# Pattern Formation of a Keller-Segel Model with the Source Termup(1-u)

Authors: Shengmao Fu; Fenli Cao
Journal: Journal of Mathematics
 (2013)

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## Abstract

Nonlinear dynamics near an unstable constant equilibrium in a Keller-Segel model with the source termup(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.


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## Body

## 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
(1)Ut=∇(Du∇U-χU∇V)+g(U),Vt=Dv∇2V+αU-βV,
where U(x,t) is the cell density, V(x,t) is the concentration of chemotactic substance, Du>0 is the amoeboid motility, χ>0 is the chemotactic sensitivity, Dv>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:(2)Ut=∇(∇U-χU∇V)+Up(1-U),x∈𝕋d,t>0,Vt=∇2V+U-βV,x∈𝕋d,t>0,
which satisfies the homogeneous Neumann boundary conditions and initial value conditions for U(x,t) and V(x,t), that is,
(3)∂U∂xi=∂V∂xi=0,xi=0,π,1≤i≤d,t>0,(4)U(x,0)=U0(x)≥(≢)0,V(x,0)=V0(x)≥(≢)0,x∈𝕋d,
where 𝕋d=(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
(5)Ut=∇2U+Up(1-U),x∈𝕋d(d=1,2,3),t>0,Vt=∇2V+U-βV,x∈𝕋d(d=1,2,3),t>0,∂U∂xi=∂V∂xi=0,atxi=0,π,for1≤i≤d,[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)=[Up(1-U),U-βV]. Then
(6)GW(W¯)=(-101-β).Theorem 1.
The positive equilibrium point[U¯,V¯] of (5) is locally asymptotically stable.Proof.
Let0=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 H1(𝕋d). Let X=[H1(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(1,1) and 𝔏=𝔇Δ+GW(W¯). The linearization of (5) at [U¯,V¯] is
(8)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¯)=(-μi-101-μi-β).
Then -μi𝔇+GW(W¯) has two negative eigenvalues -μi-1 and -μi-β. Hence, [U¯,V¯] is locally asymptotically stable (see [12]).


## 3. Growing Modes in the System (2)

Letu(x,t)=U(x,t)-U¯, v(x,t)=V(x,t)-V¯. Then
(10)ut=∇2u-χ∇2v-χ∇(u∇v)-(u+1)pu,vt=∇2v+u-βv.
The corresponding linearized system takes the form
(11)ut=∇2u-χ∇2v-u,vt=∇2v+u-βv.Letw(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 (3). To find a normal mode to the linear system (11) of the following form
(12)w(x,t)=rqexp(λqt)eq(x),
where rq is a vector depending on q, we substitute (12) into (11) to get
(13)λqrq=(-q2-1χq21-q2-β)rq,
where q2=∑i=1d‍qi2. This implies that
(14)det(λq+q2+1-χq2-1λq+q2+β)=0,(15)λq2+(2q2+β+1)λq+(q2+1)(q2+β)-χq2=0.
Thus, if there exists a q such that the linear instability criterion holds, that is,
(16)(q2+1)(q2+β)-χq2<0,
then (15) has at least one positive root λq. Obviously,
(17)(2q2+β+1)2-4(q2+1)(q2+β)+4χq2>0.
Therefore, we can denote two distinct real roots for all q by
(18)λq±=-(2q2+β+1)±(β-1)2+4χq22.
The corresponding (linearly independent) eigenvectors r-(q) and r+(q) are given by
(19)r±(q)=[λq±+q2+β,1].
It is easy to see from (16) that there exist only finitely many q, such that λq+>0. We therefore denote the largest eigenvalue by λmax>0 and define
(20)Ωmax≡{q∈Ω:λq+=λmax}.
Moreover, there is one q2 (possibly two) having λ+(q2)=λmax, when we regard λq+ as a function of q2. We also denote ν>0 to be the gap between the λmax and the rest, that is,
(21)ν=minq∈Ω∖Ωmax|λmax-λq|.
Given any initial perturbation w(x,0), that is,
(22)w(x,0)=∑q∈Ω‍wqeq(x)=∑q∈Ω‍{wq-r-(q)+wq+r+(q)}eq(x),
where
(23)wq=wq-r-(q)+wq+r+(q),
we know that the unique solution w(x,t)=[u(x,t),v(x,t)] of (11) is given by
(24)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 and 〈·,·〉 and (·,·) are the inner product of [L2(𝕋d)]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
(25)w(x,t)=[u(x,t),v(x,t)]≡e𝔏tw(x,0)
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
(26)∥w(·,t)∥≤C1exp(λmaxt)∥w(·,0)∥,∀t>0.Proof.
We prove the Lemma in the following two cases.
(1)t≥1. By analyzing (15), we have
(27)limq→∞λq±q2=-1.
It follows from (18) that
(28)λq+-λq-q2≥2χq.
By solving (23), we have
(29)|wq±|≤|r±(q)|×|wq||det[r-(q),r+(q)]|.
Clearly,
(30)|r±(q)|=(λq±q2·q2+q2+β)2+1.
Later on, we will always denote universal constants byEi(i=1,2,…). It is not hard to verify that there exist positive constants E1 and E2, such that
(31)|λq±|≤E1q2,|r±(q)|≤2E2q2,
where E2=max{E1+1,β+1}. It follows from (19) and (28) that
(32)1|det[r-(q),r+(q)]|≤12q1χ.
Combining (29)–(32), we can obtain
(33)|wq±|≤E3q|wq|,
where E3=E21/χ. From (31) and (33), we have
(34)|wq±r±(q)exp(λq±t)|≤2E2E3q3|wq|exp(λq±t).
Fort≥1, it follows from (18) that
(35)exp(λq±t)≤exp{-q2(q2+β+1-χ)+β2q2+β+1}≤exp{-q2+β+1-χβ+3}.
Therefore, for all q>0, there exists a constant E4>0, such that
(36)q3exp(λq±t)≤E4.
From (34) and (36), we obtain
(37)|wq±r±(q)exp(λq±t)|≤E5|wq|,
where E5=2E2E3E4. By (23), we have
(38)|wq|2=wq-2|r-(q)|2+2wq-wq+(r-(q),r+(q))+wq+2|r+(q)|2.
Notice that
(39)∥w(x,0)∥2=(π2)d∑q∈Ω|wq|2.
It follows from (37) and (39) that
(40)∥w(x,t)∥≤∥∑q∈Ω‍wq-r-(q)exp(λq-t)eq(x)∥+∥∑q∈Ω‍wq+r+(q)exp(λq+t)eq(x)∥≤((π2)d∑q∈Ω‍E52|wq|2)1/2+((π2)d∑q∈Ω‍E52|wq|2)1/2=2E5∥w(x,0)∥.
Therefore,
(41)∥w(x,t)∥≤2E5exp(λmaxt)∥w(x,0)∥,∀t≥1.
(2)t≤1. By (11), we have
(42)12ddt∫𝕋d‍{|u|2+A|v|2}dx+∫𝕋d‍{|∇u|2+A|∇v|2-χ∇u∇v}dx+∫𝕋d‍u2dx+Aβ∫𝕋d‍v2dx=A∫𝕋d‍uvdx.
If A=χ2, then the integrand of the second integral can be chosen nonnegative as follows:
(43)|∇u|2+A|∇v|2-χ∇u∇v≥12|∇u|2+χ22|∇v|2≥0.
Combining (42) and Young inequality together, we derive
(44)12ddt∫𝕋d{|u|2+A|v|2}dx≤A2∫𝕋d{|u|2+|v|2}dx.
By Grownwall inequality, we can obtain(45)∥w(x,t)∥≤C1exp(λmaxt)∥w(x,0)∥,
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).
Fors≥1(d=1) and s≥2(d=2,3), there exists a T0>0, such that (10) 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 χ and β.Lemma 4.
Let[u(x,t),v(x,t)] be a solution of (10). Then
(47)12ddt∑|α|=2‍∫2𝕋d‍{|Dαu|2+χ2|Dαv|2}dx+∑|α|=2‍∫2𝕋d‍{14|∇Dαu|2+χ22|∇Dαv|2}dx+Aβ2∑|α|=2‍∫2𝕋d‍|Dαv|2dx≤C{χ+2p-1(p+1)2}×(∥w∥H2+∥w∥H2p)∥∇3w∥2+C′∥u∥2,
where C is the universal constant and C′=4{A/2β-1}3.Proof.
Notice that ifw(x,t) is a solution of (10) on (0,π), then w(-xi,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 2𝕋d≡(-π,π)d.
By (10), one has
(48)12ddt∫2𝕋d‍{|∂xixju~|2dx+A|∂xixjv~|2dx}=-∫2𝕋d{|∇∂xixju~|2+A|∇∂xixjv~|2-χ∇∂xixju~∇∂xixjv~|∇∂xixju~|2}dx-∫2𝕋d‍|∂xixju~|2dx+A∫2𝕋d‍∂xixju~∂xixjv~dx-Aβ∫2𝕋d‍|∂xixjv~|2dx+χ∫2𝕋d‍∇∂xixju~·∂xixj{u~∇v~}dx+∫2𝕋d‍{|∂xixju~|2-∂xixju~∂xixj[vu~(u~+1)p]|∂xixju~|2}dx=:I1+I2,
where A=χ2,  and
(49)I1=-∫2𝕋d{|∇∂xixju~|2+A|∇∂xixjv~|2-χ∇∂xixju~∇∂xixjv~|∇∂xixju~|2}dx-Aβ∫2𝕋d‍|∂xixjv~|2dx+A∫2𝕋d‍∂xixju~∂xixjv~dx-∫2𝕋d‍|∂xixju~|2dx.
Clearly,
(50)|∇∂xixju~|2+A|∇∂xixjv~|2-χ∇∂xixju~·∇∂xixjv~≥12|∇∂xixju~|2+χ22|∇∂xixjv~|2.
Thus,
(51)I1≤-∫2𝕋d‍{12|∇∂xixju~|2+χ22|∇∂xixjv~|2}dx-Aβ2∫2𝕋d‍|∂xixjv~|2dx+(A2β-1)∫2𝕋d‍|∂xixju~|2dx.
By interpolation inequalities, one knows that
(52)∥∂xixju~∥2≤a∥∇∂xixju~∥2+14a2∥u~∥2,
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
(53)I1≤-∫2𝕋d‍{χ22|∇∂xixjv~|214|∇∂xixju~|2+χ22|∇∂xixjv~|2}dx-Aβ2∫2𝕋d‍|∂xixjv~|2dx+C′∥u~∥2,
where C′=4{A/2β-1}3.
One now estimates the second nonlinear termI2:
(54)I2=χ∫2𝕋d‍∇∂xixju~·∂xixj{u~∇v~}dx+∫2𝕋d‍{|∂xixju~|2-∂xixju~∂xixj×[u~(u~+1)p]|∂xixju~|2}dx=:I2′+I2′′.
Clearly,
(55)I2′≤χ{∥∇v~∥L∞∥∇∂xixju~∥∥∂xixju~∥+2∑i=1d‍∥∇u~∥L∞∥∂xixjv~∥∥∇∂xixju~∥+∥u~∥L∞∥∇∂xixju~∥∥∇∂xixjv~∥}.
Notice that
(56)∥g∥L∞(2𝕋d)≤E0∥g∥H2(2𝕋d),ford≤3,(57)∫2𝕋d‍∇u~dx=∫2𝕋d‍∇v~dx=0,∫2𝕋d‍∂xixju~dx=∫2𝕋d‍∂xixjv~dx=0,(58)∥g∥≤∥g∥L6(2𝕋d)≤E1∥g∥H1≤E1∥∇g∥ifd≤3,(59)∥∇g∥H2≤E2(∑|α|=2‍∥∇Dαg∥2)1/2.
From (56)–(59), one can obtain
(60)I2′≤χ{E0∥u~∥H2∥∇v~∥H2∥∇∂xixju~∥+2dE0∥v~∥H2∥∇u~∥H2∥∇∂xixju~∥+E0∥u~∥H2∥∇∂xixju~∥∥∇∂xixjv~∥}≤χ(E0E2+2dE0E2+E0)×∥w~∥H2∑|α|=2‍∥∇Dαw~∥2≤χE3∥w~∥H2∥∇3w~∥2.
Similarly, one has
(61)I2′′≤∫2𝕋d‍{|∂xixju~|2-(u~+1)p|∂xixju~|2}dx+p|p-1|∫2𝕋d‍|(u~+1)p-2u~∂xiu~∂xju~∂xixju~|dx+p∫2𝕋d‍|(u~+1)p-1u~|∂xixju~|2|dx+2p∫2𝕋d‍|(u~+1)p-1∂xiu~∂xju~∂xixju~|dx≤2p-1M2E0p{p(p+1)E22+(p+M1)E12}∥w~∥H2p∥∇3w~∥2.
Here and later, M1,M2,… are universal constants. It follows from (60) and (61) that
(62)I2≤χE3∥w~∥H2∥∇3w~∥2+2p-1M2E0p{p(p+1)E22+(p+M1)E12}×∥w~∥H2p∥∇3w~∥2≤C{χ+2p-1(p+1)2}×(∥w~∥H2+∥w~∥H2p)∥∇3w~∥2,
where C=max{E3,M2E0pE22,M2E0pE12,M1M2E0pE12}.
Recall that[u~,v~] is the even extension of [u,v]. Combining (49), (53), and (62), one has
(63)12ddt∑|α|=2‍∫2𝕋d{|Dαu|2+χ2|Dαv|2}dx+∑|α|=2‍∫2𝕋d‍{14|∇Dαu|2+χ22|∇Dαv|2}dx+Aβ2∑|α|=2‍∫2𝕋d‍|Dαv|2dx≤C{χ+2p-1(p+1)2}×(∥w∥H2+∥w∥H2p)∥∇3w∥2+C′∥u∥2,
where C=max{E0p(E12+E22)+E12,E3} and C′=4{A/2β-1}3.Lemma 5.
Letw(x,t) be a solution of (10), such that for 0≤t≤T<T0,
(64)∥w(·,t)∥H2+∥w(·,t)∥H2p≤1Cmin{14(χ+2p-1(p+1)2),χ22(χ+2p-1(p+1)2)},(65)∥w(·,t)∥≤2C1exp(λmaxt)∥w(·,0)∥.
Then
(66)∥w(·,t)∥H2p≤C3{∥w(·,0)∥H22+exp(2λmaxt)∥w(·,0)∥2}p/2,0≤t≤T,
where C3=max{((E12+1)χ2)p/2,{4C12[1+(E12+1)C′/λmax]}p/2}, if χ2≥1 and C3=max{((E12+1)/χ2)p/2,{4C12[1+(E12+1)C′/λmaxχ2]}p/2}, if χ2<1.Proof.
By (58), we have
(67)∥∇w~(·,t)∥2≤E12∑|α|=2‍∥Dαw~(·,t)∥2.
It follows from (67) that
(68)∥w(·,t)∥H2p≤(∥w(·,t)∥2+(E12+1)∑|α|=2‍∥Dαw(·,t)∥2)p/2.
Now, we estimate the second-order derivatives ofw(·,t). By Lemma 4, we have
(69)12ddt∑|α|=2‍∫𝕋d{|Dαu|2+χ2|Dαv|2}dx≤C′∥u∥2≤C′∥w(·,t)∥2.
Integrating on both sides of (69) from 0 to t, we get from (65) the following:
(70)∑|α|=2‍∫𝕋d{|Dαu(·,t)|2+χ2|Dαv(x,t)|2}dx≤∑|α|=2‍∫𝕋d{|Dαu(·,0)|2+χ2|Dαv(·,0)|2}dx+4C12C′λmax∥w(·,0)∥2exp(2λmaxt).
We will proceed in the following two cases:χ2≥1 and χ2<1.
(1) Ifχ2≥1, we have
(71)∑|α|=2‍∥Dαw(·,t)∥2≤∑|α|=2‍∫𝕋d‍{|Dαu(·,0)|2+χ2|Dαv(·,0)|2}dx+4C12C′λmax∥w(·,0)∥2exp(2λmaxt)≤χ2∑|α|=2‍∥Dαw(·,0)∥2+4C12C′λmax∥w(·,0)∥2exp(2λmaxt).
By (68) and (71), one can obtain
(72)∥w(·,t)∥H2p≤{(E12+1)χ2∑|α|=2‍∥Dαw(·,0)∥2+{4C12+(E12+1)4C12C′λmax}×∥w(·,0)∥2exp(2λmaxt)∑|α|=2‍}p/2≤C3{∥w(·,0)∥H22+∥w(·,0)∥2exp(2λmaxt)}p/2,
where C3=max{[(E12+1)χ2]p/2, {4C12[1+(E12+1)C′/λmax]p/2}.
(2) Ifχ2<1, from (70), we have
(73)χ2∑|α|=2‍∥Dαw(·,t)∥2≤∑|α|=2‍∫𝕋d‍{|Dαu(·,0)|2+χ2|Dαv(·,0)|2}dx+4C12C′λmax∥w(·,0)∥2exp(2λmaxt)≤∑|α|=2‍∥Dαw(·,0)∥2+4C12C′λmax∥w(·,0)∥2exp(2λmaxt).
Therefore,
(74)∑|α|=2‍∥Dαw(·,t)∥2≤1χ2∑|α|=2‍∥Dαw(·,0)∥2+4C12C′χ2λmax∥w(·,0)∥2exp(2λmaxt).
Combining (68) and (74), we have
(75)∥w(·,t)∥H2p≤{E12+1χ2∑|α|=2‍∥Dαw(·,0)∥2+4C12{1+(E12+1)C′λmaxχ2}×∥w(·,0)∥2exp(2λmaxt)E12+1χ2}p/2≤C3{∥w(·,0)∥H22+∥w(·,0)∥2exp(2λmaxt)}p/2,
where C3=max{[(E12+1)/χ2]p/2, {4C12[1+(E12+1)C′/λmaxχ2]}p/2}.


## 5. Main Result

Letθ be a small fixed constant, and let 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
(76)θ=δexp(λmaxTδ),
or equivalently
(77)Tδ=1λmaxlnθδ.Our main result of this paper is the following theorem.Theorem 6.
Assume that the set ofq2=∑i=1d‍qi2 satisfying instability criterion (16) is not empty for given parameters χ and β. Let
(78)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 χ 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
(79)∥wδ(·,t)-δeλmaxt∑q∈Ωmax‍wq+r+(q)eq(x)∥≤C{e-νt+δ∥w0∥H22+δp∥w0∥H2p+1+δeλmaxt+δpepλmaxt}δeλmaxt,
for 0≤t≤Tδ, and ν>0 is the gap between λmax and the rest of λq in (15).Proof.
Letwδ(x,t) be the solutions of (10) with initial data wδ(·,0)=δw0. We define
(80)T*=sup{t∣∥wδ(·,t)-δe𝔏tw0∥≤C12δexp(λmaxt)},(81)T**=sup{,χ22(χ+2p-1(p+1)2)}t∣∥wδ(·,t)∥H2+∥wδ(·,t)∥H2p128(χ+2p-1(p+1)2)≤1C0*min{14(χ+2p-1(p+1)2),χ22(χ+2p-1(p+1)2)}}.
Choose θ, such that
(82)C0*C3{θ+4ppC3(p-1)/2θp}<min{λmax4(χ+1),18(χ+2p-1(p+1)2),χ24(χ+2p-1(p+1)2)}.
Now, we establish estimates forH2 norm of wδ(x,t) for t≤min{Tδ,T*,T**}. For 0<t≤T*, by Lemma 2, we have
(83)∥wδ(·,t)∥≤∥δe𝔏tw0∥+C12δexp(λmaxt)≤C1δexp(λmaxt)∥w(·,0)∥+C12δexp(λmaxt)=32C1δexp(λmaxt).
Applying Lemma 5 and the bootstrap argument yields
(84)∥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), we can obtain
(85)∥wδ(·,t)∥H2p+1≤2pC3(p+1)/2{δp+1∥w0∥H2p+1+δp+1exp(p+1)λmaxt}.
Applying Duhamel’s principle, we know that the solution of (10) is
(86)wδ(·,t)=δe𝔏tw0-∫0t‍e𝔏(t-τ)[χ∇(uδ∇vδ)+uδ(uδ+1)p-uδ,0]dτ.
Using Lemma 2, (56), (58), and Lemma 5 yields, for 0≤t≤min{Tδ,T*,T**},
(87)∥wδ(·,t)-δe𝔏tw0∥≤C1∫0t‍eλmax(t-τ)∥uδ(τ)-uδ(τ)(uδ(τ)+1)p-χ∇(uδ(τ)∇vδ(τ))∥dτ≤C1M3∫0t‍eλmax(t-τ)∥wδ(τ)∥H2p+1dτ+C1(E12χ+E0χ)∫0t‍eλmax(t-τ)∥wδ(τ)∥H22dτ≤C1C0*{χ+1}∫0t‍eλmax(t-τ){∥wδ(τ)∥H22+∥wδ(τ)∥H2p+1}dτ,
where C0*=max{M3,E12+E0}. Notice that t≤min{Tδ,T*,T**}. Thus
(88)∥wδ(t)-δe𝔏tw0∥≤C1C0*C3(χ+1)×∫0t‍eλmax(t-τ){δ2∥w0∥H22+δ2e2λmaxτ}dτ+2pC1C0*C3(p+1)/2(χ+1)×∫0t‍eλmax(t-τ){δp+1∥w0∥H2p+1+δp+1e(p+1)λmaxτ}dτ≤C1C0*C3(χ+1)×{δ∥w0∥H22+2pC3(p-1)/2δp∥w0∥H2p+1λmax+δeλmaxt+(2p/p)C3(p-1)/2δpepλmaxtλmax}δeλmaxt.
Forδ>0 sufficiently small, we claim that
(89)Tδ=min{Tδ,T*,T**}.
If T** is the smallest, we can let t=T**≤Tδ in (84) and (85) to obtain
(90)∥wδ(·,T**)∥H2+∥w(·,T**)∥H2p≤2p-1C3p/2δp∥w0∥H2p+C3δ∥w0∥H2+C31/2(θ+2p-1C3(p-1)/2θp)<1C0*min{14(χ+2p-1(p+1)2),χ22(χ+2p-1(p+1)2)},
for 2p-1C3p/2δp∥w0∥H2p+C3δ∥w0∥H2≤(1/2C0*)min{1/4(χ+2p-1(p+1)2),χ2/2(χ+2p-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
(91)∥wδ(·,T*)-δe𝔏T*w0∥≤C1C0*C3(χ+1)×{δ∥w0∥H22+2pC3(p-1)/2δp∥w0∥H2p+1λmax+θ+(2p/p)C3(p-1)/2θpλmax}δeλmaxT*<C12δeλmaxT*,
for C0*C3(χ+1)(δ∥w0∥H22+2pC3(p-1)/2δp∥w0∥H2p+1)/λmax<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
(92)∥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)∑q∈Ω∖Ωmax‍∥=∥wδ(·,t)-δeλmaxt∑q∈Ωmax‍wq+r+(q)eq(x)∥-I1-I2,
that is,
(93)∥wδ(·,t)-δeλmaxt∑q∈Ωmax‍wq+r+(q)eq(x)∥≤∥wδ(·,t)-δe𝔏tw0∥+I1+I2.
Using (34), we deduce that
(94)I12=∥δ∑q∈Ωmax‍wq-r-(q)exp(λq-t)eq(x)∥2≤4δ2e2(λmax-ν)t(π2)d∑q∈Ωmax‍E22E32q6|wq|2.
From (18) 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 and if there are q12 and q22 satisfying λ+(q2)=λmax, we can let qmax2=max{q12,q22}. From (94), we obtain
(95)I12≤4δ2e2(λmax-ν)tE22E32qmax6(π2)d∑q∈Ωmax‍|wq|2≤4E22E32qmax6δ2e2(λmax-ν)t∥w0∥2,
that is,
(96)I1≤2E2E3qmax3δe(λmax-ν)t=C*δe(λmax-ν)t,
where C*=2E2E3qmax3. Now, we consider I2(97)I22=δ2(π2)d∑q∈Ω∖Ωmax‍{exp(2λq+t)wq+2|r+(q)|2δ2(π2)d∑q∈Ω∖Ωmax‍+2exp((λq++λq-)t)δ2(π2)d∑q∈Ω∖Ωmax‍×wq+wq-(r+(q),r-(q))δ2(π2)d∑q∈Ω∖Ωmax‍+exp(2λq-t)wq-2|r-(q)|2wq+2}.
From (39) and (41), we get
(98)I22≤δ2e2(λmax-ν)t(π2)d×∑q∈Ω∖Ωmax{wq+2|r+(q)|2+2wq+wq-(r+(q),r-(q))+wq-2|r-(q)|2}≤δ2e2(λmax-ν)t(π2)d∑q∈Ω‍|wq|2=δ2e2(λmax-ν)t,
that is,
(99)I2≤δe(λmax-ν)t.
From (88), (96), and (99), one can obtain
(100)∥wδ(·,t)-δeλmaxt∑q∈Ωmax‍wq+r+(q)eq(x)∥≤C1C0*C3(χ+1)×{δ∥w0∥H22+2pC3(p-1)/2δp∥w0∥H2p+1λmax+δeλmaxt+(2p/p)C3(p-1)/2δpepλmaxtλmax}δeλmaxt+C*δe(λmax-ν)t+δe(λmax-ν)t≤{(C*+1)e-νt+(4p/p)C1C0*C3p+1λmax(χ+1)×{δ∥w0∥H22+δp∥w0∥H2p+1+δeλmaxt+δpepλmaxt}(4p/p)C1C0*C3p+1λmax}δeλmaxt≤C{e-νt+δ∥w0∥H22+δp∥w0∥H2p+1+δeλmaxt+δpepλmaxt}δeλmaxt,
where C=max{C*+1,((4p/p)C1C0*C3p+1/λmax)(χ+1)}. This completes the proof.


## 6. Conclusion

Notice that for0≤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
(101)∥δeλmaxt∑q∈Ωmax‍wq+r+(q)eq(x)∥=δeλmaxt(π2)d/2×(∑q∈Ωmax‍|wq+|2|r+(q)|2)1/2≥δeλmaxt|wq0+||r+(q0)|,
have the dominant leading order of δeλmaxt. Theorem 6 implies that the dynamics of a general perturbation is characterized by such linear dynamics over a long time period of ɛTδ≤t≤Tδ, for any ɛ>0 (see [10]). In particular, choose a fixed q0=(q01,q02,…,q0d)∈Ωmax and let
(102)w0(x)=r+(q0)|r+(q0)|eq0(x).
Then
(103)∥w0(x)∥H2={(π2)d(1+|q0|2+|q0|4)}1/2.
Therefore, if t≤Tδ, we can obtain from (100)
(104)∥wδ(·,t)-δeλmaxTδr+(q0)|r+(q0)|eq0(x)∥≤C{e-νTδ+δ∥w0∥H22+δp∥w0∥H2p+1+δeλmaxTδ+δpepλmaxTδ}δeλmaxTδ≤C{θexp(-νλmaxlnθδ)+θδ(π2)d(1+|q0|2+|q0|4)+θδp(π2)(p+1)d/2×(1+|q0|2+|q0|4)(p+1)/2+θ2+θp+1(-νλmaxlnθδ)}.
Note that δ≤θ, θ, and ν are fixed constants, and q0 is a fixed vector. Hence, we have
(105)∥wδ(·,t)-δeλmaxTδr+(q0)|r+(q0)|eq0(x)∥≤CC4{δν/λmax+θ2+θp+1},
where C4=max{θ1-ν/λmax,(π/2)(p+1)d/2(1+|q0|2+|q0|4)(p+1)/2+1}. Thus,
(106)∥wδ(·,t)∥≥∥δeλmaxTδr+(q0)|r+(q0)|eq0(x)∥-CC4{δν/λmax+θ2+θp+1}≥θ-C**{δν/λmax+θ2+θp+1}.Letθ be sufficiently small so that θ/2C**-θ2-θp+1>0 holds. Then δ1=:(θ/2C**-θ2-θp+1)λmax/ν, and
(107)∥wδ(·,t)∥≥θ-C**{δν/λmax+θ2+θp+1}≥θ2>0,0≤δ≤δ1.
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).



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*Source: 454513-2013-10-07.xml*