convergence rates of solutions for a two-species … · 2018. 8. 28. · the lotka-volterra...

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018249DYNAMICAL SYSTEMS SERIES B

CONVERGENCE RATES OF SOLUTIONS FOR A TWO-SPECIES

CHEMOTAXIS-NAVIER-STOKES SYTSTEM WITH

COMPETITIVE KINETICS

Hai-Yang Jin

Department of Mathematics, South China University of TechnologyGuangzhou 510640, China

Tian Xiang⇤

Institute for Mathematical Sciences, Renmin University of ChinaBeijing 100872, China

(Communicated by Michael Winkler)

Abstract. We study the convergence rates of solutions to the two-specieschemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics:

8>>>>>>>>>>><

>>>>>>>>>>>:

(n1

)t + u ·rn

1

= �n

1

� �

1

r · (n1

rc) + µ

1

n

1

(1� n

1

� a

1

n

2

),

x 2 ⌦, t > 0,

(n2

)t + u ·rn

2

= �n

2

� �

2

r · (n2

rc) + µ

2

n

2

(1� a

2

n

1

� n

2

),

x 2 ⌦, t > 0,

ct + u ·rc = �c� (↵n1

+ �n

2

)c, x 2 ⌦, t > 0,

ut + (u ·r)u = �u+rP + (�n1

+ �n

2

)r�, r · u = 0,

x 2 ⌦, t > 0

under homogeneous Neumann boundary conditions for n

1

, n

2

, c and no-slipboundary condition for u in a bounded domain ⌦ ⇢ Rd(d 2 {2, 3}) with smoothboundary. The global existence, boundedness and stabilization of solutions

have been obtained in 2-D [8] and 3-D for = 0 and max{�1,�2}min{µ1,µ2}

kc0

kL1(⌦)

being su�ciently small [4]. Here, we examine further convergence and derivethe explicit rates of convergence for any supposedly given global bounded clas-sical solution (n

1

, n

2

, c, u); more specifically, in L

1-topology, we show that

(n1

(·, t), n2

(·, t), u(·, t)) t!1!

8>>>>>>>>>>><

>>>>>>>>>>>:

( 1�a11�a1a2

,

1�a21�a1a2

, 0) exponentially,

if a1

, a

2

2 (0, 1),

(0, 1, 0) exponentially, if a1

> 1 > a

2

,

(0, 1, 0) algebraically, if a1

= 1 > a

2

,

(1, , 0, 0) exponentially, if a2

> 1 > a

1

,

(1, 0, 0) algebraically, if a2

= 1 > a

1

.

In either cases, the c-solution component converges exponentially to 0.Moreover, it is shown that only the rate of convergence for u is expressed

2010 Mathematics Subject Classification. Primary: 35B40, 35K55, 35B44, 35K57; Secondary:35Q92, 92C17.

Key words and phrases. Chemotaxis-fluid system, boundedness, exponential convergence, al-gebraic convergence, convergence rates.

⇤ Corresponding author.

1

2 HAI-YANG JIN AND TIAN XIANG

in terms of the model parameters and the first eigenvalue of �� in ⌦ underhomogeneous Dirichlet boundary conditions, and all other rates of convergenceare explicitly expressed only in terms of the model parameters ai, µi,↵ and �

and the space dimension d.

1. Introduction. We consider the following two-species chemotaxis-fluid systemwith competitive terms:8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

(n1)t + u ·rn1 = �n1 � �1r · (n1rc) + µ1n1(1� n1 � a1n2), x 2 ⌦, t > 0,

(n2)t + u ·rn2 = �n2 � �2r · (n2rc) + µ2n2(1� a2n1 � n2), x 2 ⌦, t > 0,

ct + u ·rc = �c� (↵n1 + �n2)c, x 2 ⌦, t > 0,

ut + (u ·r)u = �u+rP + (�n1 + �n2)r�, r · u = 0, x 2 ⌦, t > 0,

@⌫n1 = @⌫n2 = @⌫c = 0, u = 0, x 2 @⌦, t > 0,

ni(x, 0) = ni,0(x), c(x, 0) = c0(x), u(x, 0) = u0(x), x 2 ⌦, i = 1, 2,(1.1)

where ⌦ ⇢ Rd (throughout the whole paper, d 2 {2, 3}) is a bounded domain withsmooth boundary @⌦ and @⌫ denotes di↵erentiation with respect to the outwardnormal of @⌦; 2 R, �1,�2, a1, a2 � 0 and µ1, µ2,↵,�, �, � > 0 are constants;n1,0, n2,0, c0, u0,� are known functions satisfying

0 < n1,0, n2,0 2 C(⌦), 0 < c0 2 W 1,q(⌦), u0 2 D(A#), (1.2)

� 2 C1+⌘(⌦) (1.3)

for some q > d, # 2�

34 , 1

�

, ⌘ > 0 and A is the Stokes operator.The system (1.1), an extension of the chemotaxis-fluid system introduced by

Tuval et al. [28], depicts the evolution of two competing species which react ona single chemoattractant in a liquid surrounding environment. Here, n1 and n2

denote densities of species, c means the chemical concentration, and finally, u andP represent the fluid velocity field and its associated pressure. So, it is the mixedcombination of the complex interaction between chemotaxis, the Lotka-Volterrakinetics and fluid.

Even in the absence of chemotaxis and fluid in (1.1), the reduced Lotka-Volterracompetition system has been extensively studied. It is now well-known that itspositive equilibrium (N1, N2) with

N1 :=1� a11� a1a2

, N2 :=1� a21� a1a2

(1.4)

is globally asymptotically stable in weak competition case a1 < 1 < 1a2

[5, 6]. while,

in the strong competition case a1 > 1 > 1a2, the positive equilibrium (N1, N2) is

unstable and the system admits non-constant steady states when the system pa-rameters and the domain geometry are properly balanced [11, 15, 16]. In the onecomponent chemotaxis-only context (n2 ⌘ 0 ⌘ u), the global existence, bound-edness and stabilization of solutions to (1.1) have also been studied widely, cf.[23, 24, 10, 14]; see also [30, 32, 34, 36] for classical Keller-Segel models with signalproduction instead of consumption, i.e., �↵n1c replaced with �c+↵n1 in the thirdequation.

In one-species (n2 ⌘ 0), global existence of weak (and/or eventual smoothness ofweak solutions) and classical solutions and asymptotic behavior have been investi-gated, e.g., in [31, 33, 35] without logistic source (µ1 = 0) and also the convergencerate has been explored [37] and in [13, 25, 27] with logistic source.

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 3

In two-species context, related studies first begin with fluid-free systems with sig-nal production (in which the asymptotic stability usually depends on some smallnesscondition on the chemo-sensitivities) to understand the influence of chemotaxis andthe Lotka-Volterra kinetics [1, 2, 18, 17, 20, 19, 22]. For the two-species chemotaxis-fluid system with competitive terms (1.1), the global existence, boundedness of clas-sical solutions and stabilization to equilibria were very recently studied by Hirataet al. [8] in the 2-D setting and by Cao et al. [4] in the 3-D setting for = 0. For = 1, the global existence of weak solutions to the above system, and their eventualsmoothness and stabilization were studied in [9]. We now state the bounded andclassical solutions established in [8, 4] as follows:

(B2) (Boundedness in 2-D [8]) In the case that ⌦ ⇢ R2 is a bounded domainwith smooth boundary, let �1,�2, a1, a2 � 0, µ1, µ2,↵,�, �, � > 0 and let (1.2)and (1.3) hold. The the IBVP (1.1) with = 1 possesses a unique classicalsolution (n1, n2, c, u, P ), up to addition of spatially constant functions to P ,such that

n1, n2 2 C(⌦⇥ [0,1)) \ C2,1(⌦⇥ (0,1)),

c 2 C(⌦⇥ [0,1)) \ C2,1(⌦⇥ (0,1)) \ L1loc([0,1);W 1,q(⌦)),

u 2 C(⌦⇥ [0,1)) \ C2,1(⌦⇥ (0,1)) \ L1loc([0,1);D(A#)),

P 2 C1,0(⌦⇥ (0,1)).

Moreover, there exists a constant C > 0 such that for all t > 0

kn1(·, t)kL1(⌦) + kn2(·, t)kL1(⌦) + kc(·, t)kW 1,q(⌦) + ku(·, t)kL1(⌦) C. (1.5)

(B3) (Boundedness in 3-D [4]) In the case that ⌦ ⇢ R3 is a bounded domainwith smooth boundary and that = 0, there exists a constant ⇠0 > 0 such thatwhenever max{�1,�2}kc0kL1(⌦) < ⇠0 min{µ1, µ2}, the statements of bound-edness in (B2) hold.

(UC) (Uniform Convergence [8, 4]) Let (n1, n2, c, u, P ) be the solution of (1.1)obtained from (B2) or (B3). Then it fulfills the following convergence prop-erties:(i) Assume that a1, a2 2 (0, 1). Then, with (N1, N2) defined in (1.4),

n1(·, t) ! N1, n2(·, t) ! N2, c(·, t) ! 0, u(·, t) ! 0 in L1(⌦) as t ! 1,

(ii) Assume that a1 � 1 > a2. Then

n1(·, t) ! 0, n2(·, t) ! 1, c(·, t) ! 0, u(·, t) ! 0 in L1(⌦) as t ! 1.

In this paper, we study dynamical properties for any supposedly global-in-timeand bounded classical solution to (1.1), with particular focus on the model of con-vergence as well as their explicit rates of convergence. Before proceeding to ourmain results, let us observe that, the n1- and n2-equations in (1.1) are symmetric.Thus, we should naturally have a result about stabilization to (0, 1, 0, 0). This wasnot mentioned in related works, cf. [1, 8, 17]. Now, let �P denote the Poincareconstant, cf. (4.19), and, finally, let

⇠ =1

2min

8

<

:

(1� a1a2)µ1 minn

12 ,

a1(1+a1a2)a2

o

2max{ 1N1

, a1µ1

a2µ2N2}

, (↵N1 + �N2)

9

=

;

, (1.6)


⌫ =1

4min

n

(1� a2)µ2 minn1

2,

a21 + a2

o

, 2�, (a1 � 1)µ1

o

, (1.7)

µ =1

4min{(1� a1)µ1 min

n1

2,

a11 + a1

o

, 2↵, (a2 � 1)µ2}. (1.8)

Then we are at the position to state our main results on exponential and algebraicconvergence of global-in-time bounded solutions to model (1.1).

Theorem 1.1. Let ⌦ ⇢ Rd(d 2 {2, 3}) be a bounded and smooth domain, = 0if d = 3, and let (1.2) and (1.3) be in force, finally, let (n1, n2, c, u, P ) be a givenglobal- and bounded-in time classical solution of (1.1). Then it enjoys the followingdecay properties.

(I) When a1, a2 2 (0, 1), (n1, n2, u) converges exponentially to (N1, N2, 0):8

>

>

>

>

<

>

>

>

>

:

kn1(·, t)�N1kL1(⌦) m1e� ⇠

d+2 t, 8t � 0,

kn2(·, t)�N2kL1(⌦) m2e� ⇠

d+2 t, 8t � 0,

ku(·, t)kL1(⌦) m3e� ✏

2(d+2) min{�P ,⇠}t, 8t � 0.

(1.9)

(II) When a1 = 1 > a2, (n1, n2, u) converge algebraically to (0, 1, 0):8

>

>

>

<

>

>

>

:

kn1(·, t)kL1(⌦) m4(t+ 1)�1

d+1 , 8t � 0,

kn2(·, t)� 1kL1(⌦) m5(t+ 1)�1

d+2 , 8t � 0,

ku(·, t)kL1(⌦) m6(t+ 1)�✏

d+2 , 8t � 0.

(1.10)

(III) When a1 > 1 > a2, (n1, n2, u) converges exponentially to (0, 1, 0):8

>

>

>

>

<

>

>

>

>

:

kn1(·, t)kL1(⌦) m7e� (a1�1)µ1

2(d+1) t, 8t � 0,

kn2(·, t)� 1kL1(⌦) m8e� ⌫

d+2 t, 8t � 0,

ku(·, t)kL1(⌦) m9e� ✏

2(d+2) min{�P ,⌫}, 8t � 0.

(1.11)

(II0) When a2 = 1 > a1, (n1, n2, u) converge algebraically to (1, 0, 0):8

>

>

>

<

>

>

>

:

kn1(·, t)� 1kL1(⌦) m10(t+ 1)�1

d+2 , 8t � 0,

kn2(·, t)kL1(⌦) m11(t+ 1)�1

d+1 , 8t � 0,

ku(·, t)kL1(⌦) m12(t+ 1)�✏

d+2 , 8t � 0.

(1.12)

(III0) When a2 > 1 > a1, (n1, n2, u) converge exponentially to (1, 0, 0):8

>

>

>

>

<

>

>

>

>

:

kn1(·, t)� 1kL1(⌦) m13e� µ

d+2 t, 8t � 0,

kn2(·, t)kL1(⌦) m14e� (a2�1)µ2

2(d+1) t, 8t � 0,

ku(·, t)kL1(⌦) m15e� ✏

2(d+2) min{�P ,µ}, 8t � 0.

(1.13)

(IV) In either cases, the c-solution component converges exponentially to 0:

kc(·, t)kL1(⌦) m16e� (↵N1+�N2)

2 t, 8t � 0, (1.14)

where (N1, N2) = (N1, N2) in Case (I), (N1, N2) = (0, 1) in cases (II) and

(II0), and (N1, N2) = (1, 0) in cases (III) and (III0).


Here, ✏ 2 (0, 1) is arbitrarily given, only m3,m6,m9 and m15 depend on ✏; allmi(i = 1, 2, 3, · · · , 16) are suitably large constants depending on the initial datan1,0, n2,0, c0, u0,�, the model parameters and Sobolev embedding constants but noton time t, see Section 4. Moreover, we have the following lower estimates:

mi = O(1), i = 1, 2, 7, 14, m3 � O(1)(1 + (1� a1a2)� ✏

d+2 ),

m4 � O(1)⇣

1 + (1� a2)� 1

d+1

⌘

, m5 � O(1)⇣

1 + (1� a2)� 1

d+2

⌘

,

m6 � O(1)⇣

1 + (1� a2)� ✏

d+2

⌘

, m8 � O(1)(1 + (a1 � 1)�1

d+2 + (1� a2)� 1

d+2 ),

m9 � O(1)(1 + (a1 � 1)�2✏

d+2 + (1� a2)� 2✏

d+2 ), m10 � O(1)⇣

1 + (1� a1)� 1

d+2

⌘

,

and

m11 � O(1)⇣

1 + (1� a1)� 1

d+1

⌘

, m13 � O(1)(1 + (1� a1)� 1

d+2 + (a2 � 1)�1

d+2 ),

as well as

m12 � O(1)⇣

1 + (1� a1)� ✏

d+2

⌘

, m15 � O(1)(1 + (a1 � 1)�2✏

d+2 + (1� a2)� 2✏

d+2 ).

From these estimates, we see the conditions that 1 � a1a2 > 0, 1 � a2 > 0,1� a1 > 0 and a1 > 1 > a2 etc are very crucial in their respective cases.

With certain regularity and dissipation properties of global bounded solutions,the idea for the proof of convergence is quite known and developed, cf. [1, 4, 8, 17, 25,26, 27, 33] for example. The strategy for obtaining the explicit rates of convergenceas stated in Theorem 1.1 consists mainly of four steps. In the first step, we presentstronger regularity properties, e.g., W 1,1-regularity for ni, W 1,p-regularity for uwith any finite p, and W 2,1-regularity for c, for any bounded solution of (1.1) thanthose shown in [8, 4]; these are done in Section 2. In the crucial second step donein Section 3, we use refined computations to make those widely known Lyapunovfunctionals (cf. eg. [1, 4, 8, 17]) explicit, which will enable us to derive the explicitrates of convergence. Armed with the information provided by Step two, we thenmove on to calculate precisely the rates of convergence in L1-and L2-norm for theconsidered bounded solution, and related necessary estimates are also studied ingreat details. These constitute our Step three and are conducted in Section 4.Finally, thanks to the improved regularities, we apply the well-known Gagliardo-Nirenberg interpolation inequality to pass the obtained L1- and L2-convergence tothe L1-convergence; these are our Step four and are also conducted in Section 4.

2. Regularity properties of bounded solutions. Let (n1, n2, c, u, P ) be a sup-posedly given global-in-time and bounded classical solution to (1.1) in the senseof (1.5). In this section, we provide more strong regularity properties for any suchbounded solution than those shown in [8, 4], which are needed to achieve our desiredrates of convergence in L1-norm. We start with the regularity of u and c.

Lemma 2.1. Let ⌦ ⇢ Rd be a bounded and smooth domain and = 0 if d = 3.For d < p < 1, there exists a constant C > 0 such that

ku(·, t)kW 1,p C, 8t > 1 (2.1)

and

kc(·, t)kW 1,1 + k�c(·, t)kL1 C, 8t > 1. (2.2)


Proof. Using the essentially same argument as in [33, Lemma 6.3], we obtain (2.1)for d = 2. When d = 3, we can obtain (2.1) for k = 0 by using the Stokes semigroupand noting the Lp-boundedness of �n1 + �n2r� for p > 3. The W 1,1-boundednessof c can be seen in [8, Lemma 3.9] and [27, Lemma 3.12]. With these and theL1-boundedness of n1, n2 and u, an direct application of the standard parabolicschauder theory (cf. [12, 21]) to the third equation in (1.1) yields (2.2).

With the regularity properties in Lemma 2.1 at hand, we now utilize the quitecommonly used arguments (cf. [26, 27]) to show the following W 1,1-regularity ofn1 and n2. To this end, we first establish the following L2-boundedness of thegradients of n1 and n2.

Lemma 2.2. Let the conditions be the same as Lemma 2.1. Then there exists aconstant C > 0 such that

krn1(·, t)kL2 + krn2(·, t)kL2 C, 8t > 1. (2.3)

Proof. Multiplying the first equation of system (1.1) by ��n1, integrating by partsand using the boundedness of n1, n2, u and (2.2), we get

1

2

d

dt

Z

⌦|rn1|2 +

Z

⌦|�n1|2

=

Z

⌦u ·rn1�n1 + �1

Z

⌦r · (n1rc)�n1 � µ1

Z

⌦n1(1� n1 � a1n2)�n1

1

2

Z

⌦|�n1|2 + c1

Z

⌦|rn1|2 + c2,

which yieldsd

dt

Z

⌦|rn1|2 +

Z

⌦|�n1|2 2c1

Z

⌦|rn1|2 + 2c2. (2.4)

Here and below, unless otherwise stated, ci (numbered within lemmas) or C denotesome generic positive constants which may vary from line to line.

On the other hand, thanks to the boundedness of kn1kL2 and the H2-ellipticestimate, we apply the Gagliardo-Nirenberg inequality in ⌦ ⇢ Rd to deduce

2c1krn1k2L2 c3kD2n1kL2kn1kL2 + c3kn1k2L2

c4(k�n1kL2 + kn1kL2)kn1kL2 + c3kn1k2L2 2

3k�n1k2L2 + c5.

(2.5)

Substituting (2.5) into (2.4), one has

d

dtkrn1k2L2 + c1krn1k2L2 2c2 +

3

2c5,

which trivially gives

krn1(·, t)k2L2 krn1(·, 1)k2L2e�c1(t�1) +2c2 +

32c5

c1 c6, 8t > 1.

Applying the similar arguments to the n2-equation in (1.1), one can readily obtainthat krn2(·, t)k2L2 c6 for all t > 1. We thus have shown the L2-estimate (2.3).

Lemma 2.3. There exists a constant C > 0 such that

kn1(·, t)kW 1,1 + kn2(·, t)kW 1,1 C, 8t > 1. (2.6)


Proof. First, we show there exists a constant c1 > 0 such that

kn1(·, t)kW 1,1 c1, 8t > 1. (2.7)

To this end, for any T > 2, we let

M(T ) := supt2(2,T )

krn1(·, t)kL1 .

Since clearlyrn1 is continuous on ⌦⇥[0, T ], it follows thatM(T ) is finite. Moreover,since by our universal assumption n1 is bounded in L1(⌦⇥ (0,1)), to prove (2.7),it is su�cient to derive the existence of c2 > 0 satisfying

M(T ) c2, 8T > 2. (2.8)

To achieve (2.8), for any given t 2 (2, T ), using the variation-of-constants formulato the first equation in (1.1), we get

n1(·, t) =e�n1(·, t� 1)� �

Z t

t�1e(t�s)�r · (n1(·, s)rc(·, s))ds

�Z t

t�1e(t�s)�u(·, s) ·rn1(·, s)ds

+ µ1

Z t

t�1e(t�s)�n1(·, s)(1� n1(·, s)� a1n2(·, s))ds,

which implies

krn1(·, t)kL1 kre�n1(·, t� 1)kL1 + �

Z t

t�1kre(t�s)�r · (n1(·, s)rc(·, s))kL1ds

+

Z t

t�1kre(t�s)�u(·, s) ·rn1(·, s)kL1ds

+ µ1

Z t

t�1kre(t�s)�n1(·, s)(1� n1(·, s)� a1n2(·, s))kL1ds

=I1 + I2 + I3 + I4.(2.9)

Next, we shall employ the widely known smoothing Lp-Lq type estimates of theNeumann heat semigroup {et�}t�0 in ⌦ (see [29, 3, 7] for instance) to estimateIi, i = 1, 2, 3, 4.

Thanks to the boundedness of n1, n2, u in ⌦⇥ (1,1), (2.1) and (2.2), we employthose smoothing Neumann heat semigroup estimates to obtain that

I1 = kre�n1(·, t� 1)kL1 c3kn1(·, t� 1)kL1 c4 (2.10)

and that

I2 = �

Z t

t�1kre(t�s)�r · (n1(·, s)rc(·, s))kL1ds

c5

Z t

t�1

h

1 + (t� s)�12�

d2p

i

e��1(t�s)kr · (n1(·, s)rc(·, s))kLpds

c5

Z t

t�1

h

1 + (t� s)�12�

d2p

i

e��1(t�s)krn1(·, s) ·rc(·, s)kLpds

+ c5

Z t

t�1

h

1 + (t� s)�12�

d2p

i

e��1(t�s)kn1(·, s)�c(·, s)kLpds


c6

Z t

t�1

h

1 + (t� s)�12�

d2p

i

e��1(t�s)krn1(·, s)kLpds+ c7, (2.11)

where �1(> 0) is the first nonzero eigenvalue of �� under homogeneous boundarycondition and we have used the choice of p > d to ensure the finiteness of theGamma integral. Similarly, we can estimate I3 as follows:

I3 =

Z t

t�1kre(t�s)�u(·, s) ·rn1(·, s)kL1ds

c8

Z t

t�1

h

1 + (t� s)�12�

d2p

i

e��1(t�s)krn1(·, s)kLpds+ c9.

(2.12)

At last, using the boundedness of n1 and n2 again, one has

I4 =µ1

Z t

t�1kre(t�s)�n1(·, s)(1� n1(·, s)� a1n2(·, s))kL1ds

c10

Z t

t�1

h

1 + (t� s)�12

i

e��1(t�s)ds

c10

Z 1

0(1 + ⌧�

12 )e��1⌧ds (

1

�1+ 2)c10.

(2.13)

Substituting (2.10), (2.11), (2.12) and (2.13) into (2.9), we infer that

krn1(·, t)kL1 c11

Z t

t�1

h

1 + (t� s)�12�

d2p

i

e��1(t�s)krn1(·, s)kLpds+ c12. (2.14)

Then, by the L2-boundedness of rn1(·, t) and rn2(·, t) in (2.3), the smoothness andhence boundedness of rn1 on ⌦⇥ [1, 2] and the definition of M(T ), we estimate

krn1(·, s)kLp krn1(·, s)k✓L1krn1(·, s)k1�✓L2

c13�

krn1k✓L1(⌦⇥[1,2])+M✓(T )

�

c14(1 +M✓(T )�

, 8s 2 (1, T ),

(2.15)

where ✓ = p�2p 2 (0, 1) due to p > 2.

Finally, since 12 + d

2p < 1, then a substitution of (2.15) into (2.14) entails

M(T ) c15M✓(T ) + c16, 8T > 2,

which upon a use of elementary inequality gives

M(T ) max{2c16, (2c15)1

1�✓ } = max{2c16, (2c15)p2 }, 8T > 2,

and hence (2.7) follows.The argument done for n1 can also be similarly applied to n2 to find that

kn2(·, t)kW 1,1 c17, 8t > 1.

This along with (2.7) yields simply (2.6), finishing the proof of the lemma.

3. Existence of explicit Lyapunov functionals. From boundedness to con-vergence, besides enough information on regularity, we still need some decayingestimates of bounded solutions under investigation. For the latter, the availabilityof a Lyapunov functional is crucial, see [1, 4, 8, 17] for instance. In this section,for our purpose, we particularize those known Lyapunov functionals used in thosepapers to obtain the explicit rates of convergence as stated in Theorem 1.1. Let us


start with the case of a1, a2 2 (0, 1). In this case, the explicit Lyapunov functionalthat we obtain for the chemotaxis-fluid system (1.1) reads as follows:

Lemma 3.1. Define

E1 :=

Z

⌦

✓

n1 �N1 �N1 logn1

N1

◆

+a1µ1

a2µ2

Z

⌦

✓


N2

◆

+1

2

✓

N1�21

4+

a1µ1N2�22

4a2µ2+ 1

◆

Z

⌦c2

(3.1)

and

F1 :=

Z

⌦(n1 �N1)

2 +

Z

⌦(n2 �N2)

2.

Then, in the case of a1, a2 2 (0, 1), the nonnegative functions E1 and F1 satisfy

d

dtE1(t) �(1� a1a2)µ1 min

n1

2,

a1(1 + a1a2)a2

o

F1(t) =: �⌧F1(t), 8t > 0. (3.2)

Proof. By honest di↵erentiation of E1 in (3.1) and using elementary Cauchy-Schwarzinequality, one can easily derive the dissipation estimate (3.2); or alternatively, inthe proof of [8, Lemma 4.1], by taking

k1 =a1µ1

a2µ2, l1 = (

N1�21

4+

a1µ1N2�22

4a2µ2+ 1), ✏ =

µ1

2(1� a1a2),

upon honest calculations, one can easily arrive at (3.2).

For the purpose of deriving our explicit Lyapunov functionals in Cases (II)-(IV),we wish to perform honest computations here. We illustrate it for the case thata1 � 1 > a2. In this case, from (1.1), the fact that a1 � 1 and the positivity ofn1, n2, we calculate that

d

dt

Z

⌦n1 = µ1

Z

⌦(1� n1 � a1n2)n1 �µ1

Z

⌦n21 � µ1

Z

⌦n1(n2 � 1),

d

dt

Z

⌦(n2 � 1� log n2) =�

Z

⌦

|rn2|2

n22

+ �2

Z

⌦

rn2

n2·rc

� µ2

Z

⌦(n2 � 1)2 � a2µ2

Z

⌦n1(n2 � 1)

(3.3)

as well as1

2

d

dt

Z

⌦c2 = �

Z

⌦|rc|2 �

Z

⌦(↵n1 + �n2)c

2.

Therefore, for any positive constants �1,�2 and ⌘ 2 (0, 1), in view of the positivityof ni,↵ and �, a clear linear combination of the three estimates above shows

� d

dt

Z

⌦

h

n1 + �1 (n2 � 1� log n2) +�22c2i

�Z

⌦

⇥

µ1n21 + (µ1 + a2µ2�1)n1(n2 � 1) + µ2�1(n2 � 1)2

⇤

+

Z

⌦

h

�1|rn2|2

n22

� �2�1rn2

n2·rc+ �2|rc|2

i

=

Z

⌦

⇢

hpµ2�1⌘(n2 � 1) +

(µ1 + a2µ2�1)

2pµ2�1⌘

n1

i2+h

µ1 �(µ1 + a2µ2�1)2

4µ2�1⌘

i

n21

�


+ µ2�1(1� ⌘)

Z

⌦(n2 � 1)2 +

Z

⌦

h⇣p�1

rn2

n2��2

p�1

2rc

⌘2

+⇣

�2 ��22�14

⌘

|rc|2i

� µ2�1(1� ⌘)

Z

⌦(n2 � 1)2 + [µ1 �

(µ1 + a2µ2�1)2

4µ2�1⌘]

Z

⌦n21

+ (�2 ��22�14

)

Z

⌦|rc|2.

(3.4)

With these calculations above, we obtain the next explicit decay property, which isa specification of [8, Lemma 4.3], see also [4, Section 4.2].

Lemma 3.2. Define

E2 :=

Z

⌦n1 +

µ1

a2µ2

Z

⌦(n2 � 1� log n2) +

µ1�22

8a2µ2

Z

⌦c2

and

F2 :=

Z

⌦n21 +

Z

⌦(n2 � 1)2.

Then, in the case of a1 � 1 > a2, the nonnegative functions E2 and F2 satisfy

d

dtE2(t) �(1� a2)µ1 min

n 1

2a2,

1

1 + a2

o

F2(t) =: ��F2(t), 8t > 0. (3.5)

Proof. The fact that a2 < 1 allows us to select

�1 =µ1

a2µ2, �2 =

µ1�22

4a2µ2, ⌘ =

1 + a22

2 (0, 1),

then, upon a plain calculation from (3.4), we obtain the dissipation inequality (3.5).

By the symmetry of the n1-and n2-equations in (1.1), when a2 � 1 > a1, usingsimilar arguments leading to Lemma 3.2, we have a dissipation inequality as follows:

Lemma 3.3. Define

E3 :=

Z

⌦(n1 � 1� log n1) +

a1µ1

µ2

Z

⌦n2 +

�21

8

Z

⌦c2

and

F3 :=

Z

⌦(n1 � 1)2 +

Z

⌦n22.

Then, in the case of a2 � 1 > a1, the nonnegative functions E3 and F3 satisfy

d

dtE3(t) �(1� a1)µ1 min

n1

2,

a11 + a1

o

F3(t) =: ⇢F3(t), 8t > 0. (3.6)

Proof. For any constants l1, l2 > 0 and " 2 (0, 1), using the fact that a2 � 1 andsimilar computations to the ones leading to (3.4), we infer that

� d

dt

Z

⌦

(n1 � 1� log n1) + l1n2 +l22c2�

� µ1(1� ")

Z

⌦(n1 � 1)2 + [l1µ2 �

(a1µ1 + l1µ2)2

4µ1"]

Z

⌦n22

+ (l2 ��21

4)

Z

⌦|rc|2.

(3.7)


Now, thanks to a1 < 1, we set

l1 =a1µ1

µ2, l2 =

�21

4, " =

1 + a12

2 (0, 1),

and then we easily conclude (3.6) upon trivial computations from (3.7).

4. Convergence rates. Aided by those dissipation estimates as provided in Lem-mas 3.1, 3.2 and 3.3, even weaker regularity properties than those in Section 2, usingthe quite known arguments, cf. [1, 4, 8, 25, 26, 27, 33] for example, we know thatany global-in-time and bounded classical solution of (1.1) satisfies the convergenceproperties as follows:

�

�

�

(n1(·, t), n2(·, t), c(·, t), u(·, t))� (N1, N2, 0, 0)�

�

�

L1(⌦)! 0 as t ! 1. (4.1)

Here, (N1, N2) = (N1, N2) when a1, a2 2 (0, 1), (N1, N2) = (0, 1) when a1 � 1 > a2,and (N1, N2) = (1, 0) when a2 � 1 > a1, where N1 and N2 are defined by (1.4). Inthis section, we derive the explicit rates of convergence as described in Theorem 1.1for any supposedly bounded and global-in-time classical solution to (1.1).

4.1. Exponential convergence rate of c. We first take up the convergence rateof c, which is based on a parabolic comparison argument.

Lemma 4.1. The c-solution component of any bounded solution of (1.1) stabilizesto zero exponentially, namely, there exists t0 > 1 such that

kc(·, t)kL1 kc0kL1e�↵N1+�N2

2 (t�t0), 8t � t0.

Proof. We show the proof only for the case that a1, a2 2 (0, 1). Firstly, it followsfrom the convergence of (4.1), as t ! 1, that n1(·, t) ! N1 and n2(·, t) ! N2

uniformly in ⌦. Henceforth, we can fix a t0 > 1 such that

N1

2 n1 3N1

2and

N2

2 n2 3N2

2on ⌦⇥ [t0,1). (4.2)

This simply shows

↵n1 + �n2 � ↵N1 + �N2

2on ⌦⇥ [t0,1).

This along with the third equation in (1.1) and the positivity of c gives

ct �c� u ·rc� ↵N1 + �N2

2c on ⌦⇥ [t0,1). (4.3)

Let z(t) be the solution of the following associated ODE problem:8

<

:

z0(t) + ↵N1+�N2

2 z(t) = 0, t � t0,

z(t0) = kc(·, t0)kL1 .

It is clear that z(t) satisfies (4.3) together with @⌫z = 0, and hence an applicationof the comparison principle and Hopf boundary point lemma immediately yields

c(x, t) z(t) = kc(·, t0)kL1e�↵N1+�N2

2 (t�t0) for all x 2 ⌦, t � t0.

Using the basic fact that t ! kc(·, t)kL1 is non-increasing again by comparisonprinciple, cf. [33, Lemma 2.1], one has

c(x, t) kc0kL1e�↵N1+�N2

2 (t�t0) for all x 2 ⌦, t � t0.


This completes the proof of Lemma 4.1 by noting the positivity of c.

4.2. Exponential convergence rates in Case I: a1, a2 2 (0, 1). In this case,we will show that the solution components (n1, n2, u) converge exponentially to(N1, N2, 0).

4.2.1. Convergence rates of n1 and n2 in Case I. In this subsection, we shall estab-lish the convergence rate of n1 and n2 on the basis of the convergence rate of c inLemma 4.1 and the regularity of n1 and n2 provided by Lemma 2.3. We first useLemmas 4.1 and 3.1 to obtain the exponential convergence rates of kn1 � N1kL2

and kn2 �N2kL2 .

Lemma 4.2. The n1- and n2- solution components of bounded solution of (1.1)verify

kn1(·, t)�N1k2L2 + kn2(·, t)�N2k2L2 K1e�⇠(t�t0), 8t � t0, (4.4)

where K1 = K1(t0) = O(1) > 0 is defined by

K1 =

9h

E1(t0) +2(1�a1a2)µ1|⌦|kc0k2

L1 min

n

12 ,

a1(1+a1a2)a2

o

(N1�2

14 +

a2µ2N2�22

4a1µ1+1)

min

n

2(↵N1+�N2)max{ 1N1

,a1µ1

a2µ2N2}, (1�a1a2)µ1 min{ 1

2 ,a1

(1+a1a2)a2}o

e

i

2minn

1N1

, a1µ1

a2µ2N2

o

(4.5)and the exponential decay rate ⇠ is defined by (1.6).

Proof. Applying Taylor’s formula to the function (z) = z�N1 log z at z = N1, weobtain


N1= (n1)� (N1) =

00(⇣)

2(n1 �N1)

2 =N1

2⇣2(n1 �N1)

2

(4.6)

for some ⇣ > 0 between n1 and N1. Then a combination of (4.2) and (4.6) gives

2

9N1(n1 �N1)

2 n1 �N1 �N1 logn1

N1 2

N1(n1 �N1)

2, 8t � t0,

and hence

2

9N1

Z

⌦(n1 �N1)

2 Z

⌦

✓


N1

◆

2

N1

Z

⌦(n1 �N1)

2, 8t � t0.

(4.7)Similarly, one gets for all t � t0 that

2

9N2

Z

⌦(n1 �N2)

2 Z

⌦

✓


N2

◆

2

N2

Z

⌦(n2 �N2)

2. (4.8)

On the other hand, Lemma 4.1 quickly gives rise toZ

⌦c2 |⌦|kc(·, t)k2L1 |⌦|kc0k2L1e�(↵N1+�N2)(t�t0), 8t � t0. (4.9)

A substitution of (4.7), (4.8) and (4.9) into the definition of E1 in (3.1) gives

E1(t) 2

N1

Z

⌦(n1 �N1)

2 +2a1µ1

a2µ2N2

Z

⌦(n2 �N2)

2 +mce�(↵N1+�N2)(t�t0)

✓F1 +mce�(↵N1+�N2)(t�t0), 8t � t0,

(4.10)


where

✓ = 2max{ 1

N1,

a1µ1

a2µ2N2}, mc = |⌦|kc0k2L1(

N1�21

4+

a2µ2N2�22

4a1µ1+ 1). (4.11)

Hence, from (4.10) and the dissipation estimate (3.2), we derive that

d

dtE1 +

⌧

✓E1 ⌧mc

✓e�(↵N1+�N2)(t�t0), 8t � t0,

and then, solving this Gronwall di↵erential inequality, we readily get

E1(t) E1(t0)e� ⌧

✓ (t�t0) +⌧mc

✓e(↵N1+�N2)t0e�

⌧✓ t

Z t

t0

e[⌧✓ �(↵N1+�N2)]sds

h

E1(t0) +2⌧mc

min{↵N1 + �N2,⌧✓ }e✓

i

e�min{↵N1+�N2, ⌧

✓}

2 (t�t0), 8t � t0,

(4.12)

where we have used the following algebraic calculations:

e(↵N1+�N2)t0e�⌧✓ t

Z t

t0

e[⌧✓ �(↵N1+�N2)]sds

=

8

<

:

(t� t0)e�⌧✓ (t�t0), if ⌧

✓ = (↵N1 + �N2)

1⌧✓ �(↵N1+�N2)

h

e�(↵N1+�N2)(t�t0) � e�⌧✓ (t�t0)

i

, if ⌧✓ 6= (↵N1 + �N2)

(t� t0)e�min{↵N1+�N2, ⌧✓ }(t�t0)

2

min{↵N1 + �N2,⌧✓ }e

e�min{↵N1+�N2, ⌧

✓}

2 (t�t0).

(4.13)

By the definition of E1 in (3.1) and the estimates (4.7), (4.8), we see that

E1(t) �2

9min

n 1

N1,

a1µ1

a2µ2N2

oh

Z

⌦(n1 �N1)

2 +

Z

⌦(n2 �N2)

2i

. (4.14)

Joining (4.14) and (4.12) and substituting the definitions of ⌧ , ✓ and mc in (3.2) and(4.11), we finally arrive at (4.4) with K1 and ⇠ given by (4.5) and (1.6), respectively.

Thanks to the regularity provided by Lemma 2.3, we employ the well-knownGagliardo-Nirenberg interpolation inequality to pass the L2-convergence of n1 andn2 in (4.4) to their L1-convergence.

Lemma 4.3. Let ⌦ ⇢ Rd be a bounded and smooth domain. Then the n1- andn2- solution components of any bounded solution of (1.1) decay exponentially to(N1, N2):

kn1(·, t)�N1kL1 + kn2(·, t)�N2kL1 Ce�⇠

d+2 (t�t0), 8t � t0 (4.15)

for some C > 0 independent of t. Here, the exponential decay rate ⇠ is defined by(1.6).


Proof. Due to the L2-convergence of n1, n2 in (4.4) and the uniformW 1,1-bounded-ness of n1, n2 in (2.6), the Gagliardo-Nirenberg inequality enables us to concludethat

kn1(·, t)�N1kL1 + kn2(·, t)�N2kL1

c1⇣

kn1(·, t)kd

d+2

W 1,1kn1(·, t)�N1k2

d+2

L2 + kn2(·, t)kd

d+2

W 1,1kn2(·, t)�N2k2

d+2

L2

⌘

c2⇣

kn1(·, t)�N1k2

d+2

L2 + kn2(·, t)�N2k2

d+2

L2

⌘

c3e� ⇠

d+2 (t�t0), 8t � t0.

(4.16)

This is nothing but the exponential decaying estimate (4.15).

4.2.2. Convergence rate of u in Case I. With the information gained from Lemmas4.2 and 2.1, we are now able to derive the convergence rate of u in L1-norm. Toaccomplish this goal, we again begin with its convergence rate in L2-norm.

Lemma 4.4. The u-solution component of (1.1) fulfills

ku(·, t)k2L2 ⇣

ku(·, t0)k2L2 +2K1K1

min{�P , ⇠}e

⌘

e�min{�P ,⇠}

2 (t�t0), 8t � t0, (4.17)

where K1, K1, ⇠ and �P are respectively defined by (4.5), (4.20), (1.6) and (4.19).

Proof. Recalling that r · u = 0 in ⌦ and u|@⌦ = 0, we multiply the fourth equationin (1.1) by u and integrate it over ⌦ to obtain

1

2

d

dt

Z

⌦|u|2 +

Z

⌦|ru|2 =

Z

⌦(�n1 + �n2)r� · u+

Z

⌦|u|2r · u

= �

Z

⌦(n1 �N1)r� · u+ �

Z

⌦(n2 �N2)r� · u,

(4.18)

where we also used the factR

⌦ r� · u = 0. Therefore, we apply the Poincareinequality:

�P

Z

⌦|u|2

Z

⌦|ru|2 (4.19)

for the Poincare constant �P to (4.18) deduce that

d

dt

Z

⌦|u|2 + 2�P

Z

⌦|u|2

2�

Z

⌦|n1 �N1||r� · u|+ 2�

Z

⌦|n2 �N2||r� · u|

�P

Z

⌦|u|2 + 2�2kr�k2L1

�P

Z

⌦|n1 �N1|2 +

2�2kr�k2L1

�P

Z

⌦|n2 �N2|2.

As a result, for

K1 = 2max{ �2

�P,

�2

�P}kr�k2L1 , (4.20)

it follows thatd

dt

Z

⌦|u|2 + �P

Z

⌦|u|2 K1

⇣

Z

⌦|n1 �N1|2 +

Z

⌦|n2 �N2|2

⌘

. (4.21)

Substituting (4.4) into (4.21), we derive that

d

dt

Z

⌦|u|2 + �P

Z

⌦|u|2 K1K1e

�⇠(t�t0), 8t � t0.


Solving this ODI and performing similar computations to (4.13), we readily obtain

ku(·, t)k2L2 ku(·, t0)k2L2e��P (t�t0) +K1K1e⇠t0e��P t

Z t

t0

e(�P�⇠)sds

⇣

ku(·, t0)k2L2 +2K1K1

min{�P , ⇠}e

⌘

e�min{�P ,⇠}

2 (t�t0), 8t � t0,

which is precisely the desired exponential decay estimate (4.17).

Lemma 4.5. Let ⌦ ⇢ Rd be a bounded and smooth domain. Then, for any ✏ 2(0, 1), there exists a constant

K1 � O(1)(1 + (1� a1a2)� ✏

d+2 ) (4.22)

such thatku(·, t)kL1 K1e

� ✏2(d+2) min{�P ,⇠}(t�t0), 8t � t0, (4.23)

where the exponent rate ⇠ is defined by (1.6).

Proof. Thanks to the L2-convergence of u in (4.17) and the uniform-in-time W 1,p-boundedness of u in (2.1), the Gagliardo-Nirenberg inequality allows us to infer

ku(·, t)kL1 c1ku(·, t)kdp

dp+2p�2d

W 1,p ku(·, t)k2p�2d

dp+2p�2d

L2 c2e� (p�d)

dp+2p�2d min{�P2 , ⇠2}(t�t0),

8t � t0,

which implies (4.23) upon choosing p = [d+2(1�✏)]d/[(d+2)(1�✏)](> d). Noticingthat ⇠ = O(1)(1 � a1a2) by (1.6), then the lower bound for K1 in (4.22) followsfrom (4.17).

4.3. Algebraic convergence rates in Case II: a1 = 1 > a2. Here, we will showthat the solution components (n1, n2, u) converge at least algebraically to (0, 1, 0).

4.3.1. Convergence rates of n1 and n2 in Case II. Again, we start with the deriva-tion of the L1- and L2-convergence rates of n1 and n2.

Lemma 4.6. There exists t1 � max{1, t0} such that

kn1(·, t)kL1 + kn2(·, t)� 1k2L2 K2

t+ t1, 8t � t1, (4.24)

where

K2 =max

n

2t1E2(t1),k1

⇣

k1+p

k21+2a�

⌘

�

o

min{1, 2µ1

9a2µ2}

� O(1)(1 +1

�) = O(1)(1 + (1� a2)

�1)

(4.25)

with �, k1, k2, k3 and a defined in (3.5), (4.29), (4.34) and (4.31), respectively.

Proof. Our proof makes use of the explicit Lyapunov functional provided by Lemma3.2. To proceed, we first apply the Holder inequality to find

Z

⌦n1 |⌦| 12

✓

Z

⌦n21

◆

12

. (4.26)

Next, since kn2(·, t)� 1kL1 ! 0 as t ! 1 and

limz!1

z � 1� log z

z � 1= 0,


we can take t0 > 0 such that |n2(x, t)� 1� log n2(x, t)| |n2(x, t)� 1| for all x 2 ⌦and t � t0. Accordingly, we have

Z

⌦(n2 � 1� log n2)

Z

⌦|n2 � 1| |⌦| 12

✓

Z

⌦(n2 � 1)2

◆

12

, 8t � t0. (4.27)

In this case, (N1, N2) = (0, 1), so the exponential decay of c in Lemma 4.1 warrantsthat

Z

⌦c2 |⌦|kck2L1 |⌦|kc0k2L1e��(t�t0), 8t � t0. (4.28)

From the definitions of E2 and F2 in Lemma 3.2, upon a combination of (4.26),(4.27) and (4.28) and the fact that

pA +

pB

p

2(A+B) for A,B � 0, we findtwo constants

k1 = max{1, µ1

a2µ2}(2|⌦|) 1

2 , k2 =µ1�2

2

8a2µ2|⌦|kc0k2L1 (4.29)

such that

E2(t) k1F122 (t) + k2e

��(t�t0), 8t � max{t0, t0},which further gives us

E22(t) 2k21F2(t) + 2k22e

�2�(t�t0), 8t � max{t0, t0}.A substitution of this into the dissipation inequality (3.5) entails

d

dtE2(t) +

�

2k21E2

2(t) k22�

k21e�2�(t�t0), t � max{t0, t0}. (4.30)

Now, to illustrate the algebraic decay (4.24), we first take t1 = maxn

t0, t0, 1,��1o

so that

a =k22�

k21max

n

(t+ t1)2e�2�(t�t0) : t � t0

o

=4k22t

21�

k21e�2�(t1�t0); (4.31)

and then, for any

b �k1⇣

k1 +p

k21 + 2a�⌘

�, (4.32)

we put

y(t) =b

t+ t1, t � 0. (4.33)

We use straightforward calculations from (4.33) and use (4.31) to see that

y0(t) +�

2k21y2(t)� k22�

k21e�2�(t�t0)

= (t+ t1)�2

h �

2k21b2 � b� k22�

k21(t+ t1)

2e�2�(t�t0)i

� (t+ t1)�2(

�

2k21b2 � b� a) � 0, 8t � t1.

This, upon a clear choice of b in (4.32), an ODE comparison argument to (4.30)shows

E2(t) max

n

2t1E2(t1),k1

⇣

k1+p

k21+2a�

⌘

�

o

t+ t1=:

k3t+ t1

, 8t � t1. (4.34)


Then since t1 � t0, we infer from (4.8) and the definition of E2(t) in Lemma 3.2that

min{1, 2µ1

9a2µ2}⇣

kn1(·, t)kL1 + kn2(·, t)� 1k2L2

⌘

k3t+ t1

, 8t � t1.

This, upon a substitution of the respective definitions of k1, k2, k3 and a in (4.29),(4.34) and (4.31), proves our desired algebraic decay estimate (4.24).

Lemma 4.7. There exist two constants K3 and K4 independent of t fulfilling

K3 � O(1)⇣

1 + (1� a2)� 1

d+1

⌘

, K4 � O(1)⇣

1 + (1� a2)� 1

d+2

⌘

such that

kn1(·, t)kL1 K3

(t+ t1)1

d+1

, 8t � t1 (4.35)

as well as

kn2(·, t)� 1kL1 K4

(t+ t1)1

d+2

, 8t � t1. (4.36)

Proof. Equipped with the uniform W 1,1-bounds of n1, n2 in Lemma 2.3, as before,by means of the Gagliardo-Nirenberg inequality, we readily infer, for all t � t1,

kn1(·, t)kL1 c1kn1(·, t)kd

d+1

W 1,1kn1(·, t)k1

d+1

L1 c2kn1(·, t)k1

d+1

L1 (4.37)

and

kn2(·, t)� 1kL1 c3kn2(·, t)kd

d+2

W 1,1kn2(·, t)� 1k2

d+2

L2 c4kn2(·, t)� 1k2

d+2

L2 .

These along with the L1-and L2-convergence of n1, n2 in (4.24) and the bound forK2 in (4.25) yield immediately (4.35) and (4.36).

4.3.2. Convergence rate of u in Case II.

Lemma 4.8. The u-solution component of (1.1) fulfills

ku(·, t)k2L2 K5

t+ t1, 8t � t1 (4.38)

for some positive constant K5 independent of t satisfying K5 � O(1)(1+(1�a2)�1).

Proof. Thanks to the L1-and L2-convergence of n1, n2 in (4.24), we can easily adaptthe proof of Lemma 4.4 here. Indeed, from (4.18), the Poincare inequality (4.19)and the boundedness of u, we infer that

1

2

d

dt

Z

⌦|u|2+ �P

2

Z

⌦|u|2 �kr�kL1kukL1

Z

⌦n1+

�2kr�k2L1

2�P

Z

⌦|n2�1|2. (4.39)

Thus, for

K5 = 2max{�kukL1(⌦⇥(0,1)),�2kr�kL1

2�P}kr�kL1 < 1,

from (4.39) and (4.24), we obtain an ODI for kuk2L2 as follows:

d

dt

Z

⌦|u|2 + �P

Z

⌦|u|2 K5

⇣

Z

⌦n1 +

Z

⌦|n2 � 1|2

⌘

K2K5

t+ t1, 8t � t1.


Solving this ODI, we end up with

ku(·, t)k2L2 ku(·, t1)k2L2e��P (t�t1) +K2K5e��P t

Z t

t1

e�P s

s+ t1ds

ku(·, t1)k2L2e��P (t�t1) +K2K5K5

t+ t1

⇣

ku(·, t1)k2L2e��P t1K5 +K2K5K5

⌘

(t+ t1)�1, 8t � t1,

from which (4.38) follows. Here, we have used the following facts

K5 = max{(t+ t1)e��P t : t � t1} < 1

and

K5 = maxn

(t+ t1)e��P t

Z t

t1

e�P s

s+ t1ds : t � t1

o

< 1.

The latter is due to

limt!1

h

(t+ t1)e��P t

Z t

t1

e�P s

s+ t1dsi

= limt!1

R tt1

e�P s

s+t1ds

(t+ t1)�1e�P t=

1

�p< 1.

With the L2-convergence of u in Lemma 4.8 at hand, the same argument as doneto Lemma 4.5 yields the following L1-convergence of u.

Lemma 4.9. For any ✏ 2 (0, 1), there exists a positive constant

K6 � O(1)⇣

1 + (1� a2)� ✏

d+2

⌘

such that

ku(·, t)kL1 K6

(t+ t1)✏

d+2, 8t � t1.

4.4. Algebraic convergence rates in Case (II0): a2 = 1 > a1. In this case, weshall show that (n1, n2, u) converges at least algebraically to (1, 0, 0). ComparingLemmas 3.2 and 3.3 and the n1-and n2-equations in (1.1), we see that this subsectionis fully parallel to Section 4.3, and so we simply write down their respective finaloutcomes.

Lemma 4.10. There exist t2 � max{1, t0} and positive constants K7 and K8

fulfilling

K7 � O(1)⇣

1 + (1� a1)� 1

d+2

⌘

, K8 � O(1)⇣

1 + (1� a1)� 1

d+1

⌘

such that

kn1(·, t)� 1kL1 K7

(t+ t2)1

d+2

, 8t � t2

and

kn2(·, t)kL1 K8

(t+ t2)1

d+1

, 8t � t2.

Lemma 4.11. For any ✏ 2 (0, 1), there exists C✏ � O(1)(1 + (1 � a1)� ✏

d+2 ) suchthat

ku(·, t)kL1 C✏

(t+ t2)✏

d+2, 8t � t2.


4.5. Exponential convergence rates in Case III: a1 > 1 > a2. In this section,we use the crucial fact that a1 > 1 to show that the bounded solution components(n1, n2, u) converge not only algebraically (as shown in Subsection 4.3) but alsoexponentially to (0, 1, 0) and we shall compute out explicitly their respective ratesof convergence. Armed with the knowledge from previous subsections, this sectioncan be short and thus we include their stabilization results in a single lemma.

Lemma 4.12. There exist t3 � max{1, t0} and C = O(1) > 0 such that the n1-solution component of (1.1) fulfills

kn1(·, t)kL1 Ce�(a1�1)µ12(d+1) (t�t3), 8t � t3; (4.40)

for some positive constant

K9 � O(1)(1 + (a1 � 1)�1

d+2 + (1� a2)� 1

d+2 ), (4.41)

the n2-solution component satisfies

kn2(·, t)� 1kL1 K9e� 1

4(d+2) min{ a2µ2�µ1

, 2�, (a1�1)µ1}(t�t3), 8t � t3; (4.42)

finally, for any ✏ 2 (0, 1), there exists

K10 � O(1)(1 + (a1 � 1)�2✏

d+2 + (1� a2)� 2✏

d+2 ) (4.43)

such that the u-solution component satisfies

ku(·, t)kL1 K10e� ✏

8(d+2) min{ a2µ2�µ1

, 2�, (a1�1)µ1, 4�P }(t�t3), 8t � t3. (4.44)

Here, � is defined by (3.5) in Lemma 3.2.

Proof. A simple integration of the first equation in (1.1) shows

d

dt

Z

⌦n1 = � (a1 � 1)µ1

2

Z

⌦n1 � µ1

Z

⌦

h

a1n2 � 1 + n1 �(a1 � 1)

2

i

n1. (4.45)

The uniform convergence n1 ! 0 and n2 ! 1 (c.f. Subsection 4.3 or (4.1)) alongwith the fact a1 > 1 allows us to find t3 � max{1, t0} such that

a1n2 � 1 + n1 � (a1 � 1)

2on ⌦⇥ [t3,1). (4.46)

Thus, when t > t3, based on (4.46) and (4.45), we derive a Gronwall inequality forkn1kL1 :

d

dt

Z

⌦n1 +

(a1 � 1)µ1

2

Z

⌦n1 0, 8t � t3,

which trivially yields the exponential decay of kn1kL1 :

kn1(·, t)kL1 kn1(·, t3)kL1e�(a1�1)µ1

2 (t�t3), 8t � t3. (4.47)

Then, by the uniformW 1,1-bounds of n1 in Lemma 2.3 and the Gagliardo-Nirenberginequality, similar to (4.37), we readily obtain the exponential decay estimate (4.40).

Since (N1, N2) = (0, 1), the exponential convergence of c in Lemma 4.1 entailsZ

⌦c2 |⌦|kc(·, t)k2L1 |⌦|kc0k2L1e��(t�t0), 8t � t3. (4.48)

Using the essentially same argument leading to (4.8), we get

2

9

Z

⌦(n2 � 1)2

Z

⌦(n2 � 1� log n2) 2

Z

⌦(n2 � 1)2, 8t � t3. (4.49)


By (4.47), (4.48), (4.49) and the definitions of E2 and F2 in Lemma 3.2, we bound

E2 =

Z

⌦n1 +

µ1

a2µ2

Z

⌦(n2 � 1� log n2) +

µ1�22

8a2µ2

Z

⌦c2

2µ1

a2µ2

Z

⌦(n2 � 1)2 + kn1(·, t3)kL1e�

(a1�1)µ12 (t�t3) +

µ1�22

8a2µ2|⌦|kc0k2L1e��(t�t0)

2µ1

a2µ2F2+

h

kn1(·, t3)kL1 +µ1�2

2

8a2µ2|⌦|kc0k2L1

i

e�12 min{2�, (a1�1)µ1}(t�t3)

=:2µ1

a2µ2F2 + K9e

� 12 min{2�, (a1�1)µ1}(t�t3), 8t � t3.

(4.50)

This along with the dissipation estimate (3.5) enables us to deduce that

d

dtE2(t) +

a2µ2�

2µ1E2(t)

a2µ2�

2µ1K9e

� 12 min{2�, (a1�1)µ1}(t�t3), 8t � t3,

and then, solving this Gronwall di↵erential inequality, we infer, for t � t3, that

E2(t) E2(t3)e� a2µ2�

2µ1(t�t3)

+a2µ2�

2µ1K9e

12 min{2�, (a1�1)µ1}t3e�

a2µ2�2µ1

tZ t

t3

e[a2µ2�2µ1

� 12 min{2�, (a1�1)µ1}]sds

h

E2(t3) +2K9

a2µ2�µ1

min{a2µ2�µ1

, 2�, (a1 � 1)µ1}e

i

e�14 min{ a2µ2�

µ1, 2�, (a1�1)µ1}(t�t3),

(4.51)

where we have used the following algebraic computations similar to (4.13):

e12 min{2�, (a1�1)µ1}t3e�

a2µ2�2µ1

tZ t

t3

e[a2µ2�2µ1

� 12 min{2�, (a1�1)µ1}]sds

=

8

>

>

<

>

>

:

(t� t3)e� a2µ2�

2µ1(t�t3), if a2µ2�

2µ1= 1

2 min{2�, (a1 � 1)µ1}h

e�12

min{2�,(a1�1)µ1}(t�t3)�e� a2µ2�

2µ1(t�t3)

i

a2µ2�2µ1

� 12 min{2�,(a1�1)µ1}

, if a2µ2�2µ1

6= 12 min{2�, (a1 � 1)µ1}

(t� t3)e� 1

2 min{ a2µ2�µ1

, 2�, (a1�1)µ1}(t�t3)

4

min{a2µ2�µ1

, 2�, (a1 � 1)µ1}ee�

14 min{ a2µ2�

µ1, 2�, (a1�1)µ1}(t�t3).

From the definition of E2 in Lemma 3.2 or alternatively (4.50) and the estimates(4.49) and (4.51), we conclude finally that

kn2(·, t)� 1k2L2 K9e� 1

4 min{ a2µ2�µ1

, 2�, (a1�1)µ1}(t�t3), 8t � t3, (4.52)

where

K9 =9a2µ2

2µ1

h

E2(t3) +4K9

min{a2µ2�µ1

, 2�, (a1 � 1)µ1}e

i

. (4.53)

Then, as we did in (4.16) of Lemma 4.3, we use the uniform W 1,1-boundedness ofn2 in (2.6) and the Gagliardo-Nirenberg inequality to improve the L2-convergenceof n2 in (4.52) to the L1-convergence of n2 in (4.42). The lower bound for K9 in(4.41) follows from (4.53) and the fact that � = O(1)(1� a2) by (3.5).


With the convergence rates of n1 and n2, we use the sprit of Lemma 4.4 to showfirst the L2-convergence of u. To start o↵, we utilize the L1- and L2-convergence ofn1 and n2 in (4.47) and (4.52) to bound the counterpart of (4.21) as follows:

d

dt

Z

⌦|u|2 + �P

Z

⌦|u|2

K1

⇣

Z

⌦n21 +

Z

⌦(n2 � 1)2

⌘

K1kn1kL1(⌦⇥(0,1))

Z

⌦n1 + K1

Z

⌦(n2 � 1)2

K10e� (a1�1)µ1

2 (t�t3) + K10e� 1

4 min{ a2µ2�µ1

, 2�, (a1�1)µ1}(t�t3)

2K10e� 1

4 min{ a2µ2�µ1

, 2�, (a1�1)µ1}(t�t3) =: 2K10e�⌫(t�t3),

(4.54)

where, upon notice of the fact from (3.5) that ⌫ = O(1)min{(a1 � 1), (1� a2)} andthe use of (4.20) and (4.53),

K10 = maxn

K1kn1kL1(⌦⇥(0,1)), K1K9

o

= O(1)⇣

1 + (min{(a1 � 1), (1� a2)})�1⌘

.(4.55)

Solving the ordinary di↵erential inequality (4.54) and performing similar computa-tions to the ones right after (4.51), we conclude from the fact that ⌫ = O(1)min{(a1�1), (1� a2)} by (3.5) and (4.55) that for all t � t3

ku(·, t)k2L2 ku(·, t3)k2L2e��P (t�t3) + 2K10e⌫t3e��P t

Z t

t3

e(�P�⌫)sds

⇣

ku(·, t3)k2L2 +4K10

min{�P , ⌫}e

⌘

e�min{�P ,⌫}

2 (t�t3)

O(1)⇣

1 + (a1 � 1)�2 + (1� a2)�2

⌘

e�min{�P ,⌫}

2 (t�t3).

(4.56)

With this L2-convergence and the uniform-in-time W 1,p-boundedness of u in (2.1),the Gagliardo-Nirenberg interpolation inequality allows us to deduce, as in Lemma4.5, the L1-convergence (4.44). The lower bound for K10 in (4.43) can be seenfrom (4.56).

4.6. Exponential convergence rates in Case (III0): a2 > 1 > a1. Using theLyapunov functional provided by Lemma 3.3 and the arguments parallel to Subsec-tion 4.5, we find that any bounded solution (n1, n2, u) will converge exponentiallyto (1, 0, 0). We here shall omit the details and just write down their respective finalconvergence results.

Lemma 4.13. There exist t4 � max{1, t0} and a positive constant

K11 � O(1)(1 + (1� a1)� 1

d+2 + (a2 � 1)�1

d+2 ),

such that the n1-solution component of (1.1) fulfills

kn1(·, t)� 1kL1 K11e� 1

4(d+2) min{⇢, 2↵, (a2�1)µ2}(t�t4), 8t � t4;

the n2-solution component satisfies, for some C = O(1) > 0,

kn2(·, t)kL1 Ce�(a2�1)µ22(d+1) (t�t4), 8t � t4;


finally, for any ✏ 2 (0, 1), there exists

K12 � O(1)(1 + (a1 � 1)�2✏

d+2 + (1� a2)� 2✏

d+2 )

such that the u-solution component satisfies

ku(·, t)kL1 K10e� ✏

8(d+2) min{⇢, 2↵, (a2�1)µ2, 4�P }(t�t4), 8t � t4.

Here, ⇢ is defined by (3.6) in Lemma 3.3.

Proof of Theorem 1.1. Notice that ti � max{t0, 1} > 1(i = 1, 2, 3, 4); the respectivedecay estimates and their decay rates asserted in Theorem 1.1 follow from somelemmas in this section with perhaps some large constants mi. More specifically, theexponential decay estimate (1.9) follows from Lemmas 4.3 and 4.5; the algebraicaldecay estimate (1.10) follows from Lemmas 4.7 and 4.9; the exponential decayestimate (1.11) follows from Lemma 4.12 and the decay rate ⌫ in (1.7) follows froma substitution of the definition of � in (3.5); the algebraical decay estimate (1.12)follows from Lemmas 4.10 and 4.11; the exponential decay estimate (1.13) followsfrom Lemma 4.13 and the decay rate µ in (1.8) follows from a substitution of thedefinition of ⇢ in (3.6); and, finally, the exponential decay estimate (1.14) followsfrom Lemma 4.1.

Acknowledgments. The authors are very grateful to the referee for his/her pos-itive and constructive comments/suggestions, which helped us greatly improve theexposition of the paper. The research of H.Y. Jin was supported by NSF of China(No. 11501218) and the Fundamental Research Funds for the Central Universities(No. 2017MS107), and the research of T. Xiang was supported by the NSF ofChina (No. 11601516 and 11571363) and the Research Fund of Renmin Universityof China (No. 2018030199).

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Received January 2018; revised April 2018.


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