒+22−3𝑏3𝑈+3𝑏3}. (80)
Choose 𝜃 such that 𝐶^∗₀𝐶₃𝜃1+2𝐶₃^1/2𝜃𝜆<min^max4𝜒+𝑏₂−3𝑏₃𝑈+𝑏₃,𝐷_𝑢4𝜒+2𝑏₂−3𝑏₃𝑈+3𝑏₃,𝜒𝑈²2𝐷_𝑢𝜒+2𝑏₂−3𝑏₃𝑈+3𝑏₃. (81)
We now establish a sharper 𝐿² estimate of 𝐰^𝛿(⋅,𝑡) for 0≤𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}. First of all, by the definition of 𝑇^∗ and Lemma 2, for 0<𝑡≤𝑇^∗, it is not hard to see that 𝐰^𝛿(≤3⋅,𝑡)2𝐶₁𝛿exp𝜆_max𝑡. (82)
Applying Lemma 6 and the bootstrap argument, one can prove 𝐰^𝛿(⋅,𝑡)_𝐻2≤𝐶₃𝛿𝐰_0𝐻2+𝛿exp𝜆_max𝑡. (83)
From this and (𝑎+𝑏)^𝑝≤2^𝑝−1(𝑎^𝑝+𝑏^𝑝)(𝑎≥0,𝑏≥0,𝑝≥1), it follows that 𝐰^𝛿(⋅,𝑡)³_𝐻2≤4𝐶₃^3/2𝛿³𝐰₀³_𝐻2+𝛿³exp3𝜆_max𝑡. (84)
Applying Duhamel’s principle, we can obtain 𝐰^𝛿(⋅,𝑡)=𝛿𝑒^𝔏𝑡𝐰₀−^𝑡₀𝑒^{𝔏(𝑡−𝜏)}𝜒∇𝑢^𝛿(𝜏)∇𝑣^𝛿(𝜏)−𝑏₂−3𝑏₃𝑈𝑢^𝛿(𝜏)²+𝑏₃𝑢^𝛿(𝜏)³,0𝑑𝜏. (85)
By Lemma 2, (52), (54), and Lemma 6, for 0≤𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}, we deduce that 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≤𝐶₁𝐶^∗₀𝜒+𝑏₂−3𝑏₃𝑈+𝑏₃×^𝑡₀𝑒^{𝜆max(𝑡−𝜏)}𝐰^𝛿(𝜏)²_𝐻2+𝐰^𝛿(𝜏)³_𝐻2𝑑𝜏, (86)
where 𝐶^∗₀=max{𝐸²₇,(𝐸²₁₁/𝐸²₉)+𝐸₆,𝐸³₈}. By our choice of 𝑡≤min{𝑇^𝛿,𝑇^∗,𝑇^∗∗}, it is further bounded by 𝐰^𝛿(𝑡)−𝛿𝑒^𝔏𝑡𝐰₀≤𝐶₁𝐶^∗₀𝐶₃𝜒+𝑏₂−3𝑏₃𝑈+𝑏₃×𝛿𝐰₀²_𝐻2+4𝐶₃^1/2𝛿²𝐰₀³_𝐻2𝜆_max+𝛿𝑒^𝜆max𝑡+2𝐶₃^1/2𝛿²𝑒^2𝜆max𝑡𝜆_max𝛿𝑒^𝜆max𝑡. (87)
We now 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 (83) and (84). If 𝜃 satisfies (81) with 𝐶₃≥1 and 𝛿 is sufficiently small such that 𝐶₃𝛿²‖𝐰₀‖²_𝐻2+𝐶₃𝛿‖𝐰₀‖_𝐻²≤(1/2𝐶^∗₀)min{𝐷_𝑢/(2(𝜒+2|𝑏₂−3𝑏₃𝑈|+3𝑏₃)),(𝜒𝑈)²/𝐷_𝑢(𝜒+2|𝑏₂−3𝑏₃𝑈|+3𝑏₃)}, we immediately see that 𝐰^𝛿𝑇^∗∗_𝐻2+𝐰^𝛿𝑇^∗∗2_𝐻2≤𝐶₃𝛿²𝐰₀²_𝐻2+𝐶₃𝛿𝐰_0𝐻2+𝐶₃^1/2𝜃1+𝐶₃^1/2𝜃<1𝐶^∗₀𝐷min^𝑢2𝜒+2𝑏₂−3𝑏₃𝑈+3𝑏₃,𝜒𝑈²𝐷_𝑢𝜒+2𝑏₂−3𝑏₃𝑈+3𝑏₃. (88)
This is a contradiction to the definition of 𝑇^∗∗. On the other hand, if 𝑇^∗ is the smallest, we can let 𝑡=𝑇^∗ in (87). If 𝜃 satisfies (81) and 𝛿 is sufficiently small such that 𝐶^∗₀𝐶₃(𝜒+|𝑏₂−3𝑏₃𝑈|+𝑏₃)(𝛿‖𝐰₀‖²_𝐻2+4𝐶₃^1/2𝛿²‖𝐰₀‖³_𝐻2/𝜆_max)<1/4, we also can see that 𝐰^𝛿⋅,𝑇^∗−𝛿𝑒^𝔏𝑇∗𝐰₀≤𝐶₁𝐶^∗₀𝐶₃𝜒+𝑏₂−3𝑏₃𝑈+𝑏₃×𝛿𝐰₀²_𝐻2+4𝐶₃^1/2𝛿²𝐰₀³_𝐻2𝜆_max+𝜃1+2𝐶₃^1/2𝜃𝜆_max𝛿𝑒^{𝜆max𝑇∗}<𝐶₁2𝛿𝑒^{𝜆max𝑇∗}. (89)
This again contradicts the definition of 𝑇^∗. Hence, if 𝛿 is sufficiently small, we have 𝑇^𝛿=min𝑇^𝛿,𝑇^∗,𝑇^∗∗. (90)
From (28), we have 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; (91)
that is, 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Ωmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)≤𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝔏𝑡𝐰₀+𝐼₁+𝐼₂. (92)
Using (33) and (34), we have 𝐼²₁≤𝛿²𝑒^{2(𝜆max−𝜈)𝑡}𝜋2^𝑑_𝐪∈Ωmax𝐸²₂𝐸²₃𝑞⁶𝐰_𝑞². (93)
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)=𝜆_max, we let 𝑞²_max=max{𝑞²₁,𝑞²₂}. From (93), we have 𝐼₁≤𝐸₂𝐸₃𝑞³_max𝛿𝑒^{(𝜆max−𝜈)𝑡}=𝐶^∗𝛿𝑒^{(𝜆max−𝜈)𝑡}, (94)
where 𝐶^∗=𝐸₂𝐸₃𝑞³_max. Now we consider 𝐼₂. By (38), we have 𝐼₂≤𝛿𝑒^{(𝜆max−𝜈)𝑡}. (95)
From (87), (92), (94), and (95), it follows that 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^𝜆max𝑡_𝐪∈Ωmax𝑤⁺_𝐪𝐫₊(𝐪)𝑒_𝐪(𝐱)≤𝐶𝑒^−𝜈𝑡+𝛿𝐰₀²_𝐻2+𝛿²𝐰₀³_𝐻2+𝛿𝑒^𝜆max𝑡+𝛿²𝑒^2𝜆max𝑡𝛿𝑒^𝜆max𝑡, (96)
where 𝐶=max{𝐶^∗+1,(4𝐶₁𝐶^∗₀𝐶₃^3/2/𝜆_max)(𝜒+|𝑏₂−3𝑏₃𝑈|+𝑏₃)}. 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^𝜆max𝑡𝑤⁺_𝐪0𝐫₊𝐪₀ (97)
have the dominant leading order of 𝛿𝑒^𝜆max𝑡.

Our Theorem 7 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. In particular, choose a fixed 𝐪₀=(𝑞₀₁,𝑞₀₂,…,𝑞_0𝑑)∈Ω_max and let 𝐰₀(𝐱)=𝐫₊(𝐪₀)/|𝐫₊(𝐪₀)|𝑒_𝐪0(𝐱); then 𝐰₀(𝐱)_𝐻2=𝜋2^𝑑1+𝐪₀²+𝐪₀^41/2. (98) Note that 𝛿≤𝜃, 𝜃, and 𝜈 are fixed constants and 𝐪₀ is a fixed vector. From (96) and (98), if 𝑡≤𝑇^𝛿, we have 𝐰^𝛿(⋅,𝑡)−𝛿𝑒^{𝜆max𝑇𝛿}𝐫₊𝐪₀𝐫₊𝐪₀𝑒_𝐪0(𝐱)≤𝐶^∗∗𝛿^𝜈/𝜆max+𝜃²+𝜃³, (99) where 𝐶^∗∗=𝐶𝐶₄,𝐶₄=max{𝜃^{1−(𝜈/𝜆max)},(𝜋/2)^3𝑑/2(1+|𝐪₀|²+|𝐪₀|⁴)^3/2+1}. Moreover, 𝐰^𝛿(⋅,𝑡)≥𝜃−𝐶^∗∗𝛿^𝜈/𝜆max+𝜃²+𝜃³. (100) Let 0<𝜃<(1+(2/𝐶^∗∗)−1)/2, and 𝛿₀=((𝜃/2𝐶^∗∗)−𝜃²−𝜃³)^𝜆max/𝜈. Then 𝐰^𝛿(⋅,𝑡)≥𝜃−𝐶^∗∗𝛿^𝜈/𝜆max+𝜃²+𝜃³≥𝜃2>0,0<𝛿≤𝛿₀. (101) This implies nonlinear instability as 𝛿→0. In particular, instability occurs before the possible blow-up time.

Let us point out that although our proof is based on Guo-Strauss’ bootstrap argument, the adaptation to the procedure to our problem is not trivial at all, since the appearance of a growth restriction of a cubic type, we need more delicate estimates. Notice in our theorem that each initial perturbation can be drastically different from another, which gives rise to the richness of the pattern; on the other hand, the finite number maximal growing modes determines the common characteristics of the pattern, over the time scale of ln(1/𝛿). Therefore, our result indeed provides a mathematical description for the pattern formation in the Keller-Segel model with a cubic source term.

Acknowledgments

The authors would like to thank the referees for their helpful comments. This work is supported by the China National Natural Science Foundation (nos. 11061031; 11261053), the Fundamental Research Funds for the Gansu University.