ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

LARGE TIME BEHAVIOR OF A CAHN-HILLIARD-BOUSSINESQ SYSTEM ON A BOUNDED DOMAIN

KUN ZHAO

Abstract. We study the asymptotic behavior of classical solutions to an initial-boundary value problem (IBVP) for a coupled Cahn-Hilliard-Boussinesq system on bounded domains with large initial data. A sufficient condition is established under which the solutions decay exponentially to constant states as time approaches infinity.

1. Introduction

As one of the fundamental modelling equations, the Cahn-Hilliard equation [6, 7]

plays an important role in the mathematical study of multi-phase flows, and has been studied intensively in the literature both analytically and numerically (see e.g.

[3, 4, 8, 12, 13, 19, 20, 25, 26, 27, 29]). The couplings of the Cahn-Hilliard equation with other basic modelling equations have been proposed in various situations to study complicated phenomena in fluid mechanics involving phase transition. For ex- ample, the coupled Cahn-Hilliard-Navier-Stokes (CHNS) system and its variations, which describe the motion of an incompressible two-phase flow under shear through an order parameter formulation, have been used in order to understand the phe- nomena of phase transition in incompressible fluid flows (c.f. [14, 18, 24]). Recently, a closely related model to the CHNS system has been developed in [10, 11, 15, 16]

to understand the spinodal decomposition of binary fluid in a Hele-Shaw cell, tumor growth, cell sorting, and two phase flows in porous media, which is referred as the Cahn-Hilliard-Hele-Shaw (CHHS) system. In this paper, we consider the following system of equations:

φ_{t}+U· ∇φ= ∆µ, x∈R^{n}, t >0,
µ=−α∆φ+F^{0}(φ),

Ut+U · ∇U +∇P =µ∇φ+θen,
θ_{t}+U· ∇θ=κ∆θ,

∇ ·U = 0,

(1.1)

2000Mathematics Subject Classification. 35Q35, 35B40.

Key words and phrases. Cahn-Hilliard-Boussinesq equations; classical solution;

asymptotic behavior.

c

2011 Texas State University - San Marcos.

Submitted April 11, 2010. Published April 6, 2011.

1

which is a system of strongly coupled partial differential equations obtained by cou-
pling the Cahn-Hilliard equation to the inviscid heat-conductive Boussinesq equa-
tions. It describes the motion of an incompressible inviscid two-phase flow subject
to convective heat transfer under the influence of gravitational force through or-
der parameter formulation. Here, U = (u_{1}, . . . , u_{n}),P andθ denotes the velocity,
pressure and temperature respectively. φis the order parameter andµis a chem-
ical potential derived from a coarse-grained study of the free energy of the fluid
(c.f. [17]). The constant κ >0 and α > 0 models heat conduction and diffusion
respectively, anden is the n-th unit vector inR^{n}. The functionF usually has a
physical-relevant, double-well structure, each of them representing the two phases
of the fluid. A typical example ofF takes the form (c.f. [9, 17]): F(z) = ^{1}_{4}(z^{2}−1)^{2}.
In this paper, we consider a general scenario by imposing appropriate growth con-
ditions on F. We remark that, system (1.1) reduces to the CHHS model if the
temperature equation and the hydrodynamic effect are dropped. On the other
hand, (1.1) becomes the CHNS system if the temperature equation is removed and
the viscosity of fluid is added to the velocity equation.

In the real world, flows often move in bounded domains with constraints from
boundaries, where the initial-boundary value problems appear. The solutions of
the initial-boundary value problems usually exhibit different behaviors and much
richer phenomena comparing with the Cauchy problem. In this paper, we consider
system (1.1) on a bounded domain in R^{n}. The system is supplemented by the
following initial and boundary conditions:

(φ, µ, U, θ)(x,0) = (φ_{0}, µ_{0}, U_{0}, θ_{0})(x),

∇φ·n|∂Ω= 0, ∇µ·n|∂Ω= 0, U·n|∂Ω= 0, θ|∂Ω= ¯θ, (1.2)
where Ω ⊂ R^{n} is a bounded domain with smooth boundary ∂Ω, n is the unit
outward normal to∂Ω and ¯θis a constant.

The initial-boundary value problem (1.1)–(1.2) was first studied in [30], where the global existence and uniqueness of classical solutions are established, for large initial data with finite energy in 2D. However, the large time asymptotic behavior of the solutions is not investigated due to the lack of uniform-in-time estimates of the solutions. We give definite answer to this unsolved issue in current paper for the 2D case, based on new findings of the structure of the system.

Suggested by the conservation of total mass and the boundary conditions, it is
expected that the global attractors of φand θ should be ¯φ= _{|Ω|}^{1} R

Ωφ_{0}(x)dx and
θ, respectively, due to diffusion and boundary effects. In this paper, we provide a¯
sufficient condition that guarantees the decay of the solution. We will show that
when the diffusion coefficient αpasses a threshold value determined by F and Ω,
the functions φandθwill converge exponentially in time to ¯φand ¯θ, respectively,
regardless of the magnitude of the initial perturbation. To be precise, we shall
assume thatα−F3c0>0, whereF3>0 is a constant such thatF^{00}≥ −F3, andc0

is the constant in Poincar´e inequality on Ω. This condition is crucial in our analysis
due to the fact that it produces a positive constant multiple of kφ−φk¯ ^{2}_{H}2 which
is one of the major dissipative terms controlling the exponential decay of φ. The
condition will trigger a chain reaction leading the energy estimate performed in [30]

to a whole new scenario. As consequences of the convergence ofφ and θ, we will show that the velocity and vorticity are uniformly bounded in time.

Throughout this paper, k · kL^{p}, k · kL^{∞} and k · kW^{s,p} denote the norms of the
usual Lebesgue measurable function spaces L^{p} (1 ≤ p < ∞), L^{∞} and the usual
Sobolev space W^{s,p}, respectively. Forp= 2, we denote the norm k · kL^{2} byk · k
andk · kW^{s,2} byk · kH^{s}, respectively. The function spaces under consideration are
C([0, T];H^{r}(Ω)) andL^{2}([0, T];H^{s}(Ω)), equipped with norms sup_{0≤t≤T}kΨ(·, t)kH^{r}

and RT

0 kΨ(·, τ)k^{2}_{H}sdτ^{1/2}

, respectively, where r, s are positive integers. Unless
specified,c_{i}will denote generic constants which are independent ofφ, µ, U, θandt,
but may depend onα, κ,Ω and initial data.

For the sake of completeness, we first state the results obtained in [30].

Theorem 1.1. Let Ω⊂R^{2} be a bounded domain with smooth boundary. Suppose
that F(·)satisfies the following conditions:

• F(·)is ofC^{6} class andF(·)≥0;

• There exist constants F_{1}, F_{2} > 0 such that |F^{(n)}(φ)| ≤ F_{1}|φ|^{p−n} +F_{2},
n= 1, . . . . ,6, ∀6≤p <∞ andφ∈R;

• There exists a constantF_{3}≥0such that F^{00}≥ −F_{3}.

If the initial data φ0(x) ∈ H^{5}(Ω), µ0(x) ∈ H^{3}(Ω) and (θ0(x), U0(x)) ∈ H^{3}(Ω)
are compatible with the boundary conditions, then there exists a unique solution
(φ, µ, θ, U)of (1.1)–(1.2)globally in time such that

φ∈C([0, T];H^{5}(Ω))∩L^{2}([0, T];H^{7}(Ω)),
µ∈C([0, T];H^{3}(Ω))∩L^{2}([0, T];H^{5}(Ω)),

U ∈C([0, T];H^{3}(Ω))andθ∈C([0, T];H^{3}(Ω))∩L^{2}([0, T];H^{4}(Ω))for0< T <∞.

The next theorem is the main result of this paper regarding the large-time as- ymptotic behavior of the solution obtained in Theorem 1.1.

Theorem 1.2. Suppose that the assumptions in Theorem 1.1 are in force and as- sume that the constantα−F3c0>0, wherec0 is the constant in Poincar´e inequality onΩ. Then the solution to (1.1)–(1.2)satisfies

kφ(·, t)−φk¯ _{H}5+kµ(·, t)−F^{0}( ¯φ)k_{H}3+kθ(·, t)−θk¯ _{H}3 ≤γe^{−βt};
kU(·, t)kW^{1,p}≤γ(p), ∀1≤p <∞; kω(·, t)kL^{∞} ≤¯γ, ∀t≥0,
for some constantsγ, β, γ(p),γ >¯ 0independent oft, whereφ¯= _{|Ω|}^{1} R

Ωφ0(x)dxand
ω=v_{x}−u_{y} is the 2D vorticity.

Remark 1.3. It should be pointed out that, in the theorems obtained above, no smallness restriction is put upon the initial data.

Remark 1.4. We observe that, by assuming small initial perturbation around the equilibrium state and by exploring the structure of the functionF, one can show the exponential decay of the solution. However, the asymptotic result obtained in Theorem 1.2 has an obvious advantage over the case for small perturbation. Indeed, Theorem 1.2 provides a convenient criterion for determining whether the solution collapses to a constant state as time evolves. Based on the result, one only needs to measure the volume of the domain, instead of measuring the “smallness” of the initial perturbation which is usually laborious to perform, to determine whether the solution decays or not when other system parameters are fixed.

Remark 1.5. It is well-known that the Cahn-Hilliard equation is an effective model in the study of sharp interfaces in two-phase fluid flows. However, based on our results, the order parameterφtends to a uniform constant ¯φinstead of ±1. This suggests that under the conditions of Theorem 1.1 and Theorem 1.2, the modelling equations (1.1) indeed fail to model the sharp interfacial phenomenon. Therefore, our results exhibit some bifurcation phenomena on the effectiveness of the modelling equations.

The proof of Theorem 1.2 involves a series of accurate combinations of energy estimates. The estimates are delicate mainly due to the coupling between the equa- tions by convection, gravitational force and boundary effects. Great efforts have been made to simplify the proof. Current proof involves intensive applications of Sobolev embeddings and Ladyzhenskaya type inequalities, see Lemma 2.1. Roughly speaking, because of the lack of the spatial derivatives of the solution at the bound- ary, our energy framework proceeds as follows: We first apply the standard energy estimate on the solution and the temporal derivatives of the solution. We then apply standard results on elliptic equations to recover estimates of the spatial derivatives.

Such a process will be repeated up to third order, and then with the aid of the assumption on α, the carefully coupled estimates will be composed into a desired one leading to the exponential decay of the solution.

The rest of the paper is organized as follows. In Section 2, we give some basic facts that will be used in the proof of Theorem 1.2. In Section 3, we prove some uniform-in-time energy estimates of the solution based on which the combinations of energy estimates will be performed. We then complete the proof of Theorem 1.2 in Section 4.

2. Preliminaries

In this section, we shall collect several facts which will be used in the proof of Theorem 1.2. First, we recall some inequalities of Sobolev and Ladyzhenskaya type (c.f. [1, 21]).

Lemma 2.1. LetΩ⊂R^{2}be any bounded domain with smooth boundary ∂Ω. Then
(i) kfkL^{∞} ≤c1kfkH^{2};

(ii) kfkL^{∞} ≤c_{2}kfkW^{1,p} for all p >2;

(iii) kfkL^{p}≤c_{3}kfkH^{1} for all1≤p <∞;

(iv) kfk^{2}_{L}4 ≤c_{4} kfkk∇fk+kfk^{2}

;
(v) kfk^{2}_{L}8 ≤c5 kfkk∇fk_{L}4+kfk^{2}

, for some constants ci=ci(p,Ω),i= 1, . . . ,5.

Next, we recall some classical results on elliptic equations (c.f. [2, 22, 23]), which are useful in the estimation ofθ.

Lemma 2.2. Let Ω⊂R^{2} be any bounded domain with smooth boundary∂Ω. Con-
sider the Dirichlet problem

κ∆Θ =f inΩ, Θ = 0 on∂Ω.

Iff ∈W^{m,p}, thenΘ∈W^{m+2,p}and there exists a constantc_{6}=c_{6}(p, κ, m,Ω)such
that

kΘkW^{m+2,p}≤c_{6}kfkW^{m,p}

for any p∈(1,∞)and the integer m≥ −1.

The next lemma is useful for the estimation of the velocity field (see [5]).

Lemma 2.3. Let Ω⊂R^{2} be any bounded domain with smooth boundary∂Ω, and
letF ∈W^{s,p}(Ω) be a vector-valued function satisfyingF·n|∂Ω= 0, wherenis the
unit outward normal to∂Ω. Then there exists a constantc7=c7(s, p,Ω)such that

kFkW^{s,p} ≤c7(k∇ ×Fk_{W}s−1,p+k∇ ·Fk_{W}s−1,p+kFkL^{p})
for any s≥1 andp∈(1,∞).

Finally, we recall some Poincar´e type inequalities, which will be used in the estimation ofφ, whose proof is straightforward.

Lemma 2.4. LetΩ⊂R^{n} be any bounded domain with smooth boundary∂Ω. Then,
for any functionH^{s}(Ω)3f : Ω→R, there exists a constantc_{8}=c_{8}(s,Ω)>0such
that

(1) kf−f¯k_{H}2s ≤c8k∆^{s}fkandkf−f¯k_{H}2s+1≤c8k∇∆^{s}fk,s≥1, if∇f·n|∂Ω=
0;

(2) kfk_{H}2s≤c8k∆^{s}fk andkfk_{H}2s+1≤c8k∇∆^{s}fk,s≥1, iff|∂Ω= 0,
wheref¯= _{|Ω|}^{1} R

Ωf dx.

3. Uniform energy estimates

In this section, we establish some uniform-in-time energy estimates of the solu- tion under the conditionα−F3c0>0, based on which the exponential decay rate of the solution will be proved. The results are stated as a sequence of lemmas and the proofs are carried out by carefully exploring the condition α−F3c0 > 0 and delicate applications of Cauchy-Schwarz and Gronwall inequalities.

To study the asymptotic behavior, we first reformulate the original problem to get the one for the perturbations. For this purpose, let Φ =φ−φ¯and Θ =θ−θ.¯ After plugging Φ and Θ into (1.1) we obtain

Φ_{t}+U· ∇Φ = ∆µ,
µ=−α∆Φ +F^{0}(φ),
U_{t}+U· ∇U+∇P˜ =µ∇Φ + Θe_{2},

Θt+U· ∇Θ =κ∆Θ,

∇ ·U = 0,

(3.1)

which is equivalent to (1.1) for sufficiently smooth solutions, where ˜P =P −θy,¯ and the initial and boundary conditions become

(Φ, µ, U,Θ)(x,0) = (Φ0, µ0, U0,Θ0)(x)≡(φ0−φ, µ¯ 0, U0, θ0−θ)(x),¯

∇Φ·n|_{∂Ω}= 0, ∇µ·n|_{∂Ω}= 0, U ·n|_{∂Ω}= 0, Θ|_{∂Ω}= 0. (3.2)
We begin with the uniform estimate ofkΦkL^{2}.

Lemma 3.1. Under the assumptions of Theorem 1.2, it holds that

kΦ(·, t)k^{2}+
Z t

0

kΦ(·, τ)k^{2}_{H}2dτ ≤c9, ∀t≥0. (3.3)

Proof. TakingL^{2}inner product of (3.1)1with Φ, after integrating by parts we have
1

2 d

dtkΦk^{2}=
Z

Ω

Φ∆µx=− Z

Ω

∇µ· ∇Φdx=−αk∆Φk^{2}−
Z

Ω

F^{00}(φ)|∇Φ|^{2}dx, (3.4)
which gives

1 2

d

dtkΦk^{2}+αk∆Φk^{2}≤F3k∇Φk^{2}, (3.5)
where we have used the condition onF. Since Φ =φ−φ¯and ¯φis a constant, using
the boundary conditions, Cauchy-Schwarz and Poincar´e inequalities we have

k∇Φk^{2}=−
Z

Ω

Φ∆Φdx≤ 1 2c0

kΦk^{2}+c_{0}

2k∆Φk^{2}≤ 1

2k∇Φk^{2}+c_{0}

2k∆Φk^{2}, (3.6)
which implies

k∇Φk^{2}≤c0k∆Φk^{2}, (3.7)

where c_{0} is the constant in Poincar´e inequality on Ω. Let α_{1} ≡ α−F_{3}c_{0} > 0.

Substituting (3.7) in (3.5) we have 1 2

d

dtkΦk^{2}+α1k∆Φk^{2}≤0. (3.8)
Upon integrating (3.8) in time over [0, t] and using Lemma 2.4 we obtain (3.3).

This completes the proof.

Remark 3.2. The estimate (3.8) already implies the decay of kΦk^{2}. However,
our ultimate goal is to show the decay rate of the higher order derivatives of the
solution. Hence, for the sake of completeness, we leave the proof of the decay rate
in the next section.

Next, we prove uniform estimates of Θ, which will be used to settle down the
uniform bound ofkUk_{H}1.

Lemma 3.3. Under the assumptions of Theorem 1.2, there exists a constant β_{0}
independent oft such that for anyt≥0, it holds that

kΘ(·, t)k^{2}≤ kΘ0k^{2}e^{−2β}^{0}^{t},
Z t

0

k∇Θ(·, τ)k^{2}e^{β}^{0}^{τ}dτ ≤ 1

κkΘ0k^{2}. (3.9)
Proof. Taking theL^{2}inner product of (3.1)4 with Θ we have

1 2

d

dtkΘk^{2}+κk∇Θk^{2}= 0. (3.10)

Since Θ|∂Ω= 0, Poincar´e’s inequality implies d

dtkΘk^{2}+2κ

c_{0}kΘk^{2}≤0, (3.11)

which yields immediately

kΘ(·, t)k^{2}≤ kΘ0k^{2}e^{−2β}^{0}^{t}, (3.12)
whereβ_{0}=κ/c_{0}. This proves the first part of (3.9).

Next, we multiply (3.10) bye^{β}^{0}^{t}and use (3.12) to obtain
d

dt e^{β}^{0}^{t}kΘk^{2}

+ 2κe^{β}^{0}^{t}k∇Θk^{2}≤β_{0}e^{−β}^{0}^{t}kΘ_{0}k^{2}. (3.13)

For anyt >0, upon integrating (3.13) in time we obtain
e^{β}^{0}^{t}kΘ(·, t)k^{2}− kΘ0k^{2}+ 2κ

Z t

0

e^{β}^{0}^{τ}k∇Θ(·, τ)k^{2}dτ ≤ 1−e^{−β}^{0}^{t}

kΘ0k^{2}, (3.14)
which implies the second part of (3.9) immediately. This completes the proof.

With the help of Lemma 3.3 we now prove the uniform estimates ofkUkH^{1} and
kΦkH^{1}.

Lemma 3.4. Under the assumptions of Theorem 1.2, for allt≥0,
kU(·, t)k^{2}_{H}1+kΦ(·, t)k^{2}_{H}1+

Z t

0

k∇µ(·, τ)k^{2}+kΦ(·, τ)k^{2}_{H}3

dτ ≤c10. (3.15)
Proof. Step 1. Note that due to Lemma 2.3 and the boundary condition onU, it
suffices to estimate kUk^{2} and kωk^{2}, in order estimate kU(·, t)k^{2}_{H}1. Taking the L^{2}
inner product of (3.1)3 withU we have

1 2

d

dtkUk^{2}=
Z

Ω

µ(∇Φ·U)dx+ Z

Ω

Θe_{2}·U dx. (3.16)
TakingL^{2} inner product of (3.1)_{1}withµwe have

d dt

α

2k∇Φk^{2}+
Z

Ω

F(φ)dx

+k∇µk^{2}=−
Z

Ω

µ(∇Φ·U)dx. (3.17) Adding (3.16) and (3.17), we obtain

d dt

1

2kUk^{2}+α

2k∇Φk^{2}+
Z

Ω

F(φ)dx

+k∇µk^{2}=
Z

Ω

Θe_{2}·U dx. (3.18)
Applying Cauchy-Schwarz inequality to the right-hand side of (3.18) and using
(3.12), we obtain

d dt

1

2kUk^{2}+α

2k∇Φk^{2}+
Z

Ω

F(φ)dx

+k∇µk^{2}≤e^{−β}^{0}^{t}kUk^{2}+e^{−β}^{0}^{t}kΘ0k^{2}. (3.19)
After droppingk∇µk^{2}from the left hand side (LHS) of (3.19), we have

d dt

1

2kUk^{2}+α

2k∇Φk^{2}+
Z

Ω

F(φ)dx

≤e^{−β}^{0}^{t}kUk^{2}+e^{−β}^{0}^{t}kΘ0k^{2}. (3.20)
SinceF ≥0, Gronwall’s inequality then gives

1

2kUk^{2}+α

2k∇Φk^{2}+
Z

Ω

F(φ)dx≤c11, ∀t≥0. (3.21) Applying (3.21) to (3.19) and integrating with respect tot we obtain

kUk^{2}+kΦk^{2}_{H}1+
Z

Ω

F(φ)dx+ Z t

0

k∇µ(·, τ)k^{2}dτ ≤c12, ∀t≥0. (3.22)
Step 2. By the definition of Φ, Lemma 2.4 and (3.1)2, we observe that

kΦk^{2}_{H}3 ≤c8k∇(∆Φ)k^{2}≤c13 k∇µk^{2}+kF^{00}(φ)∇Φk^{2}

. (3.23)

Using the condition onF, H¨older inequality, Lemma 2.1 (iii) and (3.22) we have
kF^{00}(φ)∇Φk^{2}≤c14 kφk^{2(p−2)}_{L}4(p−2)k∇Φk^{2}_{L}4+k∇Φk^{2}

≤c_{15} (kΦk^{2(p−2)}_{H}_{1} +kφk¯ ^{2(p−2)}_{H}_{1} )k∇Φk^{2}_{H}1+k∇Φk^{2}

≤c_{16}kΦk^{2}_{H}2.

(3.24)

substituting (3.24) in (3.23) we have

kΦk^{2}_{H}3≤c_{17} k∇µk^{2}+kΦk^{2}_{H}2

, (3.25)

which, together with (3.3) and (3.22), implies Z t

0

kΦ(·, τ)k^{2}_{H}3dτ ≤c18, ∀t≥0. (3.26)
Step 3. By taking the curl of the velocity equation, we obtain

ωt+U· ∇ω=µxΦy−µyΦx+ Θx, (3.27)
where ω =v_{x}−u_{y} is the 2D vorticity. Taking L^{2} inner product of (3.27) withω
and applying H¨older inequality we have

d

dtkωk ≤2k∇µkk∇ΦkL^{∞}+k∇Θk. (3.28)
Upon integrating (3.28) in time using H¨older and Sobolev inequalities we have

kω(·, t)k ≤ Z t

0

2k∇µkk∇ΦkL^{∞}+k∇Θk

dτ +kω0k

≤c19

Z t

0

k∇µk^{2}dτ1/2Z t
0

kΦk^{2}_{H}3dτ1/2

+Z t 0

e^{β}^{0}^{τ /2}k∇Θk^{2}dτ^{1/2}Z t
0

e^{−β}^{0}^{τ /2}dτ^{1/2}

+kω0k.

(3.29)

Since the right hand side of (3.29) is uniformly bounded in time by virtue of previous estimates, we have

kω(·, t)k ≤c20, ∀t≥0. (3.30) Thus, (3.15) follows from (3.22), (3.26) and (3.30). This completes the proof.

With the aid of Lemma 3.4 we are now able to improve the estimates of Φ andµ.

Due to the lack of spatial derivatives of the solution on∂Ω, we shall alternatively work on the temporal derivatives and use an iteration program to recover the spatial derivatives.

Lemma 3.5. Under the assumptions of Theorem 1.2, it holds that

kΦ(·, t)k^{2}_{H}2+kµ(·, t)k^{2}+
Z t

0

kΦtk^{2}+k∇µk^{2}_{H}1

dτ ≤c21, ∀t≥0. (3.31)
Proof. Step 1. By takingL^{2} inner product of (3.1)1 with Φt we have

kΦtk^{2}+
Z

Ω

Φt(U · ∇Φ)dx= Z

Ω

Φt∆µdx. (3.32)

Using the boundary conditions we calculate the RHS of (3.32) as:

Z

Ω

Φ_{t}∆µdx=−d
dt

α

2k∆Φk^{2}+1
2

Z

Ω

F^{00}(φ)|∇Φ|^{2}dx
+1

2 Z

Ω

F^{000}(φ)Φ_{t}|∇Φ|^{2}dx.

(3.33) Substituting (3.33) in (3.32) we obtain

d dt

α

2k∆Φk^{2}+1
2
Z

Ω

F^{00}(φ)|∇Φ|^{2}dx

+kΦtk^{2}

= 1 2 Z

Ω

F^{000}(φ)Φt|∇Φ|^{2}dx−
Z

Ω

Φt(U· ∇Φ)dx.

(3.34)

Using Cauchy-Schwarz inequality and Lemma 3.4 we estimate the first term on the RHS of (3.34) as

1 2

Z

Ω

F^{000}(φ)Φt|∇Φ|^{2}dx
≤1

4kΦtk^{2}+1
4

Z

Ω

|F^{000}(φ)|^{2}|∇Φ|^{4}dx

≤1

4kΦtk^{2}+c22kφk^{2(p−3)}_{L}4(p−3)k∇Φk^{4}_{L}8+c22k∇Φk^{4}_{L}4

≤1

4kΦtk^{2}+c23 k∇Φk^{4}_{L}8+k∇Φk^{4}_{L}4

.

(3.35)

Lemma 2.1 (iii)–(v) and Lemma 3.4 then give

k∇Φk^{4}_{L}4+k∇Φk^{4}_{L}8≤c24 k∇Φk^{2}k∇^{2}Φk^{2}+k∇Φk^{4}+k∇Φk^{2}k∇^{2}Φk^{2}_{L}4+k∇Φk^{4}

≤c25 k∇^{2}Φk^{2}+k∇Φk^{2}+k∇^{2}Φk^{2}_{H}1

≤c26kΦk^{2}_{H}3.

(3.36) So we update (3.35) as

1 2

Z

Ω

F^{000}(φ)Φ_{t}|∇Φ|^{2}dx
≤1

4kΦ_{t}k^{2}+c_{27}kΦk^{2}_{H}3. (3.37)
The second term on the RHS of (3.34) is estimated as

−

Z

Ω

Φt(U · ∇Φ)dx ≤ 1

4kΦtk^{2}+c28kUk^{2}_{H}1k∇Φk^{2}_{H}1

≤ 1

4kΦ_{t}k^{2}+c_{29}kΦk^{2}_{H}2,

(3.38)

where we have used Lemma 3.4. Combining (3.34), (3.37) and (3.38) we have d

dt α

2k∆Φk^{2}+1
2

Z

Ω

F^{00}(φ)|∇Φ|^{2}dx
+1

2kΦtk^{2}≤c30kΦk^{2}_{H}3. (3.39)
Upon integrating (3.39) in time and using (3.26) we have

α

2k∆Φk^{2}+1
2

Z

Ω

F^{00}(φ)|∇Φ|^{2}dx+1
2

Z t

0

kΦtk^{2}dτ ≤c31. (3.40)
SinceF^{00}≥ −F3, we have

Z

Ω

F^{00}(φ)|∇Φ|^{2}dx≥ −F3k∇Φk^{2}≥ −F3c0k∆Φk^{2}, (3.41)
where we have used (3.6). Substituting (3.41) in (3.40) we have

α1

2 k∆Φk^{2}+1
2

Z t

0

kΦtk^{2}dτ ≤c31,
which, together with Lemma 2.4, implies

kΦk^{2}_{H}2+
Z t

0

kΦtk^{2}dτ ≤c32. (3.42)
Step 2. We derive some consequences of (3.42). From (3.1)_{2} and Lemma 2.1
(i) we see that

kµ(·, t)k^{2}≤c33 k∆Φk^{2}+kF^{0}(φ)k^{2}

≤c34 k∆Φk^{2}+kφk^{2(p−1)}_{L}∞ + 1

≤c_{35} k∆Φk^{2}+kΦk^{2(p−1)}_{H}_{2} +kφk¯ ^{2(p−1)}_{L}∞ + 1
.

(3.43)

Therefore, using (3.42) we have

kµ(·, t)k^{2}≤c_{36}, ∀t≥0. (3.44)
Since∇µ·n|∂Ω= 0, by Lemma 2.4 and Lemma 3.4 we have

k∇µk^{2}_{H}1 ≤c37k∆µk^{2}≤c38 kΦtk^{2}+kU · ∇Φk^{2}

≤c39 kΦtk^{2}+kUk^{2}_{H}1k∇Φk^{2}_{H}1

≤c40 kΦtk^{2}+kΦk^{2}_{H}2

,

(3.45)

which, together with (3.15) and (3.42), implies that Z t

0

k∇µk^{2}_{H}1dτ ≤c41. (3.46)

Therefore, (3.31) follows from (3.42), (3.44) and (3.46). This completes the proof.

The next lemma gives the uniform estimate of kUt(·, t)k^{2} whose proof requires
more careful examination of the energy estimate for the temperature.

Lemma 3.6. Under the assumptions of Theorem 1.2, there exists a constantβ1>0 independent oft such that

e^{β}^{1}^{t}kΘ(·, t)k^{2}_{H}1+
Z t

0

e^{β}^{1}^{τ /2}kΘ(·, τ)k^{2}_{H}2dτ ≤c_{42}, ∀t≥0. (3.47)
Proof. Step 1. TakingL^{2} inner product of (3.1)4 with Θt we have

κ 2

d

dtk∇Θk^{2}+kΘ_{t}k^{2}≤ kU· ∇Θk^{2}+1

4kΘ_{t}k^{2}. (3.48)
Using Lemma 3.4 we have

kU · ∇Θk^{2}≤c43kUk^{2}_{H}1k∇Θk^{2}_{L}4 ≤c44k∇Θk^{2}_{L}4. (3.49)
So we update (3.48) as

κ 2

d

dtk∇Θk^{2}+3

4kΘtk^{2}≤c44k∇Θk^{2}_{L}4. (3.50)
The estimate of the RHS of (3.50) is tricky. First, applying Lemma 2.1 (iv) to

∇Θ to obtain

c44k∇Θk^{2}_{L}4 ≤c45 k∇ΘkkD^{2}Θk+k∇Θk^{2}

≤c46(δ)k∇Θk^{2}+δkD^{2}Θk^{2}, (3.51)
where δ is a number to be determined. Since Θ|∂Ω = 0, by the elliptic estimate
(c.f. Lemma 2.2), we have

kΘk^{2}_{H}2 ≤c_{47} kΘtk^{2}+kU· ∇Θk^{2}

. (3.52)

For the second term on the RHS of (3.52), we use (3.49) and (3.51) to get
kU· ∇Θk^{2}≤c48 k∇ΘkkD^{2}Θk+k∇Θk^{2}

. (3.53)

Then, using Cauchy-Schwarz inequality we update (3.52) as
kΘk^{2}_{H}2 ≤c_{49} kΘtk^{2}+k∇ΘkkD^{2}Θk+k∇Θk^{2}

≤c_{50} kΘ_{t}k^{2}+k∇Θk^{2}
+1

2kΘk^{2}_{H}2, (3.54)
which implies

kΘk^{2}_{H}2≤c_{51} kΘtk^{2}+k∇Θk^{2}

. (3.55)

By choosingδ= 1/(4c51) in (3.51), and coupling the result with (3.55) we have
c44k∇Θk^{2}_{L}4 ≤c52k∇Θk^{2}+1

4kΘtk^{2}. (3.56)
Combining (3.50) and (3.56) we obtain

κ 2

d

dtk∇Θk^{2}+1

2kΘtk^{2}≤c52k∇Θk^{2}. (3.57)
Step 2. We multiply (3.10) by 2c52/κand add the result to (3.57) to obtain

d dt

c_{52}

κ kΘk^{2}+κ

2k∇Θk^{2}

+c52k∇Θk^{2}+1

2kΘtk^{2}≤0. (3.58)
It is clear that, by Poincar´e inequality, there exists a constantc53>0 such that

c53

c52

κ kΘk^{2}+κ

2k∇Θk^{2}

≤c52k∇Θk^{2}. (3.59)
Substituting (3.59) in (3.58) we have

d dt

c52

κ kΘk^{2}+κ

2k∇Θk^{2}
+c53

c52

κ kΘk^{2}+κ

2k∇Θk^{2}
+1

2kΘtk^{2}≤0, (3.60)
which implies (by dropping ^{1}_{2}kΘtk^{2}from the LHS) that

c52

κ kΘ(·, t)k^{2}+κ

2k∇Θ(·, t)k^{2}

≤c52

κ kΘ0k^{2}+κ

2k∇Θ0k^{2}

e^{−c}^{53}^{t}. (3.61)
Therefore, fort≥0,

kΘ(·, t)k^{2}_{H}1≤

min{c_{52}/κ, κ/2}−1c52

κ kΘ_{0}k^{2}+κ

2k∇Θ_{0}k^{2}

e^{−c}^{53}^{t}. (3.62)
Using (3.58) and (3.61), by repeating the same procedure as in Lemma 3.3, we
have

e^{c}^{53}^{t/2}c_{52}

κ kΘ(·, t)k^{2}+κ

2k∇Θ(·, t)k^{2}
+

Z t

0

e^{c}^{53}^{τ /2}

c52k∇Θ(·, τ)k^{2}+1

2kΘt(·, τ)k^{2}
dτ

≤2c_{52}

κ kΘ0k^{2}+κ

2k∇Θ0k^{2}
,

(3.63)

which yields Z t

0

e^{c}^{53}^{τ /2}

c52k∇Θ(·, τ)k^{2}+1

2kΘt(·, τ)k^{2}

dτ ≤2c52

κ kΘ0k^{2}+κ

2k∇Θ0k^{2}
. (3.64)
In view of (3.55) we see that

Z t

0

e^{c}^{53}^{τ /2}kΘ(·, τ)k^{2}_{H}2dτ ≤c_{54}. (3.65)
Therefore, (3.47) follows from (3.62) and (3.65). This completes the proof.

With the help of Lemma 3.6, we have the following result.

Lemma 3.7. Under the assumptions of Theorem 1.2, for allt≥0,

kµ(·, t)k^{2}_{H}1+kΦt(·, t)k^{2}+kU(·, t)k^{2}_{W}1,4+kU(·, t)k^{2}_{L}∞+kUt(·, t)k^{2}≤c_{55}. (3.66)

Proof. Step 1. Taking L^{2} inner product of (3.27) with |ω|^{2}ω and using H¨older
inequality we have

d

dtkωkL^{4} ≤2k∇µkL^{8}k∇ΦkL^{8}+k∇ΘkL^{4}≤c56(k∇µkH^{1}kΦkH^{2}+kΘkH^{2}). (3.67)
Integrating (3.67) in time and using the previous lemmas, we have

kω(·, t)k_{L}4≤c56

Z t

0

k∇µk^{2}_{H}1dτ1/2Z t
0

kΦk^{2}_{H}2dτ1/2

+c56

Z t

0

e^{−β}^{1}^{τ /2}dτ^{1/2}Z t
0

e^{β}^{1}^{τ /2}kΘk^{2}_{H}2dτ^{1/2}

+kω0k_{L}4

≤c57,

(3.68)

which, together with Lemma 2.3 and Sobolev embedding, implies

kU(·, t)k_{W}1,4+kU(·, t)kL^{∞} ≤c58, ∀t≥0. (3.69)
For the estimate ofkUtk^{2}, taking L^{2} inner product of (3.1)3withUtwe have
kUtk^{2}≤c59 kUk^{2}_{L}∞k∇Uk^{2}+kµk^{2}_{H}1kΦk^{2}_{H}2+kΘ0k^{2}

≤c60 k∇µk^{2}+ 1

, (3.70) where we have used (3.42), (3.44) and (3.69).

Step 2. We now deal withµand Φt. TakingL^{2} inner product of (3.1)_{1}withµt

we have 1 2

d

dtk∇µk^{2}+αk∇Φtk^{2}=−
Z

Ω

F^{00}(φ)Φ^{2}_{t}+µt(U· ∇Φ)

dx. (3.71) From (3.42) and Sobolev embedding we know that

kΦkL^{∞}≤c61, (3.72)

which according to the condition onF implies
kF^{(p−n)}(φ)kL^{∞} ≤F1c(p, n) kΦk^{p−n}_{L}∞ +kφk¯ ^{p−n}_{L}∞

+F2≤c62, n= 1, . . . . ,6. (3.73) Using (3.73) we estimate the RHS of (3.71) as follows:

−

Z

Ω

F^{00}(φ)Φ^{2}_{t}+µt(U · ∇Φ)
dx

≤ kF^{00}(φ)k_{L}^{∞}kΦ_{t}k^{2}+ 2αkUk^{2}_{L}∞k∇Φk^{2}+ 1
8αkµ_{t}k^{2}

≤c_{62}kΦ_{t}k^{2}+c_{63}k∇Φk^{2}+ 1

8α 2α^{2}k∆Φ_{t}k^{2}+ 2kF^{00}(φ)k^{2}_{L}∞kΦ_{t}k^{2}

≤c64kΦtk^{2}+c65k∆Φk^{2}+α

4k∆Φtk^{2},

where we have used (3.69) and Lemma 2.4. We update (3.71) as 1

2 d

dtk∇µk^{2}+αk∇Φtk^{2}≤c64kΦtk^{2}+c65k∆Φk^{2}+α

4k∆Φtk^{2}. (3.74)
Differentiating (3.1)1 with respect totwe have

Φtt+Ut· ∇Φ +U· ∇Φt= ∆µt. (3.75)
TakingL^{2} inner product of (3.75) with Φtwe have

1 2

d

dtkΦtk^{2}+αk∆Φtk^{2}=
Z

Ω

F^{00}(φ)Φt∆Φtdx+
Z

Ω

Φ(Ut· ∇Φt)dx. (3.76)

Using (3.73) and (3.70) we estimate the RHS of (3.76) as follows:

Z

Ω

F^{00}(φ)Φt∆Φtdx+
Z

Ω

Φ(Ut· ∇Φt)dx

≤ α

4k∆Φtk^{2}+1

αkF^{00}(φ)k^{2}_{L}∞kΦtk^{2}+ 1

2αkΦk^{2}_{L}∞kUtk^{2}+α

2k∇Φtk^{2}

≤ α

4k∆Φtk^{2}+c66kΦtk^{2}+c67kΦk^{2}_{H}2(k∇µk^{2}+ 1) +α

2k∇Φtk^{2}

≤ α

4k∆Φtk^{2}+c66kΦtk^{2}+c68 k∇µk^{2}+k∆Φk^{2}
+α

2k∇Φtk^{2}.

(3.77)

So we update (3.76) as 1

2 d

dtkΦ_{t}k^{2}+3α

4 k∆Φ_{t}k^{2}≤c_{66}kΦ_{t}k^{2}+c_{68} k∇µk^{2}+k∆Φk^{2}
+α

2k∇Φ_{t}k^{2}. (3.78)
Combining (3.74) and (3.78) we obtain

d dt

k∇µk^{2}+kΦtk^{2}
+α

k∇Φtk^{2}+k∆Φtk^{2}

≤c69 kΦtk^{2}+k∇µk^{2}+k∆Φk^{2}
. (3.79)
After integrating (3.79) in time and using (3.3) and (3.31) we have

k∇µ(·, t)k^{2}+kΦt(·, t)k^{2}+
Z t

0

k∇Φtk^{2}+k∆Φtk^{2}

dτ ≤c_{70}, ∀t≥0. (3.80)
Substituting (3.80) in (3.70) we have kUt(·, t)k^{2} ≤c71. This completes the proof.

As consequences of previous lemmas, we have the following result.

Lemma 3.8. Under the assumptions of Theorem 1.2, it holds

kΦ(·, t)k^{2}_{H}4+kµ(·, t)k^{2}_{H}2 ≤c72, ∀t≥0. (3.81)
Proof. First, by (3.45) we have

kµ(·, t)k^{2}_{H}2 ≤c40 kΦt(·, t)k^{2}+kΦ(·, t)k^{2}_{H}2

+kµ(·, t)k^{2}. (3.82)
Then the uniform estimate ofkµ(·, t)k^{2}_{H}2 follows from Lemma 3.5 and Lemma 3.7.

Second, by Lemma 2.4 and (3.1)2we have

kΦ(·, t)k^{2}_{H}4≤c73 kµ(·, t)k^{2}_{H}2+kF^{0}(φ)(·, t)k^{2}_{H}2

. (3.83)

Using the second condition (H2) on F, (3.31) and (3.73), it is straightforward
to show that kF^{0}(φ)(·, t)k^{2}_{H}2 ≤ c74, which together with the uniform bound of
kµ(·, t)k^{2}_{H}2 imply thatkΦ(·, t)k^{2}_{H}4 ≤c_{75}. This completes the proof.

4. Large time asymptotic behavior

In this section we prove Theorem 1.2, based on a sequence of accurate combina- tions of energy estimates. For the convenience of the reader, we first collect some uniform-in-time estimates. From (3.31), (3.66) and (3.81) we have, for anyt≥0:

kΦk^{2}_{H}4+kµk^{2}_{H}2+kΦtk^{2}+kF^{(p−n)}(φ)k^{2}_{L}∞+kUk^{2}_{W}1,4+kUk^{2}_{L}∞+kUtk^{2}

(t)≤c_{76}.
(4.1)

4.1. Decay of (Φ, µ). Step 1. First, by (3.1)2we have

kµ−F^{0}( ¯φ)k^{2}_{H}3 =k −α∆Φ +F^{0}(φ)−F^{0}( ¯φ)k^{2}+k∇µk^{2}_{H}2

≤2α^{2}k∆Φk^{2}+ 2kF^{0}(φ)−F^{0}( ¯φ)k^{2}+k∇µk^{2}_{H}2. (4.2)
Using (4.1) and Lemma 2.3 we estimate the last two terms on the RHS of (4.2) as
follows:

2kF^{0}(φ)−F^{0}( ¯φ)k^{2}+k∇µk^{2}_{H}2

≤2kF^{00}(ζ)k^{2}_{L}∞kΦk^{2}+c77k∇∆µk^{2}

≤c78kΦk^{2}+c79 k∇Φtk^{2}+k∇(U· ∇Φ)k^{2}

≤c78kΦk^{2}+c80 k∇Φtk^{2}+k∇Uk^{2}k∇Φk^{2}_{∞}+kUk^{2}_{∞}k∇^{2}Φk^{2}

≤c81 k∇Φtk^{2}+kΦk^{2}_{H}3

.

(4.3)

Combining (4.2) and (4.3) we have

kµ−F^{0}( ¯φ)k^{2}_{H}3 ≤c_{82} k∇Φtk^{2}+kΦk^{2}_{H}3

. (4.4)

Combining (3.25) and (4.4) we then have

kµ−F^{0}( ¯φ)k^{2}_{H}3≤c83 kΦtk^{2}_{H}1+kΦk^{2}_{H}2+k∇µk^{2}

. (4.5)

Second, by Lemma 2.4 and (3.1)2we have

kΦk^{2}_{H}5 ≤c8k∇∆^{2}Φk^{2}≤c84k∇∆µk^{2}+c85k∇∆F^{0}(φ)k^{2}. (4.6)
By direct calculations and Sobolev embeddings we can show that

k∇F^{0}(φ)k^{2}_{H}2 ≤c86kF^{000}(φ)k^{2}_{L}∞ k∇Φk^{4}_{H}1+ 2k∇Φk^{2}_{H}1k∇Φk^{2}_{H}2

+c87kF^{0000}(φ)k^{2}_{L}∞k∇Φk^{4}_{H}2k∇Φk^{2}. (4.7)
Using (4.1) we obtain from (4.7) thatk∇F^{0}(φ)k^{2}_{H}2 ≤c88k∇Φk^{2}_{H}2, which, together
with (4.6), implies that

kΦk^{2}_{H}5≤c89 k∇Φtk^{2}+kΦk^{2}_{H}3

. (4.8)

Combining (4.5), (4.8) and (3.25) we have

kµ−F^{0}( ¯φ)k^{2}_{H}3+kΦk^{2}_{H}5 ≤c90 kΦtk^{2}_{H}1+kΦk^{2}_{H}2+k∇µk^{2}

. (4.9)

Therefore, it suffices to show the decay of RHS of (4.9) in order to prove the decay of Φ andµ.

Step 2. We recall (3.17), d

dt α

2k∇Φk^{2}+
Z

Ω

F(φ)dx

+k∇µk^{2}=−
Z

Ω

µ(∇Φ·U)dx. (4.10) Due to the structure of the function F(·), there may be a constant term in the integralR

ΩF(φ)dxin general, which is impossible to decay. In order to resolve this issue, we observe, sinceR

Ω(φ−φ)dx= 0, it holds that Z

Ω

F(φ)−F(φ)dx= Z

Ω

F^{0}(φ)(φ−φ)dx+1
2

Z

Ω

F^{00}(ξ)(φ−φ)^{2}dx

=1 2

Z

Ω

F^{00}(ξ)Φ^{2}dx, for someξ betweenφand ¯φ.

Then we update (4.10) as d

dt α

2k∇Φk^{2}+1
2
Z

Ω

F^{00}(ξ)Φ^{2}dx

+k∇µk^{2}=−
Z

Ω

µ(∇Φ·U)dx. (4.11) Using (4.1) and Lemma 2.4 we estimate the RHS of (4.11) as

−

Z

Ω

µ(∇Φ·U)dx =

Z

Ω

ΦU· ∇µdx

≤ 1

2k∇µk^{2}+1

2kUk^{2}_{L}4kΦk^{2}_{L}4

≤ 1

2k∇µk^{2}+c91kΦk^{2}_{H}1

≤ 1

2k∇µk^{2}+c92k∆Φk^{2}.

(4.12)

Substituting (4.12) in (4.11), we have d

dt

αk∇Φk^{2}+
Z

Ω

F^{00}(ξ)Φ^{2}dx

+k∇µk^{2}≤c93k∆Φk^{2}. (4.13)
Step 3. Recalling (3.34) and using (4.1) and Lemma 2.4 we have

d dt

α

2k∆Φk^{2}+1
2

Z

Ω

F^{00}(φ)|∇Φ|^{2}dx

+kΦtk^{2}

=1 2

Z

Ω

F^{000}(φ)Φt|∇Φ|^{2}dx−
Z

Ω

Φt(U· ∇Φ)dx

≤1

2kΦ_{t}k^{2}+c_{94}(k∇Φk^{4}_{H}1+k∇Φk^{2}_{H}1)

≤1

2kΦ_{t}k^{2}+c_{95}k∆Φk^{2},
which yields

d dt

αk∆Φk^{2}+
Z

Ω

F^{00}(φ)|∇Φ|^{2}dx

+kΦ_{t}k^{2}≤c_{96}k∆Φk^{2}. (4.14)
Step 4. From (3.77) and (4.1) we have

1 2

d

dtkΦ_{t}k^{2}+αk∆Φ_{t}k^{2}≤ α

4k∆Φ_{t}k^{2}+α

2k∇Φ_{t}k^{2}+c_{97}(kΦ_{t}k^{2}+k∆Φk^{2}). (4.15)
Combining (3.74) and (4.15), we have

d

dt k∇µk^{2}+kΦtk^{2}

+α k∇Φtk^{2}+k∆Φtk^{2}

≤c98 kΦtk^{2}+k∆Φk^{2}

. (4.16)
Step 5. TakingL^{2} inner product of (3.75) withµ_{t}we have

α 2

d

dtk∇Φtk^{2}+k∇µtk^{2}=
Z

Ω

(UtΦ +UΦt)· ∇µtdx− Z

Ω

F^{00}(φ)ΦtΦttdx. (4.17)
For the last term on the RHS of (4.17), we have

− Z

Ω

F^{00}(φ)ΦtΦttdx=−1
2

d dt

Z

Ω

F^{00}(φ)Φ^{2}_{t}dx+1
2

Z

Ω

F^{000}(φ)Φ^{3}_{t}dx.

So we update (4.17) as d dt

αk∇Φtk^{2}+
Z

Ω

F^{00}(φ)Φ^{2}_{t}dx

+ 2k∇µtk^{2}

= 2 Z

Ω

(UtΦ +UΦt)· ∇µtdx+ Z

Ω

F^{000}(φ)Φ^{3}_{t}dx.

(4.18)

Using (4.1) and Lemma 3.6 we estimate the first two terms on the RHS of (4.18) as

2

Z

Ω

(UtΦ +UΦt)· ∇µtdx

≤ k∇µtk^{2}+kUtk^{2}kΦk^{2}_{L}∞+kUk^{2}_{L}∞kΦtk^{2}

≤ k∇µ_{t}k^{2}+c_{99}kΦk^{2}_{H}2+c_{76}kΦ_{t}k^{2}

≤ k∇µ_{t}k^{2}+c_{100}k∆Φk^{2}+c_{101}k∆Φ_{t}k^{2}.

(4.19)

Similarly, for the term involving Φ^{3}_{t}, we have

Z

Ω

F^{000}(φ)Φ^{3}_{t}dx

≤ kF^{000}(φ)k_{∞}kΦtk^{3}_{L}3

≤c_{76}kΦtk^{2}_{∞}kΦtkL^{1}

≤c102k∆Φtk^{2}kΦtk ≤c103k∆Φtk^{2}.

(4.20)

Substituting (4.19) and (4.20) in (4.18), we have d

dt

αk∇Φtk^{2}+
Z

Ω

F^{00}(φ)Φ^{2}_{t}dx

+k∇µtk^{2}≤c104 k∆Φtk^{2}+k∆Φk^{2}

. (4.21) Step 6. In this step, we make combinations of energy estimates, which will be used to prove the exponential decay of Φ andµ. First, we collect energy inequalities from Steps 2–5:

(4.13) _{dt}^{d}

αk∇Φk^{2}+R

ΩF^{00}(ξ)Φ^{2}dx

+k∇µk^{2}≤c_{93}k∆Φk^{2}
(4.14) _{dt}^{d}

αk∆Φk^{2}+R

ΩF^{00}(φ)|∇Φ|^{2}dx

+kΦtk^{2}≤c96k∆Φk^{2}
(4.16) _{dt}^{d} k∇µk^{2}+kΦ_{t}k^{2}

+α k∇Φ_{t}k^{2}+k∆Φ_{t}k^{2}

≤c_{98} kΦ_{t}k^{2}+k∆Φk^{2}
(4.21) _{dt}^{d}

αk∇Φtk^{2}+R

ΩF^{00}(φ)Φ^{2}_{t}dx

+k∇µtk^{2}≤c_{104} k∆Φtk^{2}+k∆Φk^{2}
First, multiply (4.16) by ^{2c}_{α}^{104} and then add (4.21) to obtain

d dt

J_{0}(t)

+K_{0}(t)≤c_{105} kΦ_{t}k^{2}+k∆Φk^{2}

, (4.22)

where

J0(t)≡ 2c104

α k∇µk^{2}+kΦtk^{2}

+αk∇Φtk^{2}+
Z

Ω

F^{00}(φ)Φ^{2}_{t}dx,
K_{0}(t)≡c_{104} 2k∇Φ_{t}k^{2}+k∆Φ_{t}k^{2}

+k∇µ_{t}k^{2}.
Second, multiply (4.14) by 2c105then add (4.22) to obtain

d dt

J_{1}(t)

+K_{1}(t)≤c_{106}k∆Φk^{2}, (4.23)
where

J1(t)≡J0(t) + 2c105

αk∆Φk^{2}+
Z

Ω

F^{00}(φ)|∇Φ|^{2}dx
,
K_{1}(t)≡c_{105}kΦ_{t}k^{2}+K_{0}(t).