Asymptotic stability of a composite wave of two traveling waves to a hyperbolic-parabolic system modeling chemotaxis

Research Article

Received 6 January 2012 Published online 12 February 2013 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.2731MOS subject classification: 35M30; 35B35

Asymptotic stability of a composite wave oftwo traveling waves to a hyperbolic–parabolicsystem modeling chemotaxis

Jingyu Lia,b*†, Lina Wanga and Kaijun Zhangb

Communicated by Z. Xin

In this paper, we study the asymptotic stability of a composite wave consisting of two traveling waves to a hyperbolic–parabolic system modeling repulsive chemotaxis. On the basis of elementary energy estimates, we show that thecomposite wave is asymptotically stable under general initial perturbations, which are not necessarily zero integral. Asan application, we obtain a similar result for this system in the presence of a boundary. Copyright © 2013 John Wiley &Sons, Ltd.

Keywords: chemotaxis; Keller–Segel model; asymptotic stability; composite wave; traveling waves; nonlinear kinetics

1. Introduction

In this paper, we consider the following hyperbolic–parabolic system arising in chemotaxis:�pt � .pq/x D Dpxx , x 2R, t > 0,qt � px D 0, x 2R, t > 0

(1)

with initial data

.p, q/.x, 0/D .p0, q0/.x/ �!�

p˙, q˙�

as x!˙1, (2)

where p0.x/ > 0 and p˙ > 0. This model is derived from the following simplified Keller–Segel model [1–3]:(Opt D D

�Opxx C ˛

�Op OwxOw

�x

�, x 2R, t > 0,

Owt D �Op Ow �� Ow, x 2R, t > 0,(3)

where Op denotes the density of bacteria and Ow represents the chemoattractant concentration at time t in position x, D > 0 is the dif-fusion rate of bacteria, and � > 0 and � � 0 are constants representing the reaction rates. If ˛ < 0, the chemotaxis is attractive; and if

˛ > 0, the chemotaxis is repulsive. In this paper, we consider the repulsive case ˛ > 0. Taking pD �Op, qD OwxOwD @@x ln Ow and normalizing

D˛ D 1, the system (3) is then transformed to (1).System (3) was first studied in [3], where Levine and Sleeman showed various qualitative properties of the solution, including the

blowup, the collapse, and the aggregation. In the attractive case ˛ < 0, Yang et al. [4] generalized some of the results of [3] to the mul-tidimensional case with zero-flux boundary condition. Later, Yang et al. [5] also studied the life-span problem for (3). In the repulsivecase ˛ > 0, when the initial data are sufficiently small, Zhang and Zhu [6] proved the global well-posedness of weak solutions to (1) ona bounded interval with zero-flux boundary condition. Later, Guo et al. [7] extended the results of [6] to the large solutions.

Concerning system (1), which is equivalent to (3) in the repulsive case, Wang and Hillen [8] proved the existence of traveling wavesmotivated by the numerical results in [3]. Using the elementary energy method, Li and Wang [9] obtained the nonlinear stability ofsingle traveling waves to (1) under the initial perturbations of zero integral but with large wave amplitude. On the basis of many more

a Center for Partial Differential Equations, East China Normal University , Minhang, Shanghai, 200241, Chinab School of Mathematics and Statistics, Northeast Normal University , Changchun, 130024, China*Correspondence to: Jingyu Li, Center for Partial Differential Equations, East China Normal University , Minhang, Shanghai, 200241, China.†E-mail: [email protected]

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Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2013, 36 1862–1877

J. LI, L. WANG AND K. ZHANG

involved estimates, Li and Wang [10] generalized their results in [9] to the model with more general chemical kinetic functions. Recently,they [11] also studied the stability of single traveling waves to a parabolic–parabolic chemotaxis model with small chemical diffusionrate. In the multidimensional case, Li et al. [12] established the blowup criterion and the global existence of classical solutions to (1).Especially, when the initial data are sufficiently close to a positive constant steady state, they proved the exponentially convergence ofthe solutions to that constant steady state as time tends to infinity. Very recently, Li et al. [13] generalized the results in [12] to the caseof bounded domain. For other works on chemotaxis models, we refer the interested readers to [1, 14].

In this paper, we are interested in the stability of a composite wave consisting of two traveling waves under general perturbationsfor system (1). More precisely, denote by .P1, Q1/.x, t/ the 1-traveling wave connecting the left state .p�, q�/ with an intermediatestate .pm, qm/ with speed s1 < 0, and by .P2, Q2/.x, t/ the 2-traveling wave connecting the intermediate state .pm, qm/ with theright state .pC, qC/ with speed s2 > 0 ((12) and (16)). The composite wave consisting of these two traveling waves is then givenby .Qp, Qq/ .x, t/ D .P1 C P2 � pm, Q1 C Q2 � qm/. Unlike the works in [9–11], we show that such composite wave .Qp, Qq/ .x, t/ to system(1) is asymptotically stable under general initial perturbations without zero integral assumption.

This problem is actually motivated by the study of the following initial boundary value problem on the half space for system (1):

8̂̂ˆ̂̂̂<ˆ̂̂̂̂̂:

pt � .pq/x D Dpxx , x > 0, t > 0,

qt � px D 0, x > 0, t > 0,

q.0, t/D 0, t > 0,

.p, q/.x, 0/D .p0, q0/.x/, x > 0,

limx!C1

.p0, q0/.x/D .pC, qC/,

(4)

where pC > 0, qC > 0 and the initial data satisfy the compatible condition. Note that system (1) is invariant under the followingtransformation:

.p, q/.x, t/D .p,�q/.�x, t/, (5)

which satisfies the boundary condition q.0, t/ D 0. Thus when we study the stability of 2-traveling wave to the half space problem (4),which connects .pA, 0/ with

�pC, qC

�with speed s2 > 0, by taking the transformation (5), we change the problem (4) into the initial

value problem (1) with .p�, q�/D�

pC,�qC�‘ and .pm, qm/D .pA, 0/.

Another motivation of this problem arises from the following perfect inviscid system related to system (1):�pt � .pq/x D 0, x 2R, t > 0,qt � px D 0, x 2R, t > 0.

(6)

It was shown by Wang and Hillen [8] that the solution of the general Riemann problem for system (6) connecting two states .p�, q�/and

�pC, qC

�is generally a composite wave consisting of two shock waves. More precisely, there exists an intermediate state .pm, qm/

such that .pm, qm/ is connected to the left state .p�, q�/ by a 1-shock wave with speed s1 < 0, whereas�

pC, qC�

is connected to.pm, qm/ by a 2-shock wave with speed s2 > 0. Because system (6) is actually the vanishing viscosity limit (D! 0) of system (1) [8], itis natural to study the asymptotic behavior of solutions to the viscous system (1) toward the viscous version of such composite wave.It is remarkable that such type of stability problem for a composite wave was also studied by Huang and Matsumura [15] for the fullcompressible Navier–Stokes equations and by Hsiao et al. [16] for the Jin–Xin relaxation hyperbolic system.

The organization of this paper is as follows. In Section 2, we introduce some preliminary notations and state the main results forsystem (1). In Section 3, we reformulate system (1) to an integrated system and establish various a priori estimates for this integratedsystem. In the end of this section, we prove the main results for system (1). In Section 4, we consider a case where the chemical kineticfunction is nonlinear, and we obtain similar results.

Notations. Throughout this paper, C denotes a generic positive constant, which can change from one line to another. An integrallacking limits of integration means an integral over the whole real line. Hk.�/ denotes the usual kth order Sobolev space on � � Rwith the norm given by

kfkHk.�/ :D

0@ kX

jD0

k@jxfk2

L2.�/

1A

1=2

.

Denote k � k :D k � kL2.�/ and k � kk :D k � kHk.�/ for simplicity. Denote by ı :D jpC � p�j C jqC � q�j the wave amplitude.

2. Preliminaries and main results

Before the statement of our main results, we present some basic notations and calculations. Let us first recall the general Riemannproblem for system (6) with Riemann initial data

.p, q/.x, 0/D

�.p�, q�/, x < 0,�

pC, qC�

, x > 0.(7)


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63


It was shown by Wang and Hillen [8] that because p > 0, system (6) has two eigenvalues satisfying

�1 D�q

2�

pq2C 4p

2< 0< �

q

2C

pq2C 4p

2D �2

with eigenvectors given by

r1.p, q/D .��1, 1/> and r2.p, q/D .�2,�1/>.

Obviously, r�1 � r1 D �q=p

q2C 4p � 1 < 0 and r�2 � r2 D q=p

q2C 4p � 1 < 0, which implies that the two characteristic fields areboth genuinely nonlinear. Thus the Rankine–Hugoniot jump condition takes the form(

s�

pC � p��D�pCqCC p�q�,

s�

qC � q��D�pCC p�,

(8)

where s are the shock speeds. From (8), it is easy to compute that

� s2�

qC � q��D�pCqCC p�q�. (9)

By .8/2, we have

p� D pCC s�

qC � q��

,

which together with (9) leads to

s2C sq� � pC D 0. (10)

Therefore, the shock speeds are given by

s1 D�q�

2�

p.q�/2C 4pC

2< 0< �

q�

2C

p.q�/2C 4pC

2D s2. (11)

In view of these facts, Wang and Hillen [8] proved that the solution to the general Riemann problem (6) and (7) consists of two shockwaves. Precisely speaking, there exists an intermediate state .pm, qm/ such that .pm, qm/ must be connected to .p�, q�/ by a 1-shockwave with speed s1, whereas

�pC, qC

�is connected to .pm, qm/ by a 2-shock wave with speed s2.

Let us next recall the definitions of traveling waves to system (1) corresponding to the aforementioned shock waves [8]. The1-traveling wave corresponding to the 1-shock wave is a solution to (1), which has the form

.p, q/.x, t/D .P1, Q1/.z1/with z1 :D x � s1t.

Thus .P1, Q1/ satisfies (�s1P1z1 � .P1Q1/z1 D DP1z1z1 ,

�s1Q1z1 � P1z1 D 0(12)

with boundary condition

.P1, Q1/.�1/D .p�, q�/ and .P1, Q1/.C1/D .pm, qm/.

Lemma 2.1The 1-traveling wave .P1, Q1/ has the following explicit formula:

P1.z1/D p�Cp� � pm

C1e1

Ds1.pm�p�/z1 � 1

and Q1.z1/D1

s1.A1 � P1/. (13)

Here s1 < 0 is given by (11), A1 :D s1q�C p� D s1qmC pm, C1 < 0 is a constant, P1z1 > 0 and pm > p�.

ProofThis formula was first obtained by Wang and Hillen [8, p. 66]. However, for the convenience of the reader, we present the proof here.Integrating the equation .12/2 on .�1, z1/, we obtain

Q1 D1

s1.A1 � P1/ (14)1

86

4



with A1 D s1q�C p�. Substituting (14) into .12/1, integrating the resultant equation over .�1, z1/, and using (10), we have

s1DP1z1 D P21 �

�s2

1C A1

�P1C p�

�s2

1C s1q��

D P21 �

�s2

1C A1

�P1C p�pm.

(15)

One can then easily prove that the solution of (15) is given by (13). �

Similarly, the 2-traveling wave .P2, Q2/.x � s2t/ satisfies(�s2P2z2 � .P2Q2/z2 D DP2z2z2 ,

�s2Q2z2 � P2z2 D 0(16)

with z2 :D x � s2t and boundary condition

.P2, Q2/.�1/D .pm, qm/ and .P2, Q2/.C1/D�

pC, qC�

.

Furthermore, .P2, Q2/ satisfies the following lemma.

Lemma 2.2The 2-traveling wave .P2, Q2/ has the following explicit formula:

P2.z2/D pmCpm � pC

C2e1

Ds2.pC�pm/z2 � 1

and Q2.z2/D1

s2.A2 � P2/. (17)

Here s2 > 0 is given by (11), A2 :D s2qmC pm D s2qCC pC, C2 < 0 is a constant, P2z2 < 0 and pC < pm.

The composite traveling wave consisting of the 1-traveling wave and the 2-traveling wave has the form

Qp.x, t/D P1.x � s1t/C P2.x � s2t/� pm,

Qq.x, t/D Q1.x � s1t/CQ2.x � s2t/� qm.(18)

We assume that the initial data .p0, q0/ satisfy�p0.�/� Qp.�, 0/, q0.�/� Qq.�, 0/

�2�

H1 � H1�\�

L1 � L1�

. (19)

Denote by ı :D jpC�p�jCjqC�q�j the wave amplitude. It is easy to see that if ı is sufficiently small, then the two vectors

�pC � pm

qC � qm

�

and

�pm � p�

qm � q�

�are linearly independent in R2. Hence, in view of (19), suppose there are two constants ˛1 and ˛2, which are in

general not zero, such that Z C1�1

�p0.x/� Qp.x, 0/q0.x/� Qq.x, 0/

�dx D ˛1

�pm � p�

qm � q�

�C ˛2

�pC � pm

qC � qm

�.

If we take

Np.x, t/D P1.x � s1tC ˛1/C P2.x � s2tC ˛2/� pm,

Nq.x, t/D Q1.x � s1tC ˛1/CQ2.x � s2tC ˛2/� qm,(20)

we then have Z C1�1

�p0.x/� Np.x, 0/q0.x/� Nq.x, 0/

�dx D

Z C1�1

�p0.x/� Qp.x, 0/q0.x/� Qq.x, 0/

�dxC

Z C1�1

�Qp.x, 0/� Np.x, 0/Qq.x, 0/� Nq.x, 0/

�dx

D

Z C1�1

�p0.x/� Qp.x, 0/q0.x/� Qq.x, 0/

�dxC

Z C1�1

�P1.x/� P1.xC ˛1/

Q1.x/�Q1.xC ˛1/

�dx

C

Z C1�1

�P2.x/� P2.xC ˛2/

Q2.x/�Q2.xC ˛2/

�dx

D

Z C1�1

�p0.x/� Qp.x, 0/q0.x/� Qq.x, 0/

�dx � ˛1

�pm � p�

qm � q�

�� ˛2

�pC � pm

qC � qm

�

D

�00

�.

(21)

Set

�0.x/ :D�

Z C1x

Œp0.y/� Np.y, 0/�dy and 0.x/ :D�

Z C1x

Œq0.y/� Nq.y, 0/�dy. (22)


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65


By (21), it is clear that

�0.˙1/D 0D 0.˙1/. (23)

Our first main result is stated as follows.

Theorem 2.3Suppose that the Riemann solution of (6) and (7) consists of two shock waves. Assume that the initial data .p0, q0/ satisfy (19) and that.�0, 0/ 2 L2. Then there exist positive constants ı0, �0, and �0, such that if ı < ı0 and

k .p0 � Qp.�, 0/, q0 � Qq.�, 0// kH1\L1 Ck.�0, 0/k � �0,

then the system (1) and (2) has a unique global solution satisfying p.x, t/� �0 for all .x, t/ 2R�RC and

.p� Np, q� Nq/ 2 C�Œ0,C1/; H1

�\ L2

�.0,C1/; H1

�, .px � Npx/ 2 L2

�.0,C1/; H1

�.

Furthermore,

supx2Rj�

p� Np, q� Nq�.x, t/j ! 0 as t!1. (24)

Remark 2.1In [9], Li and Wang proved the nonlinear asymptotic stability of single traveling waves to system (1) without diffusion waves but withlarge wave amplitude. In the current paper, the initial perturbation for the composite wave is generally not zero integral. More precisely,RC1�1 .p�Qp, q�Qq/.x, 0/dx is in general not zero. By taking two translations˛1 and˛2 in the composite wave (see (20)), which are uniquely

determined by the initial perturbation, the integralRC1�1 .p� Np, q� Nq/ .x, 0/dx is zero. As a consequence, the anti-derivative technique

is also applicable for the unknowns .p� Np, q� Nq/. In other words, when we consider the composite wave for system (1), the assump-tion for the zero diffusion wave made in [9] for the single traveling wave is not needed anymore. To study the asymptotic stability ofcomposite waves, the difficulties now arise from the nonlinear interactions of different traveling waves. To control such interactions, wemake the assumption of small wave amplitude. The question of the stability of composite waves with large wave amplitude is still open,and we shall investigate this problem in the future. We should also mention that the stability of single traveling waves with diffusionwaves to system (1) remains open.

To prove Theorem 2.3, we employ the elementary energy method originally adopted by Matsumura and Nishihara [17] (see also[18, 19]). Using the anti-derivative technique, we first reformulate the original system to the integrated system (27). Then we establishvarious a priori estimates for this integrate system. Finally, the Sobolev imbedding theorem gives the desired result. In such energymethod, the most important step is to establish the basic L2 energy estimate (Lemma 3.3). To achieve this, the key observation is thatthe interactions of traveling waves with the same speed are good terms, whereas the interactions of traveling waves with differentspeeds are exponentially decay.

As we mentioned in Section 1, through the transformation (5), the initial boundary value problem (4) can be reduced to a specialcase of the system (1) and (2) with .p�, q�/ D

�pC,�qC

�. Moreover, owing to the results of Wang and Hillen [8], we know that when

qC > 0 is sufficiently small, the Riemann problem (6) and (7) with .p�, q�/D�

pC,�qC�

has a unique solution consisting of two shockwaves with the intermediate state given by .pA, 0/; and the 2-traveling wave to (4) corresponding to the 2-shock wave to (6) connectsthe state .pA, 0/ to the state

�pC, qC

�. After extending (4) to the whole space through (5), it is easy to see that .Np.x, t/, Nq.x, t// in (20) is

still well defined with the shifts ˛1 and ˛2 satisfying ˛1 D�˛2. Now our second result can be stated as follows.

Theorem 2.4Assume that qC > 0 and that

�p0 � pC, q0 � qC

�2 H1

�RC

�\ L1

�RC

�and .�0, 0/ 2 L2

�RC

�. Then there exist positive constants ı0,

�0 and �0, such that if 0< qC < ı0 and

k�

p0 � pC, q0 � qC�kH1.RC/\L1.RC/Ck.�0, 0/kL2.RC/ � �0,

then the initial boundary value problem (4) has a unique global solution satisfying p.x, t/� �0 for all .x, t/ 2RC �RC and

.p� Np, q� Nq/ 2 C�Œ0,C1/; H1

�RC

��\ L2

�.0,C1/; H1

�RC

��, .px � Npx/ 2 L2

�.0,C1/; H1

�RC

��.

Furthermore,

supx2RC

j.p� P2.x � s2tC ˛2/, q�Q2.x � s2tC ˛2//j ! 0 as t!1.18

66



3. Proofs of the main results

Following the standard energy estimate approach [9, 15, 17, 18], we first reformulate the original system (1) to an integrated system. Todo so, note that the system (1) is of conserved form, in view of (22), we can expect that the following new unknown functions �.x, t/and .x, t/ are well defined in some Sobolev space for all t > 0:

�.x, t/D�

Z C1x

Œp.y, t/� Np.y, t/�dy and .x, t/D�

Z C1x

Œq.y, t/� Nq.y, t/�dy,

which implies that we shall seek a solution of the form

.p, q/.x, t/D .Np, Nq/ .x, t/C .�x , x/.x, t/. (25)

Substituting (25) into (1) and using (12) and (16), we obtain��xt � ŒNp x C Nq�x C �x x C E�x D D�xxx , tx D �xx ,

(26)

where E :D .P1 � pm/.Q2 � qm/C .P2 � pm/.Q1 � qm/. By (23), it is also natural to expect that �.�1, t/ D .�1, t/ D 0. Integrating(26) with respect to x, we then have �

�t D D�xx C �x x C Np x C Nq�x C E, t D �x

(27)

with initial data (22). Clearly, the conclusion that k.�x , x/.x, t/kL1x ! 0 as t ! 1 implies the asymptotic stability of the compositewave .Np, Nq/. Therefore, we look for solutions to the reformulated system (27) and (22) in the following solution space:

X.0, T/ :Dn.�.x, t/, .x, t// : .�, / 2 C

�Œ0, T�; H2

�,�x 2 L2

�.0, T/; H2

�, x 2 L2

�.0, T/; H1

�o,

where Œ0, T� is a time interval on which the solution is assumed to exist. Define

N.T/ :D supt2Œ0,T�

k.�, /.�, t/k2.

By the Sobolev embedding theorem, it holds

supt2Œ0,T�

fk�kL1x , k�xkL1x , k kL1x , k xkL1x g � N.T/. (28)

For system (27) and (22), we have the following proposition.

Proposition 3.1There exist positive constants ı0 and �1, such that if

ı < ı0 and k.�0, 0/k2 � �1,

then the Cauchy problem (27) and (22) has a unique global solution .�, / 2 XŒ0,C1/ satisfying

k.�, /k22C

Z t

0

�k�xk

22Ck xk

21

�d � Ck.�0, 0/k

22C Cı1=2

0 (29)

for all t 2 Œ0,C1/.

In view of (25) and the Sobolev embedding theorem, Theorem 2.3 is a consequence of Proposition 3.1. To prove Proposition 3.1, bya standard method (e.g., [20–22]), we can first prove the local existence of a unique solution to system (27) and (22); and then by thestandard continuation process, the global existence of .�, / follows directly from the following a priori estimates.

Proposition 3.2There exist positive constants ı0 and �0, such that if ı < ı0 and .�, / 2 X.0, T/ is a solution to (27) for some positive T and N.T/ � �0,then the solution .�, / of (27) and (22) satisfies (29) for any t 2 Œ0, T�.

Note that when the wave amplitude ı is sufficiently small, by Lemmas 2.1 and 2.2, we first have

Np >p�

4> 0. (30)

Then the proof of Proposition 3.2 is given by the following series of lemmas.


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67


Lemma 3.3Assume that ı0 and �1 are sufficiently small, then there exists a constant C > 0 such that

k.�, /k2C

Z t

0k�xk

2d � Ck.�0, 0/k2C N.T/

Z t

0k xk

2d C Cı1=20 . (31)

ProofMultiplying .27/1 by �=Np and .27/2 by , integrating the resultant equations on R and adding them, we have

1

2

d

dt

Z ��2

NpC 2

�dx D D

Z�xx

�

NpdxC

1

2

Z�2�

1

Np

�t

dxC1

2

ZNq

Np

��2�

xdxC

Z x�x

�

NpdxC

ZE�

Npdx. (32)

It is easy to see that

�xx�

NpD

��x�

Np

�x��2

x

Np� �x�

�1

Np

�x

D

��x�

Np

�x��2

x

Np�

1

2

��2�

1

Np

�x

�xC

1

2�2�

1

Np

�xx

.

We then conclude that

1

2

d

dt

Z ��2

NpC 2

�dxC D

Z�2

x

Npdx D

1

2

Z�2�

1

Np

�tC

�D

Np

�xx�

�Nq

Np

�x

dxC

Z x�x

�

NpdxC

ZE�

Npdx. (33)

The last term of (33) can be estimated as follows. Note that P1.C1/D pm, s1 < 0 and pm > p�, by (13), we have

jP1 � pmj D jpm � p�j �

ˇ̌̌C1e

1Ds1.pm�p�/.xC˛1/ � e�

1D .pm�p�/t

ˇ̌̌ˇ̌̌C1e

1Ds1.pm�p�/.xC˛1/ � e�

1D .pm�p�/t � 1

ˇ̌̌

�

�Cıe�Cı.jxjCt/ for x > 0,Cı.p�C ı/ for x � 0.

(34)

Similarly, note that s2 > 0 and pC < pm, by (17), we have

jQ2 � qmj D1

s2jP2 � pmj �

�Cı.pCC ı/ for x > 0,Cıe�Cı.jxjCt/ for x � 0.

(35)

Therefore

jEj � Cı2e�Cı.jxjCt/ for .x, t/ 2R�RC. (36)

By (30) and Cauchy’s inequality, the last term of (33) satisfiesˇ̌̌ˇZ

E ��

Npdx

ˇ̌̌ˇ� C

�ZE2dx

�1=2 �Z�2dx

�1=2

� Cı3=2e�Cıtk�k. (37)

We now estimate the first term on the right hand side of (33). For convenience, we denote

I1 :D

�1

Np

�tC

�D

Np

�xx�

�Nq

Np

�x

. (38)

By (20), (12), and (16),

I1 D�1

Np2

2P1t C 2P2t � .P1Q1/x � .P2Q2/x �

2D

Np.P1x C P2x/

2

C.P1C P2 � pm/.Q1x CQ2x/� .Q1CQ2 � qm/.P1x C P2x/�

D�1

Np2

"�2s1P01 � 2P01Q1 �

2D.P01/2

Np

� 2s2P02 � 2P02Q2 �2D.P02/

2

Np�

4DP01P02Np

C.P1 � pm/Q02C .P2 � pm/Q

01C .qm �Q1/P

02C .qm �Q2/P

01

�D:

2

Np2

"s1P01C P01Q1C

D.P01/2

Np

#C

2

Np2

"s2P02C P02Q2C

D.P02/2

Np

#C I2,

(39)

18

68



where P0i :D P0i .zi/ with zi D x � sitC ˛i for i D 1, 2, and I2 consists of interaction of traveling waves with different speeds. By (30) andsimilar to (36), we have

jI2j � Cı2e�Cı.jxjCt/. (40)

We can estimate the first term in the last equality of (39) as follows. By (30) and a similar argument to (36),

2

Np2

"s1P01C P01Q1C

D.P01/2

Np

#

D2

Np3

hs1P01P1C s1P01.P2 � pm/C P01Q1P1C P01Q1.P2 � pm/C D

�P01�2i

�2

Np3

hs1P01P1C P01Q1P1C D

�P01�2iC Cı2e�Cı.jxjCt/

D2P01s1 Np3

hs2

1P1C P1A1 � P21 C s1DP01

iC Cı2e�Cı.jxjCt/

D2P01s1 Np3

� p�pmC Cı2e�Cı.jxjCt/,

(41)

where we have used (15) in the last equality. Similarly, the second term in the last equality of (39) satisfies

2

Np2

"s2P02C P02Q2C

D.P02/2

Np

#�

2P02s2 Np3

� pCpmC Cı2e�Cı.jxjCt/. (42)

By Lemma 2.1, Lemma 2.2, and (11), we obtain

P01s1< 0 and

P02s2< 0. (43)

Because pm > p� > 0, it follows from (38)–(43) that

I1 � Cı2e�Cı.jxjCt/,

which, together with (28), (30), (33), and (37), leads to

Z�2dxC

Z 2dxC

Z t

0

Z�2

x dxd

� C

Z�2

0 dxC C

Z 2

0 dxC .CC N.T//ı3=2Z t

0e�Cı�k�kd C N.T/

Z t

0

Zj�x xjdxd

� k.�0, 0/k2C CN.T/ı1=2C N.T/

Z t

0

Zj�xj

2dxd C N.T/

Z t

0

Zj xj

2dxd ,

where we have used Cauchy’s inequality in the first inequality. Choosing N.T/ suitably small, consequently, we obtain thedesired (31). �

Lemma 3.4If ı0 and �1 are sufficiently small, then the following inequalities hold:

k xk2C

Z t

0k xk

2d � Ck 0xk2C Ck.�0, 0/k

2C Cı1=20 , (44)

k.�, /k2C

Z t

0k�xk

2d � Ck 0xk2C Ck.�0, 0/k

2C Cı1=20 . (45)

ProofWe differentiate .27/2 with respect to x and use .27/1 to obtain

D tx D �t � �x x � Np x � Nq�x � E. (46)

Multiplying this equation by x , integrating the resultant equation on R�RCt , observing that

�t x D .� x/t � � xt D .� x/t � ��xx D .� x/t � .��x/x C �2x ,


18

69


and using Cauchy’s inequality, (28), (36), and Lemma 3.3, we obtain

D

2

Z 2

x dxC

Z t

0

ZNp 2

x dxd

�D

2

Z 2

0xdxC

Z� xdx �

Z�0 0xdxC

Z t

0

Z�2

x dxd

C N.T/

Z t

0

Z 2

x dxd C CN.T/ı3=2Z t

0e�Cı�d

�D

2

Z 2

0xdxCD

4

Z 2

x dxC Ck.�0, 0/k2C N.T/

Z t

0k xk

2d C Cı1=2N.T/.

(47)

Taking N.T/ small enough, we obtain (44), which in combination with (31) yields (45). �

Lemma 3.5If ı0 and �1 are sufficiently small, it follows that

k�xk2C

Z t

0k�xxk

2d � Ck.�0, 0/k21C N.T/

Z t

0

Z 2

xxdxd C Cı1=20 . (48)

ProofWe differentiate .27/1 with respect to x to obtain

�tx D D�xxx C .�x x/x C .Np x/x C .Nq�x/x C Ex . (49)

Multiplying (49) by �x and integrating the resulting equation on R�RCt , notice that

�xxx�x D .�xx�x/x � �2xx ,

.�x x/x�x D��2

x x

�x�

1

2 x

��2

x

�xD��2

x x

�x�

1

2

� x�

2x

�xC

1

2 xx�

2x ,

.Np x/x �x D .Np x�x/x � Np x�xx ,

.Nq�x/x �x D�Nq�2

x

�x�

1

2Nq��2

x

�xD

1

2

�Nq�2

x

�xC

1

2Nqx�

2x ,

Ex�x D .E�x/x � E�xx ,

and using Lemma 3.4, we thus have

1

2

Z�2

x dxC D

Z t

0

Z�2

xxdxd

�1

2

Z�2

0xdxC N.T/

Z t

0

Zj�x xxjdxd C C

Z t

0

Zj x�xxjdxd

C C

Z t

0

Z�2

x dxd C C

Z t

0

ZjE�xxjdxd

� Ck.�0, 0/k21C N.T/

Z t

0

Z 2

xxdxd CD

4

Z t

0

Z�2

xxdxd C Cı1=2.

This gives the desired (48). �

We next estimate the second-order derivatives of .�, /.

Lemma 3.6For t 2 Œ0, T�,

k xxk2C

Z t

0k xxk

2d � k 0xxk2C Ck.�0, 0/k

21C Cı1=2

0 , (50)

k�xk2C

Z t

0k�xxk

2d � Ck.�0, 0/k21C Ck 0xxk

2C Cı1=20 . (51)

ProofDifferentiating (46) with respect to x gives

D txx D �tx � .�x x/x � .Np x/x � .Nq�x/x � Ex . (52)

18

70



Multiplying (52) by xx , integrating the resulting equation on R�RCt , and notice that

�tx xx D .�x xx/t � �x txx D .�x xx/t � �x�xxx D .�x xx/t � .�x�xx/x C �2xx ,

and

.�x x/x xx D �xx x xx C �x 2xx ,

.Np x/x xx D Npx x xx C Np 2xx ,

.Nq�x/x xx D Nqx�x xx C Nq�xx xx ,

(53)

we obtain Z 2

xxdxC

Z t

0

ZNp 2

xxdxd

� C

Z 2

0xxdxC C

Z�2

0xdxC C

Z�2

x dxC

�p�

8C 2N.T/

�Z t

0

Z 2

xxdxd

C C

Z t

0

Z�2

xxdxd C C

Z t

0

Z 2

x dxd C C

Z t

0

Z�2

x dxd C C

Z t

0

ZE2

x dxd .

(54)

Choosing N.T/ suitably small and using Lemmas 3.4 and 3.5, we obtainZ 2

xxdxC

Z t

0

Z 2

xxdxd �

Z 2

0xxdxC Ck.�0, 0/k21C Cı1=2,

which in combination with Lemma 3.5 gives (51). �

Lemma 3.7For t 2 Œ0, T�,

k�xxk2C

Z t

0k�xxxk

2d � k.�0, 0/k22C Cı1=2

0 . (55)

ProofMultiplying (49) by �xxx , we have

�tx�xxx D D�2xxx C Œ.�x x/x C .Np x/x C .Nq�x/x� �xxx C Ex�xxx . (56)

Note that

�tx�xxx D .�tx�xx/x � �txx�xx D .�tx�xx/x �1

2

��2

xx

�t

.

Integrating (56) on R�RCt , we derive

1

2

Z�2

xxdxCD

Z t

0

Z�2

xxxdxd

�

Z�2

0xxdxC

�N.T/C

D

2

�Z t

0

Z�2

xxxdxd C .1C N.T//

Z t

0

Z 2

xxdxd

C .1C N.T//

Z t

0

Z�2

xxdxd C C

Z t

0

Z 2

x dxd C C

Z t

0

Z�2

x dxd C C

Z t

0

ZE2

x dxd .

Choosing N.T/ small enough, and using Lemmas 3.4 and 3.6, we obtain (55). �

Proof of Theorem 2.3We differentiate the second equation of (27) with respect to x, multiply the resultant equation by x , and integrate it over Rx

to obtain

d

dtk x.t/k

2 D 2

Z�xx xdx,

which, in combination with (29), yieldsZ t

0

ˇ̌̌ˇ d

dtk x.t/k

2ˇ̌̌ˇd � C

Z t

0

�k�xxk

2Ck xk2�

d � Ck.�0, 0/k22C Cı1=2

0 . (57)


18

71


Similarly, from the first equation of (27), we haveZ t

0

ˇ̌̌ˇ d

dtk�x.t/k

2ˇ̌̌ˇd � Ck.�0, 0/k

22C Cı1=2

0 . (58)

Thus together (29) and (57) with (58), we obtain

k.�x , x/k! 0 as t!1,

which, in combination with the Sobolev inequality, gives

supx2Rj.�x , x/j

2 � C.k�x.t/kk�xx.t/kC k x.t/kk xx.t/k/! 0 as t!1. (59)

On the other hand, by (25), the smallness assumptions on j.pC � p�, qC � q�/j and the initial perturbation imply the smallness of ı0

and k.�0, 0/k2, and the desired (24) follows directly from (59).It is left to show the positivity of p. Owing to (25), (30), and (29),

p.x, t/D Np.x, t/C �x.x, t/�p�

4� C�1 � Cı1=4

0 > 0,

when �1 and ı0 are sufficiently small. �

Proof of Theorem 2.4Because through the transformation (5), the initial boundary value problem (4) can be reduced to a special case of the Cauchy prob-lem (1) and (2) with .p�, q�/ D

�pC,�qC

�, and when qC > 0 is sufficiently small, the Riemann problem (6) and (7) with .p�, q�/ D�

pC,�qC�

has a unique solution consisting of two shock waves with the intermediate state given by .pA, 0/, and Theorem 2.4 followsdirectly from Theorem 2.3. �

4. The case of nonlinear chemical kinetic function

In this section, we consider the following hyperbolic–parabolic system with nonlinear chemical kinetic function:�pt � .pq/x D Dpxx , x 2R, t > 0,qt � f .p/x D 0, x 2R, t > 0

(60)

with initial data

.p, q/.x, 0/D .p0, q0/.x/ �!�

p˙, q˙�

as x!˙1, (61)

where p0.x/ > 0 and p˙ > 0. One can refer to [10] for the derivation of system (60). For simplicity, we assume that the nonlinear kineticfunction f is smooth and satisfies

f .p/� 0, f 0.p/ > 0, f 00.p/� 0 (62)

for all p under considerations. Similar to (8), the Rankine–Hugoniot jump condition takes the form�s�

pC � p��D�pCqCC p�q�,

s.qC � q�/D�f .pC/C f .p�/,

where s are the shock speeds. Canceling qC, we have

s2C sq� � pC �f .pC/� f .p�/

pC � p�D 0. (63)

Clearly, the condition f 0.p/ > 0 ensures that f.pC/�f.p�/pC�p�

> 0. Thus it is easy to calculate that

s1 D�q�

2�

1

2

s.q�/2C 4pC �

f�

pC�� f .p�/

pC � p�< 0< �

q�

2C

1

2

s.q�/2C 4pC �

f�

pC�� f .p�/

pC � p�D s2. (64)

The i-traveling wave .Pi , Qi/.zi/with zi D x � sit and iD 1, 2 satisfies(�siPizi � .PiQi/zi D DPizi zi ,

�siQizi � f .Pi/zi D 0(65)1

87

2



with boundary condition

.P1, Q1/.�1/D .p�, q�/, .P1, Q1/.C1/D .pm, qm/, and

.P2, Q2/.�1/D .pm, qm/, .P2, Q2/.C1/D�

pC, qC�

.

The standard arguments for the ordinary differential equations (see [10, Theorem 2.1] for details) assert the existence of two travelingwaves to (60) satisfying the following.

Lemma 4.1Let (62) hold. Then the 1-traveling wave .P1, Q1/.z1/with z1 D x�s1t, which connects .p�, q�/ and .pm, qm/, satisfies P1z1 > 0, Q1z1 > 0and �

s1DP1z1 D P1f .P1/��

s21C %1

�P1C %2,

s1Q1 D %1 � f .P1/,(66)

where %1 :D s1q� C f .p�/ D s1qm C f .pm/, %2 :D�

s21C s1q�

�p� D

�s2

1C s1qm�

pm; and the 2-traveling wave .P2, Q2/.z2/ withz2 D x � s2t, which connects .pm, qm/ and

�pC, qC

�, satisfies P2z2 < 0, Q2z2 > 0 and�

s2DP2z2 D P2f .P2/��

s22C 1

�P2C 2,

s2Q2 D 1 � f .P2/,

where 1 :D s2qmC f .pm/D s2qCC f�

pC�, 2 :D

�s2

2C s2qm�

pm D�

s22C s2qC

�pC.

The composite wave consisting of the 1-traveling wave and the 2-traveling wave has the form

Qp.x, t/D P1.x � s1t/C P2.x � s2t/� pm, and Qq.x, t/D Q1.x � s1t/CQ2.x � s2t/� qm.

Denote by .Np, Nq/ the composite wave with translations given by (20). Let .�0, 0/ satisfy (22). Similar to (23), we have �0.˙1/ D 0 D 0.˙1/.

Our main result in this section can be stated as follows.

Theorem 4.2Suppose that (62) holds, and that the Riemann solution of the corresponding inviscid system of (60) consists of two shock waves.Assume that the initial data .p0, q0/ satisfy (19), and that .�0, 0/ 2 L2. Then there exist positive constants ı0, �0 and �0, such that ifı < ı0 and

k.p0 � Qp.�, 0/, q0 � Qq.�, 0//kH1\L1 Ck.�0, 0/k � �0,

then the system (60) and (61) has a unique global solution satisfying p.x, t/� �0 for all .x, t/ 2R�RC and

.p� Np, q� Nq/ 2 C�Œ0,C1/; H1

�\ L2

�.0,C1/; H1

�, .px � Npx/ 2 L2

�.0,C1/; H1

�.

Furthermore,

supx2Rj.p� Np, q� Nq/.x, t/j ! 0 as t!1.

Remark 4.1Just for simplicity, we assume that f 00.p/ � 0. By a more careful argument as that used in [10], this condition can be generalized to

f 00.p/ > �maxp2I2f 0.p/

p for all p under considerations (see (2.4) in [10]).

Similar to Theorem 2.4, we also derive the stability of traveling wave for the following initial boundary value problem:8̂̂ˆ̂̂̂<ˆ̂̂̂̂̂:

pt � .pq/x D Dpxx , x > 0, t > 0,

qt � f .p/x D 0, x > 0, t > 0,

q.0, t/D 0, t > 0,

.p, q/.x, 0/D .p0, q0/.x/, x > 0,

limx!C1

.p0, q0/.x/D .pC, qC/.

(67)

More precisely, on the basis of the result of Theorem 4.2, we have the following.

Corollary 4.3Assume that (62) holds, and that qC > 0. Suppose that

�p0 � pC, q0 � qC

�2 H1

�RC

�\ L1

�RC

�and .�0, 0/ 2 L2

�RC

�. Then there

exist positive constants ı0, �0 and �0, such that if 0< qC < ı0 and

k�

p0 � pC, q0 � qC�kH1.RC/\L1.RC/Ck.�0, 0/kL2.RC/ � �0,


18

73


then the initial boundary value problem (67) has a unique global solution satisfying p.x, t/� �0 for all .x, t/ 2RC �RC and

.p� Np, q� Nq/ 2 C�Œ0,C1/; H1

�RC

��\ L2

�.0,C1/; H1

�RC

��, .px � Npx/ 2 L2

�.0,C1/; H1

�RC

��.

Furthermore,

supx2RC

j.p� P2.x � s2tC ˛2/, q�Q2.x � s2tC ˛2//j ! 0 as t!1.

The proof of Theorem 4.2 is similar to that of Theorem 2.3. We first introduce the following two unknown functions �.x, t/ and .x, t/:

�.x, t/ :D�

Z C1x

Œp.y, t/� Np.y, t/�dy and .x, t/ :D�

Z C1x

Œq.y, t/� Nq.y, t/�dy,

and we seek a solution of the form

.p, q/.x, t/D .Np, Nq/.x, t/C .�x , x/.x, t/.

Similar to (27), one can see that .�, / satisfies

��t D D�xx C �x x C Np x C Nq�x C E, t D f 0.Np/�x C F.P1, P2,�x/

(68)

with initial data (22) and F.P1, P2,�x/D f .NpC�x/� f .P1/� f .P2/C f .pm/� f 0.Np/�x . By an argument similar to that used in Theorem 2.3,to finish the proof of Theorem 4.2, we only establish the following a priori estimates.

Lemma 4.4There exists a positive constant �0, such that if N.T/� �0, then the solution .�, / of (68) satisfies

k.�, /k22C

Z t

0

�k�xk

22Ck xk

21

�d � Ck.�0, 0/k

22C Cı1=2

0 (69)

for any t 2 Œ0, T�.

ProofStep 1: L2 estimate.

The proof of this claim is a slight modification of that of Lemma 3.3. Multiplying .68/1 by �=Np and .68/2 by =f 0.Np/, integrating theresultant equations on R, and adding them, similar to (33), we have

1

2

d

dt

Z ��2

NpC

2

f 0.Np/

�dxCD

Z�2

x

Npdx D

1

2

Z�2�

1

Np

�tC

�D

Np

�xx�

�Nq

Np

�x

dxC

Z x�x

�

NpdxC

ZE�

Npdx

C1

2

Z 2

�1

f 0.Np/

�t

dxC

ZF

f 0.Np/dx.

(70)

By (62), Lemma 4.1, and Taylor’s expansion, similar to (34) and (35), it is easy to calculate that

jQ1 � qmj D O.1/jP1 � pmj �

�Cıe�Cı.jxjCt/ for x > 0,Cı for x � 0,

and

jQ2 � qmj D O.1/jP2 � pmj �

�Cı for x > 0,Cıe�Cı.jxjCt/ for x � 0.

Thus inequality (37) still holds, that is, ˇ̌̌ˇZ

E ��

Npdx

ˇ̌̌ˇ� Cı3=2e�Cıtk�k. (71)

In view of (62) and Lemma 4.1, �1

f 0.Np/

�tD�f 00.Np/Npt

Œf 0.Np/�2D

f 00.Np/.P1z1 s1C P2z2 s2/

Œf 0.Np/�2� 0. (72)1

87

4



To estimate the first term on the right hand side of (70), we only need to observe that (41) is replaced by

2

Np2

"s1P01C P01Q1C

D.P01/2

Np

#�

2

Np3

hs1P01P1C P01Q1P1CD.P01/

2iC Cı2e�Cı.jxjCt/

D2P01s1 Np3

hs2

1P1C P1.%1 � f .P1//C s1DP01

iC Cı2e�Cı.jxjCt/

D2P01s1 Np3

hs2

1C s1q�i

p�C Cı2e�Cı.jxjCt/

D2P01s1 Np3

� pC �f�

pC�� f .p�/

pC � p�� p�C Cı2e�Cı.jxjCt/,

where we have used (66) in the first and second equalities and (63) in the last equality. The remaining calculations will be the same asin the proof of Lemma 3.3. Thus we have �

1

Np

�tC

�D

Np

�xx�

�Nq

Np

�x� Cı2e�Cı.jxjCt/. (73)

Consequently, by (70)–(73), Cauchy’s inequality and choosing N.T/ suitably small, we have

k.�, /k2C

Z t

0

Z�2

x dxd � k.�0, 0/k2C CN.T/

Z t

0

Zj xj

2dxd C Cı1=20 C C

Z t

0

ZjF jdxd . (74)

Step 2: H1 estimates.We slightly modify the proofs of Lemmas 3.4 and 3.5. We differentiate .68/2 with respect to x, multiply the resultant equation by

D=f 0.Np/, and use .68/1 to obtain

D tx

f 0.Np/D

Df 00.Np/Npx

f 0.Np/�x C

DFx

f 0.Np/C �t � �x x � Np x � Nq�x � E. (75)

Multiplying this equation by x , integrating the resultant equation on R�RCt , observing that f is smooth and

D tx

f 0.Np/� x D

�D 2

x

2f 0.Np/

�t�

D 2x

2

�1

f 0.Np/

�t

,

�t x D .� x/t � � xt D .� x/t � �.f0.Np/�x C F/x D .� x/t � .�f 0.Np/�x C �F/x C f 0.Np/�2

x C F�x ,

and using Cauchy’s inequality, (28), (36), (72), (74), and a similar argument as that used in (47), we obtain

k xk2C

Z t

0k xk

2d � Ck 0xk2C Ck.�0, 0/k

2C Cı1=20 C C

Z t

0

ZŒjFj .j j C j�xj/C jFx xj�dxd . (76)

(76) in combination with (74) gives

k.�, /k2C

Z t

0k�xk

2d � Ck 0xk2C Ck.�0, 0/k

2C Cı1=20 C C

Z t

0


On the other hand, by the same argument as that used in Lemma 3.5, it is easy to see that

k�xk2C

Z t

0k�xxk

2d � Ck.�0, 0/k21C N.T/

Z t

0

Z 2

xxdxd C Cı1=20 C C

Z t

0


Step 3: H2 estimates.Similar to the proof of Lemma 3.6, we differentiate (75) with respect to x to obtain

D txx

f 0.Np/D�D tx

�1

f 0.Np/

�xC

�Df 00.Np/Npx�x

f 0.Np/

�xC

�DFx

f 0.Np/

�xC �tx � .�x x/x � .Np x/x � .Nq�x/x � Ex .

Multiplying this equation by xx , integrating the resultant equation on R�RCt , observing that

D txx

f 0.Np/� xx D

�D 2

xx

2f 0.Np/

�t�

D 2xx

2

�1

f 0.Np/

�t

,

D tx

�1

f 0.Np/

�x xx D D.f 0.Np/�x C F/x

�1

f 0.Np/

�x xx ,�

DFx

f 0.Np/

�x xx D DFx

�1

f 0.Np/

�x xx C

DFxx xx

f 0.Np/,


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75


�tx xx D .�x xx/t � �x txx D .�x xx/t � �x.f0.Np/�x C F/xx

D .�x xx/t � .�x.f0.Np/�x C F/x/x C �xx.f

0.Np/�x C F/x

D .�x xx/t � .�x.f0.Np/�x C F/x/x C �

2xx C f 00.Np/Npx�x�xx C Fx�xx ,

and that (53) still holds, and using an argument similar to that used in (54), we obtain

Z 2

xxdxC

Z t

0

Z 2

xxdxd �

Z 2

0xxdxC Ck.�0, 0/k21C Cı1=2

0 C C

Z t

0

ZŒjFj .j j C j�xj/�dxd

C C

Z t

0

ZŒjFxj.j xj C j xxj C j�xxj/C jFxx xxj�dxd ,

(79)

after choosing N.T/ suitably small. This inequality, in combination with (78), leads to

k�xk2C

Z t

0k�xxk

2d �Ck.�0, 0/k21C Ck 0xxk

2C Cı1=20 C C

Z t

0


C C

Z t

0

Z �jFxj.j xj C j xxj C j�xxj/C jFxx xxj

�dxd .

(80)

By the same argument as that used in the proof of Lemma 3.7, one can easily see that

k�xxk2C

Z t

0k�xxxk

2d �k.�0, 0/k22C Cı1=2

0 C C

Z t

0


C C

Z t

0

ZŒjFxj.j xj C j xxj C j�xxj/C jFxx xxj�dxd .

This inequality together with (76), (77), (79), and (80) gives

k.�, /k22C

Z t

0

�k�xk

22Ck xk

21

�d �k.�0, 0/k

22C Cı1=2

0 C C

Z t

0


C C

Z t

0

ZŒjFxj.j xj C j xxj C j�xxj/C jFxx xxj�dxd .

Note that by Taylor’s expansion,

F D f .NpC �x/� f .P1/� f .P2/C f .pm/� f 0.Np/�x

D f .NpC �x/� f .Np/� f 0.Np/�x C Œf .Np/� f .P1/�� Œf .P2/� f .pm/�

D

Z 1

0.1� s/f 00.NpC s�x/ds�2

x C

Z 1

0f 0.P1C s.P2 � pm//ds.P2 � pm/�

Z 1

0f 0.pmC s.P2 � pm//ds.P2 � pm/

D

Z 1

0.1� s/f 00.NpC s�x/ds�2

x C

Z 1

0f 00.pmC .P1 � pm/C s.P2 � pm//dds.P1 � pm/.P2 � pm/

D O.1/�2x CO.1/.P1 � pm/.P2 � pm/.

(81)

Similar to (71), we then obtain

Z t

0

ZŒjFj .j j C j�xj/�dxd � CN.T/

Z t

0

Zj�xj

2dxC CN.T/ı1=2.

Similarly,

Z t

0

ZŒjFxj.j xj C j xxj C j�xxj/C jFxx xxj�dxd � CN.T/

Z t

0k�xk

2Ck�xxk2Ck xxk

2Ck�xxxk2/d C Cı1=2.

Therefore, the desired (69) holds. We complete the proof now. �

Acknowledgements

The authors would like to thank the referee for the important suggestions on improvement of the paper. Li’s work is partially supportedby the National Natural Science Foundation of China (no. 11101073), the Fundamental Research Funds for the Central Universities(no. 10QNJJ001), and the China Postdoctoral Science Foundation funded project. Zhang’s work is partially supported by the ChineseNSF (no. 11071034) and the Fundamental Research Funds for the Central Universities (no. 111065201).

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Asymptotic stability of a composite wave of two traveling waves to a hyperbolic-parabolic system modeling chemotaxis