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
(1)Ut=∇(Du∇U-χU∇V),Vt=Dv∇2V+αU-βV,
where U(x,t) is cell density, V(x,t) is chemoattractant concentration, Du is the amoeboid motility, χ is the chemotactic sensitivity, Dv 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 𝕋d=(0,π)d(d=1,2,3); that is,
(2)∂U∂xi=∂V∂xi=0,atxi=0,π,for1≤i≤d.
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:
(3)ut=Du∇2u-χU-∇2v-χ∇(u∇v),vt=Dv∇2v+αu-βv.
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:
(4)Ut=∇(Du∇U-χU∇V)+f(U),Vt=Dv∇2V+αU-βV.

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 rc. For r>rc, 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 Du. 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,
(5)Ut=∇(Du∇U-χU∇V)+(b1+b2U-b3U2)U,Vt=Dv∇2V+αU-βV,
where Du, χ, Dv, 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 (<xref ref-type="disp-formula" rid="EEq1.1">5</xref>) without Chemotaxis

The corresponding semilinear system of (5) without chemotaxis is as follows:
(6)Ut=Du∇2U+(b1+b2U-b3U2)U,x∈𝕋d,t>0,Vt=Dv∇2V+αU-βV,x∈𝕋d,t>0,∂U∂xi=∂V∂xi=0,xi=0,π,1≤i≤d,U(x,0)=U0(x)≥0,V(x,0)=V0(x)≥0,x∈𝕋d(d=1,2,3).

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 𝕋d(d=1,2,3) with the homogeneous Neumann boundary condition, and let E(μi) be the eigenspace corresponding to μi in L2(𝕋d). Let X=[L2(𝕋d)]2, {ϕij:j=1,…,dimE(μi)} be an orthonormal basis of E(μi), and Xij={c·ϕij∣c∈ℝ2}. Then
(7)X=⊕i=1∞Xi,Xi=⊕j=1dim[E(μi)]Xij.

Let 𝔇= diag(Du,Dv) and 𝔏=𝔇Δ+GW(W-), where
(8)GW(W-)=(-3b3U-2+2b2U-+b10α-β).
Then the linearization of (6) at [U-,V-] is Wt=𝔏(W-W-).

For each i≥1, Xi is invariant under the operator 𝔏, and λ is an eigenvalue of 𝔏 on Xi if and only if it is an eigenvalue of the matrix
(9)-μi𝔇+GW(W-)=(-μiDu-3b3U-2+2b2U-+b10α-μiDv-β).
Notice that (b2+b22+3b1b3)/3b3 is the positive root of -3b3x2+2b2x+b1=0 and
(10)U->b2+b22+3b1b33b3.
Thus, -μi𝔇+GW(W-) has two negative eigenvalues -μiDu-3b3U-2+2b2U-+b1 and -μiDv-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
(11)0≤U≤max{U-,|U0(x)|∞},0≤V≤max{V-,|V0(x)|∞},∀t≥0.
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
(12)∥U(·,t)∥C2,α(𝕋-d)≤C,∥V(·,t)∥C2,α(𝕋-d)≤C,∀t≥1.

(i) If b1>0, b2∈ℝ, then b3U--b2>0. We define the Lyapunov function
(13)E(t)=∫𝕋d[p(U-U--U-lnUU-)+(V-V-)2]dx,
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
(14)dE(u,v)dt≤-∫𝕋d{pDuU-U2|∇U|2+2Dv|∇V|2}dx-∫𝕋d{p(b3U+b3U--b2)(U-U-)2-2α(U-U-)(V-V-)+2β(V-V-)2}dx≤-∫𝕋d{p(b3U--b2)(U-U-)2-2α(U-U-)(V-V-)+2β(V-V-)2}dx≤-α2β∫𝕋d(U-U-)2dx-β∫𝕋d(V-V-)2dx.

(ii) If b1=0, b2>0, then b2=b3U-. We define the Lyapunov function
(15)E(t)=∫𝕋d[qU(U-U-)2(V-V-)2]dx,
where q=2α2/βb3U-.

Calculating in the same way as (14), we have
(16)dE(u,v)dt≤-∫𝕋d2qDuU-2u3|∇U|2+Dv|∇V|2dx-∫𝕋dqb3(U+U-)(U-U-)2-2α(U-U-)(V-V-)+2β(V-V-)2dx≤-∫𝕋dqb3U-(U-U-)2-2α(U-U-)(V-V-)+2β(V-V-)2dx≤-α2β∫𝕋d(U-U-)2dx-β∫𝕋d(V-V-)2dx.

Combining (12)–(16) and Lemma 3.2 in [11], we conclude that
(17)limt→∞∥U(·,t)-U-∥L2(𝕋d)=0,limt→∞∥V(·,t)-V-∥L2(𝕋d)=0.

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 (<xref ref-type="disp-formula" rid="EEq1.1">5</xref>)

Let u(x,t)=U(x,t)-U-,v(x,t)=V(x,t)-V-. Then
(18)ut=Du∇2u-χU-∇2v-(3b32U-2-2b2U--b1)u-χ∇(u∇v)+(b2-3b3U-)u2-b3u3,vt=Dv∇2v+αu-βv.
The corresponding linearized system takes the form
(19)ut=Du∇2u-χU-∇2v-(3b32U-2-2b2U--b1)u,vt=Dv∇2v+αu-βv.
Let w(x,t)≡[u(x,t),v(x,t)], q=(q1,…,qd)∈Ω=ℕd, and eq(x)=∏i=1dcos(qixi). Then {eq(x)}q∈Ω forms a basis of the space of functions in 𝕋d that satisfy Neumann boundary conditions. We look for a normal mode to the linearized system (19) of the following form
(20)w(x,t)=rqexp(λqt)eq(x),
where rq is a vector depending on q. Plugging (20) into (19) we have the following dispersion formula for λq(21)λq2+{q2(Du+Dv)+β+3b3U-2-2b2U--b1}λq+q2{DuDvq2+βDu+(3b3U-2-2b2U--b1)Dv-αχU-{q2}}+(3b3U-2-2b2U--b1)β=0.
Thus we deduce the following well-known aggregation (i.e., linear instability) criterion by requiring there exists a q such that
(22)q2{DuDvq2+βDu+(3b3U-2-2b2U--b1)Dv-αχU-}+(3b3U-2-2b2U--b1)β<0
to ensure that (21) has at least one positive root λq. This implies that for q,
(23){q2(Du+Dv)+β+3b3U-2-2b2U--b1}2-4{q2{DuDvq2+βDu+(3b3U-2-2b2U--b1)Dv-αχU-{q2}}+(3b3U-2-2b2U--b1)β}={q2(Du-Dv)-(β-3b3U-2+2b2U-+b1)}2+4αχU-q2≥0.
If q=0, then (21) has two negative roots -β and -3b3U-2+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 λq± for all q≠0 to the quadratic equation (21). We denote the corresponding (linearly independent) eigenvectors by r-(q) and r+(q), such that
(24)r±(q)=[λq±+Dvq2+βα,1].
Clearly, for q large
(25)q2{DuDvq2+βDu+(3b3U-2-2b2U--b1)Dv-αχU-}+(3b3U-2-2b2U--b1)β>0.
Hence, there are only finitely many q such that λq+>0. We denote the largest eigenvalue by λmax>0 and define Ωmax≡{q∈Ω∣λq+=λmax}. It is easy to see that there is one q2 (possibly two) having λ+(q2)=λmax if we regard λq+ as a function of q2. We also denote ν>0 to be the gap between the λmax and the rest. Given any initial perturbation w(x,0), we can expand it as
(26)w(x,0)=∑q∈Ωwqeq(x)=∑q∈Ω{wq-r-(q)+wq+r+(q)}eq(x),
so that
(27)wq=wq-r-(q)+wq+r+(q).
The unique solution w(x,t)=[u(x,t),v(x,t)] to (19) is given by
(28)w(x,t)=∑q∈Ω{wq-r-(q)exp(λq-t)+wq+r+(q)exp(λq+t)}eq(x)≡e𝔏tw(x,0).
For any g(·,t)∈[L2(𝕋d)]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)≡e𝔏tw(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 Du, χ, U-, b1, b2, b3, Dv, α, and β, such that
(29)∥w(·,t)∥≤C1exp(λ
max
t)∥w(·,0)∥,∀t≥0.

Proof.

We first consider the case for t≥1. By analyzing (21), for q large, we have
(30)limq→∞λq±q2=-Du,-Dv,
respectively. From the quadratic formula of (21), we can see that
(31)λq+-λq-q2≥2αχU-q.
It follows from (27) that
(32)|wq±|≤|r±(q)|×|wq||det[r-(q),r+(q)]|.
Later on we will always denote universal constants by Ei(i=1,2,…). Note that q∈ℕd and q≠0. From (24) and (30), for all q>0, there exists a positive constant E1 and E2, such that |λq±/αq2|≤E1, and
(33)|r±(q)|≤E2q2.
By (24), (31), (32), and (33), we deduce that
(34)|wq±|≤E3q|wq|.
Thus, it is clear from (33) and (34) that
(35)|wq±r±(q)exp(λq±t)|≤E2E3q3|wq|exp(λq±t).
For t≥1, it is not hard to verify that there exists a constant E4>0, such that
(36)q3exp(λq±t)≤E4.
It follows from (35) and (36) that
(37)|wq±r±(q)exp(λq±t)|≤E5|wq|.
Denote by 〈·,·〉 and (·,·) the inner product of [L2(𝕋d)]2 and the scaler product of ℝ2, respectively. A simple computation shows that
(38)∥w(x,0)∥2=(π2)d∑q∈Ω|wq|2.
From (28), (37) and (38), we have
(39)∥w(x,t)∥≤2E5exp(λmaxt)∥w(x,0)∥,fort≥1.
On the other hand, for t<1, it is sufficient to derive the standard energy estimate in L2. By (19), we have
(40)12ddt∫𝕋d{|u|2+A|v|2}dx+∫𝕋d{Du|∇u|2+ADv|∇v|2-χU-∇u∇v}dx+(3b3U-2-2b2U--b1)∫𝕋du2dx+Aβ∫𝕋dv2dx=Aα∫𝕋duvdx.
Let
(41)A=(χU-)2DuDv.
Then the integrand of the second term on the left side of (40) satisfies
(42)Du|∇u|2+ADv|∇v|2-χU-∇u∇v≥Du2|∇u|2+(χU-)22Du|∇v|2≥0.
By (40), Young's inequality, and A≥1, we deduce that
(43)12ddt∫𝕋d{|u|2+A|v|2}dx≤Aα2∫𝕋d{|u|2+A|v|2}dx.
Using Grownwall's inequality and noticing A≥1 and t<1, we can obtain
(44)∥w(x,t)∥≤(Aexp(Aα))1/2exp(λmaxt)∥w(x,0)∥.
If 0<A<1, by (43), t<1, and Grownwall inequality, we have
(45)∥w(x,t)∥≤(expαA)1/2exp(λmaxt)∥w(x,0)∥.
Let C1=max{2E5,(Aexp(Aα))1/2}≥1 if 0<A<1. Then ∥w(x,t)∥≤C1exp(λmaxt)∥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)∈Hs has a unique solution w(·,t) on (0,T0) which satisfies
(46)∥w(t)∥Hs≤C∥w(0)∥Hs,0<t<T0,
where C is a positive constant depending on Du, χ, U-, b1, b2, b3, Dv, 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 2𝕋d=(-π,π)d(d=1,2,3) is also the solution of (18) which satisfies homogeneous Neumann boundary conditions and periodical boundary conditions on 2𝕋d=(-π,π)d(d=1,2,3).

Lemma 5.

Let [u(x,t),v(x,t)] be a solution of (18). Then
(47)12ddt∑|α|=2∫𝕋d{|Dαu|2+(χU-)2DuDv|Dαv|2}dx+∑|α|=2∫𝕋d{Du4|∇Dαu|2+(χU-)22Du|∇Dαv|2}dx+Aβ2∑|α|=2∫𝕋d|Dαv|2dx+3b3U-2-2b2U--b12×∑|α|=2∫𝕋d|Dαu|2dx≤C0(χ2+|b2-3b3U-|+32b3)×(∥w∥H2+∥w∥H22)∥∇3w∥2+C2∥u∥2,
where C2=E15χ12U-12α12/2Du8Dv6β6(3b3U-2-2b2U--b1)3.

Proof.

It is known by Lemma 4 that
(48)u~t=Du∇2u~-χU-∇2v~-(3b3U-2-2b2U--b1)u~-χ∇(u~∇v~)+(b2-3b3U-)u~2-b3u~3,v~t=Dv∇2v~+αu~-βv~,∂u~∂xi=∂v~∂xi=0,atxi=-π,0,π,for1≤i≤d,
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 xi,xj and making inner product with ∂xixju~ and A∂xixjv~, respectively, on both sides then adding the two equations together, we have
(49)12ddt∫2𝕋d{|∂xixju~|2+A|∂xixjv~|2}dx+∫2𝕋d{Du|∇∂xixju~|2+ADv|∇∂xixjv~|2-χU-∇∂xixju~·∇∂xixjv~{|∇∂xixjv~|2}}dx+Aβ∫2𝕋d|∂xixjv~|2dx+(3b3U-3-2b2U--b1)∫2𝕋d|∂xixju~|2dx=χ∫2𝕋d∇∂xixju~·∂xixj(u~·∇v~)dx+Aα∫2𝕋d∂xixju~·∂xixjv~dx+2(b2-3b3U-)×∫2𝕋d∂xixju~(∂xiu~·∂xju~+u~·∂xixju~)dx-3b3∫2𝕋d∂xixju~(2u~·∂xiu~·∂xju~+u~2·∂xixju~)dx≡I1+I2+I3+I4.
Clearly
(50)Du|∇∂xi,xju~|2+ADv|∇∂xi,xjv~|2-χU-∇∂xixju~·∇∂xixjv~≥Du2|∇∂xixju~|2+(χU-)22Du|∇∂xixjv~|2.
The nonlinear term I1 is bounded by
(51)I1≤χ{∥∇v~∥L∞∥∇∂xixju~∥∥∂xixju~∥+2∑i=1d∥∇u~∥L∞∥∂xixjv~∥×∥∇∂xixju~∥+∥u~∥L∞∥∇∂xixju~∥∥∇∂xixjv~∥{∑i=1d}}.
We know that
(52)∥g∥L∞(2𝕋d)≤E6∥g∥H2(2𝕋d),∥g∥L4(2𝕋d)≤E7∥g∥H2(2𝕋d).∥g∥L6(2𝕋d)≤E8∥g∥H2(2𝕋d),
for d≤3, and
(53)∫2𝕋d∇u~dx=∫2𝕋d∇v~dx=0,∫2𝕋d∂xixju~dx=∫2𝕋d∂xixjv~dx=0.
Applying the Poincaré inequality, we have
(54)∥g∥≤E9∥g∥L4(2𝕋d)≤E10∥g∥H1≤E11∥∇g∥,d≤3.
It follows from (53) and (54) that ∥∂xig∥≤E11∥∇∂xig∥. Thus
(55)∥∇g∥≤E11(∑i,j=1d∥∂xixjg∥2)1/2.
Furthermore,
(56)1d2∑i,j=1d∥∂xixjg∥2≤∑|α|=2∥Dαg∥2≤∑i,j=1d∥∂xixjg∥2.
This implies that (∑i,j=1d∥∂xixjg∥2)1/2 is equivalent to (∑|α|=2∥Dαg∥2)1/2. From (53)–(56), we have
(57)∥∂xixjg∥≤E11∥∇∂xixjg∥,∥∇g∥≤E11d(∑|α|=2∥Dαg∥2)1/2≤E112d(∑|α|=2∥∇Dαg∥2)1/2.
It follows from (56) and (57) that
(58)∥∇g∥H2≤E12(∑|α|=2∥∇Dαg∥2)1/2,E12=(E114d2+E112d2+1)1/2.
By (51), (52), (57), and (58), we have
(59)∑|α|=2I1≤χE13∥w~∥H2∥∇3w~∥2.
Now we consider I2. From Gagliardo-Nirenberg inequality and Young inequality, we obtain
(60)∥∂xixju~∥2≤8E1429(a∥∇∂xixju~∥2+∥u~∥24a2).
Let a=9Du3Dv2β2(3b3U-2-2b2U--b1)/4E142χ4U-4α4. Then
(61)∑|α|=2I2≤3b3U-2-2b2U--b12∑|α|=2∥Dαu~∥2+Aβ2∑|α|=2∥Dαv~∥2+Du4∑|α|=2∥∇Dαu~∥2+E15χ12U-12α12Du8Dv6β6(3b3U-2-2b2U--b1)3∥u~∥2.
It follows from (52), (57), and (58) that
(62)∑|α|=2I3≤2|b2-3b3U-|E16∥w~∥H2∥∇3w~∥2,∑|α|=2I4≤3b3E17∥w~∥2∥∇3w~∥.
Combining (49), (50), (59), (61), and (62), we have
(63)12ddt∑|α|=2∫𝕋d{|Dαu|2+(χU-)2DuDv|Dαv|2}dx+∑|α|=2∫𝕋d{Du4|∇Dαu|2+(χU-)22Du|∇Dαv|2}dx+Aβ2∑|α|=2∫𝕋d|Dαv|2dx+3b3U-2-2b2U--b12×∑|α|=2∫𝕋d|Dαu|2dx≤C0χ+2|b2-3b3U-|+3b32×(∥w∥H2+∥w∥H22)∥∇3w∥2+C2∥u∥2,
where C0=max{E13,E16,E17}, C2=E15χ12U-12α12/2Du8Dv6β6(3b3U-2-2b2U--b1)3.

Lemma 6.

Let w(x,t) be a solution of (18) such that for 0≤t≤T<T0(64)∥w(·,t)∥H2+∥w(·,t)∥H22≤1C0min{Du2(χ+2|b2-3b3U-|+3b3),(χU-)2Du(χ+2|b2-3b3U-|+3b3)},(65)∥w(·,t)∥≤2C1exp(λ
max
t)∥w(·,0)∥.
Then
(66)∥w(·,t)∥H22≤C3{∥w(·,0)∥H22+exp(2λ
max
t)∥w(·,0)∥2},0≤t≤T,
where C3=
max
{(E112d2+1)((χU-)2/DuDv),4C12{1+(E112d2+1)(C2/λ
max
)}} if (χU-)2/DuDv≥1,C3=
max
{(E112d2+1)(DuDv/(χU-)2),4C12{1+(E112d2+1)(C2DuDv/λ
max
(χU-)2)}} if (χU-)2/DuDv<1.

Proof.

It is clear from (57) that
(67)∥∇w(·,t)∥2≤E112d2∑|α|=2∥Dαw(·,t)∥2.
It follows from (67) that
(68)∥w(·,t)∥H22≤∥w(·,t)∥2+(E112d2+1)∑|α|=2∥Dαw(·,t)∥2.
Now we estimate the second-order derivatives of w(·,t). By (65) and Lemma 5, we immediately see that
(69)12ddt∑|α|=2∫𝕋d{|Dαu|2+(χU-)2DuDv|Dαv|2}dx≤C2∥u∥2≤C2∥w(·,t)∥2.
Integrating on both sides of (69) from 0 to t and from (65), we have
(70)∑|α|=2∫𝕋d{|Dαu(·,t)|2+(χU-)2DuDv|Dαv(x,t)|2}dx≤∑|α|=2∫𝕋d{|Dαu(·,0)|2+(χU-)2DuDv|Dαv(·,0)|2}dx+4C12C2λmax∥w(·,0)∥2exp(2λmaxt).
We will proceed in the following two cases: (χU-)2/DuDv≥1, (χU-)2/DuDv<1.

(1) If (χU-)2/DuDv≥1, it follows from (70) that
(71)∑|α|=2∥Dαw(·,t)∥2≤(χU-)2DuDv∑|α|=2∥Dαw(·,0)∥2+4C12C2λmax∥w(·,0)∥2exp(2λmaxt).
By (68) and (71), we have
(72)∥w(·,t)∥H22≤C3{∥w(·,0)∥H22+∥w(·,0)∥2exp(2λmaxt)},
where C3=max{(E112d2+1)((χU-)2/DuDv),4C12{1+(E112d2+1)(C2/λmax)}}.

(2) If (χU-)2/DuDv<1, it follows from (71) that
(73)∑|α|=2∥Dαw(·,t)∥2≤DuDv(χU-)2∑|α|=2∥Dαw(·,0)∥2+DuDv(χU-)2·4C12C2λmax∥w(·,0)∥2exp(2λmaxt).

By (68) and (73), we have
(74)∥w(·,t)∥H22≤C3{∥w(·,0)∥H22+∥w(·,0)∥2exp(2λmaxt)},
where C3=max{(E112d2+1)(DuDv/(χU-)2),4C12{1+(E112d2+1)(C2DuDv/λmax(χU-)2)}}.

5. Main Result

Let θ be a small fixed constant, and λmax be the dominant eigenvalue which is the maximal growth rate. We also denote the gap between the largest growth rate λmax and the rest by ν>0. Then for δ>0 arbitrary small, we define the escape time Tδ by
(75)θ=δexp(λmaxTδ),
or equivalently
(76)Tδ=1λmaxlnθδ.
Our main result is as follows.

Theorem 7.

Assume that the set of q2=∑i=1dqi2 satisfying instability criterion (22) is not empty for given parameters Du, χ, U-, b1, b2, b3, Dv, α, β. Let
(77)w0(x)=∑q∈Ω{wq-r-(q)+wq+r+(q)}eq(x)∈H2,
such that ∥w0∥=1. Then there exist constants δ0>0,C>0 and θ>0 depending on Du, χ, U-, b1, b2, b3, Dv, α, and β, such that for all 0<δ≤δ0, if the initial perturbation of the steady state [U-,V-] is wδ(·,0)=δw0, then its nonlinear evolution wδ(·,t) satisfies
(78)∥wδ(·,t)-δeλ max t∑q∈Ω max wq+r+(q)eq(x)∥≤C{e-νt+δ∥w0∥H22+δ2∥w0∥H23+δeλ max t+δ2e2λ max t}δeλ max t
for 0≤t≤Tδ, and ν>0 is the gap between λ
max
and the rest of λq in (21).

Proof.

Let wδ(x,t) be the solutions to (18) with initial data wδ(·,0)=δw0. We define
(79)T*=sup{t∣∥wδ(·,t)-δe𝔏tw0∥≤C12δexp(λmaxt)}.

We also define
(80)T**=sup×{t∣∥wδ(·,t)∥H2+∥wδ(·,t)∥H22≤1C0*min{Du2(χ+2|b2-3b3U-|+3b3),(χU-)2Du(χ+2|b2-3b3U-|+3b3)}}.

Choose θ such that
(81)C0*C3θ(1+2C31/2θ)<min{{(χU-)22Du}λmax4(χ+|b2-3b3U-|+b3),Du4(χ+2|b2-3b3U-|+3b3),(χU-)22Du(χ+2|b2-3b3U-|+3b3)}.

We now establish a sharper L2 estimate of wδ(·,t) for 0≤t≤min{Tδ,T*,T**}. First of all, by the definition of T* and Lemma 2, for 0<t≤T*, it is not hard to see that
(82)∥wδ(·,t)∥≤32C1δexp(λmaxt).

Applying Lemma 6 and the bootstrap argument, one can prove
(83)∥wδ(·,t)∥H2≤C3{δ∥w0∥H2+δexp(λmaxt)}.

From this and (a+b)p≤2p-1(ap+bp)(a≥0,b≥0,p≥1), it follows that
(84)∥wδ(·,t)∥H23≤4C33/2{δ3∥w0∥H23+δ3exp(3λmaxt)}.

Applying Duhamel’s principle, we can obtain
(85)wδ(·,t)=δe𝔏tw0-∫0te𝔏(t-τ)[{(uδ(τ))3}χ∇(uδ(τ)∇vδ(τ))-(b2-3b3U-)(uδ(τ))2+b3(uδ(τ))3,0{(uδ(τ)∇vδ(τ))]]dτ.

By Lemma 2, (52), (54), and Lemma 6, for 0≤t≤min{Tδ,T*,T**}, we deduce that
(86)∥wδ(·,t)-δe𝔏tw0∥≤C1C0*{χ+|b2-3b3U-|+b3}×∫0teλmax(t-τ){∥wδ(τ)∥H22+∥wδ(τ)∥H23}dτ,

where C0*=max{E72,(E112/E92)+E6,E83}. By our choice of t≤min{Tδ,T*,T**}, it is further bounded by
(87)∥wδ(t)-δe𝔏tw0∥≤C1C0*C3(χ+|b2-3b3U-|+b3)×{δ∥w0∥H22+4C31/2δ2∥w0∥H23λmax+δeλmaxt+2C31/2δ2e2λmaxtλmax}δeλmaxt.

We now prove by contradiction that for δ sufficiently small, Tδ=min{Tδ,T*,T**}. If T** is the smallest, we can let t=T**≤Tδ in (83) and (84). If θ satisfies (81) with C3≥1 and δ is sufficiently small such that C3δ2∥w0∥H22+C3δ∥w0∥H2≤(1/2C0*)min{Du/(2(χ+2|b2-3b3U-|+3b3)),(χU-)2/Du(χ+2|b2-3b3U-|+3b3)}, we immediately see that
(88)∥wδ(T**)∥H2+∥wδ(T**)∥H22≤C3δ2∥w0∥H22+C3δ∥w0∥H2+C31/2θ(1+C31/2θ)<1C0*min{{(χU-)2Du(χ+2|b2-3b3U-|+3b3)}Du2(χ+2|b2-3b3U-|+3b3),(χU-)2Du(χ+2|b2-3b3U-|+3b3)}.

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 C0*C3(χ+|b2-3b3U-|+b3)(δ∥w0∥H22+4C31/2δ2∥w0∥H23/λmax)<1/4, we also can see that
(89)∥wδ(·,T*)-δe𝔏T*w0∥≤C1C0*C3(χ+|b2-3b3U-|+b3)×{δ∥w0∥H22+4C31/2δ2∥w0∥H23λmax+θ(1+2C31/2θ)λmax}δeλmaxT*<C12δeλmaxT*.

This again contradicts the definition of T*. Hence, if δ is sufficiently small, we have
(90)Tδ=min{Tδ,T*,T**}.

From (28), we have
(91)∥wδ(·,t)-δe𝔏tw0∥≥∥wδ(·,t)-δeλmaxt∑q∈Ωmaxwq+r+(q)eq(x)∥-∥δ∑q∈Ωmaxwq-r-(q)exp(λq-t)eq(x)∥-∥δ∑q∈Ω∖Ωmax{wq-r-(q)exp(λq-t)+wq+r+(q)exp(λq+t)}eq(x)∥≡∥wδ(·,t)-δeλmaxt∑q∈Ωmaxwq+r+(q)eq(x)∥-I1-I2;

that is,
(92)∥wδ(·,t)-δeλmaxt∑q∈Ωmaxwq+r+(q)eq(x)∥≤∥wδ(·,t)-δe𝔏tw0∥+I1+I2.

Using (33) and (34), we have
(93)I12≤δ2e2(λmax-ν)t(π2)d∑q∈ΩmaxE22E32q6|wq|2.

We know that there is one (or two) q2 satisfying λ+(q2)=λmax. If there is only one q2 satisfying λ+(q2)=λmax, we denote it by qmax2. If there are q12 and q22 satisfying λ+(q2)=λmax, we let qmax2=max{q12,q22}. From (93), we have
(94)I1≤E2E3qmax3δe(λmax-ν)t=C*δe(λmax-ν)t,

where C*=E2E3qmax3. Now we consider I2. By (38), we have
(95)I2≤δe(λmax-ν)t.

From (87), (92), (94), and (95), it follows that
(96)∥wδ(·,t)-δeλmaxt∑q∈Ωmaxwq+r+(q)eq(x)∥≤C{e-νt+δ∥w0∥H22+δ2∥w0∥H23+δeλmaxt+δ2e2λmaxt}δeλmaxt,

where C=max{C*+1,(4C1C0*C33/2/λmax)(χ+|b2-3b3U-|+b3)}. Notice that for 0≤t≤Tδ, δeλmaxt≤θ is sufficiently small. As long as wq0+≠0 for at least one q0∈Ωmax, which is generic for perturbations, the corresponding fastest growing modes
(97)∥δeλmaxt∑q∈Ωmaxwq+r+(q)eq(x)∥≥δeλmaxt|wq0+||r+(q0)|

have the dominant leading order of δeλmaxt.

Our Theorem 7 implies that the dynamics of a general perturbation is characterized by such linear dynamics over a long time period of εTδ≤t≤Tδ, for any ε>0. In particular, choose a fixed q0=(q01,q02,…,q0d)∈Ωmax and let w0(x)=r+(q0)/|r+(q0)|eq0(x); then
(98)∥w0(x)∥H2={(π2)d(1+|q0|2+|q0|4)}1/2.
Note that δ≤θ, θ, and ν are fixed constants and q0 is a fixed vector. From (96) and (98), if t≤Tδ, we have
(99)∥wδ(·,t)-δeλmaxTδr+(q0)|r+(q0)|eq0(x)∥≤C**{δν/λmax+θ2+θ3},
where C**=CC4,C4=max{θ1-(ν/λmax),(π/2)3d/2(1+|q0|2+|q0|4)3/2+1}. Moreover,
(100)∥wδ(·,t)∥≥θ-C**{δν/λmax+θ2+θ3}.
Let 0<θ<(1+(2/C**)-1)/2, and δ0=((θ/2C**)-θ2-θ3)λmax/ν. Then
(101)∥wδ(·,t)∥≥θ-C**{δν/λmax+θ2+θ3}≥θ2>0,0<δ≤δ0.
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.

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