Insensitizing controls for the Cahn–Hilliard type equation

2014, No. 35, 1–22;

Insensitizing controls for the Cahn–Hilliard type equation

Peng Gao

Institute of Mathematics, Jilin University, 2699 Qianjin Street, Changchun, 130012, P. R. China Received 30 December 2013, appeared 29 July 2014

Communicated by Vilmos Komornik

Abstract. This paper is addressed to showing the existence of insensitizing controls for the one-dimensional Cahn–Hilliard type equation with a superlinear nonlinearity. We solve this problem by reducing the original problem to a controllability problem. The crucial point in this paper is an observability estimate for a linearized cascade system of the Cahn–Hilliard type equation. In order to obtain this observability estimate, we establish a global Carleman estimate for a fourth order parabolic operator.

Keywords: Cahn–Hilliard type equation, insensitizing control, controllability, observ- ability, superlinear nonlinearity, global Carleman estimate.

2010 Mathematics Subject Classification: 35K35, 93B05, 93B07.

1 Introduction

Set I = (0, 1), T > 0, and Q = I ×(0,T). Let ω and O be nonempty open subsets of I. We consider the Cahn–Hilliard type equation posed on the finite interval I satisfying some homogeneous boundary conditions and an initial condition, namely















y_t+y_xxxx+ f(y) =ξ+hχω inQ, y(0,t) =0=y(1,t) in(0,T), y_x(0,t) =0=y_x(1,t) in(0,T), y(x, 0) =y⁰(x) +τz⁰(x) in I,

(1.1)

where f is aC¹ function defined onRverifying f⁰⁰ ∈ L^∞_loc(R), f(0) =0 and

|slim|→_∞

f⁰(s)

log(1+|s|) =0, (1.2)

ξ ∈ L²(Q)and y⁰ ∈ L²(I) are given, z⁰ ∈ L²(I) is unknown with kz⁰k_L2(I) = 1, τ is a small unknown real number, andh∈L²(Q)is a control function to be determined. Hereχ_ωdenotes the characteristic function ofω.

BCorresponding author. Email: gaopengjilindaxue@126.com

The Cahn–Hilliard equation is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. It arises as a phenomenological model for isothermal phase separation in a binary alloy, see Cahn [7, 8] and Hilliard [20] for a derivation, [15,23,27] for general analysis, and the reviews given in [16].

Let us define

Φ(y(·,·,τ,h)) = ¹ 2

Z _T

0

Z

O |y(x,t,τ,h)|² dx dt, wherey(·_,·_,τ,h)is the solution of (1.1) associated toτ.

The following control problem is addressed: Does there exist a control h ∈ L²(Q) such

that dΦ(y(·_,·_,τ,h))

dτ

τ=0

=0

holds? The problem is interesting, and attracts many authors’ attention. We call it insensitiz- ing control problem. Next, we investigate the existence of insensitizing controls forΦ about the system (1.1), their definitions are as follows.

Definition 1.1. The control h is said to insensitize the functional Φ if for every z⁰ satisfying kz⁰k_L2(I) =1, the corresponding solutionyof (1.1) satisfies

d

dτΦ(y(·,·,τ,h)) τ=0

=0.

The insensitizing control problem consists in finding a control function such that some functional of the state is locally insensitive to the perturbations of these initial and boundary data. The concept of insensitizing control was introduced by J. L. Lions [21]. Later on, Bodart and Fabre proposed the weakened notion of ε−insensitizing control in [2]. A similar result was proved by Teresa [12] in unbounded domains. The first results on the existence and non- existence of insensitizing controls were proved in [13]. For more general nonlinearities, see [3,4,5]. A similar result for wave equations was obtained in [11,26].

The main purpose in our paper is to study the existence of insensitizing controls for the Cahn–Hilliard equation. As far as we know, there is no insensitivity result for this equation.

In this sense, this is the first attempt to consider insensitizing controls problem for the Cahn–

Hilliard equation. In order to solve this problem, we establish a new observability estimate (see Theorem1.2).

Following the methods introduced in [21] and developed in [2,11,13,26], one gets that the existence of a controlhinsensitizing the functionalΦalong the solutions of (1.1) is equivalent to the existence of a controlhsuch that the solution(y,q)of the cascade system (1.3)–(1.4)















y_t+y_xxxx+ f(y) =ξ+hχ_ω in Q, y(0,t) =0=y(1,t) in (0,T), y_x(_0,t) =₀= y_x(_1,t) _in (_0,T)_, y(x, 0) =y⁰(x) in I

(1.3)















−q_t+q_xxxx+ f⁰(y)q=yχ_O inQ, q(0,t) =0=q(1,t) in(0,T), q_x(_0,t) =₀=q_x(_1,t) _in(_0,T)_,

q(x,T) =0 in I

(1.4)

satisfies

q(x, 0) =0.

Namely, system (1.3)–(1.4) is null controllable. The null controllability has been widely in- vestigated for the heat equation and there has been a great number of results (see for in- stance [6, 22] and the references therein for a detailed account). To our best knowledge, there have been limited publications on the controllability of higher order parabolic equations.

Among them, Díaz [14] considered the approximate controllability and non-approximate con- trollability of higher order parabolic equations. The null boundary controllability for a one- dimensional fourth order parabolic equation was studied in [6,10]. Cerpa [9] considered the local boundary controllability for an especial one-dimensional fourth order parabolic equation (Kuramoto–Sivashinsky equation). Recently, Zhou [28] considered the null controllability for one-dimensional semilinear fourth order parabolic equations.

In order to investigate system (1.3)–(1.4), we firstly consider the linearized system of (1.3)–

(1.4)















yt+yxxxx+ay=ξ+hχω in Q, y(0,t) =0= y(1,t) in (0,T), yx(0,t) =0=yx(1,t) in (0,T), y(x, 0) =y⁰(x) _in I

(1.5)















−q_t+q_xxxx+bq=yχ_O inQ, q(0,t) =0=q(1,t) in(0,T), q_x(_0,t) =₀=q_x(_1,t) _in(_0,T)_,

q(x,T) =0 in I

(1.6)

where a,b∈ L^∞(Q).

The adjoint system of (1.5)–(1.6) is















p_t+p_xxxx+b(x,t)p=0 inQ, p(0,t) =0= p(1,t) in(0,T), px(_0,t) =₀= _px(_1,t) _in(_0,T)_, p(x, 0) = p⁰(x) in I

(1.7)















−zt+zxxxx+a(x,t)z = pχO in Q, z(0,t) =0=z(1,t) in (0,T), zx(0,t) =0=zx(1,t) in (0,T),

z(x,T) =_{0 in} I.

(1.8)

According to the duality argument, the observability estimate of (1.7)–(1.8) is important for the insensitizing control problem.

Theorem 1.2. For every p⁰ ∈ L²(I), if (p,z)is the solution to (1.7)–(1.8), there exist M > 0 and C(T)>0,such that

Z

Q

e^−Mt z²dx dt ≤e^C(T)(kakL∞(Q)+kbkL∞(Q)+1)^Z

ω×(0,T)

z²dx dt. (1.9)

The following duality identity for the solutions of (1.5)–(1.6) and (1.7)–(1.8) holds Z

Q

(ξ+hχω)z dx dt=

Z

I q(x, 0)p⁰(x)−z(x, 0)y⁰(x)dx (1.10) for everyh ∈ L²(Q),y⁰ ∈ L²(_Ω), p⁰ ∈ L²(_Ω). Indeed, multiplying (1.5) by z and integrating by parts in account of the boundary and initial (final) conditions in (1.5)–(1.6) and (1.7)–(1.8), we can get (1.10).

By the observability estimate (1.9) of the linearized system (1.5)–(1.6) and the fixed point theorem, we have the following result:

Theorem 1.3. LetOandωbe nonempty open subsets of I satisfyingω∩ O 6=_∅ and y⁰ =0.Then for anyξ ∈ L²(Q)verifying e^M2tξ ∈ L²(Q),one can find a control function h ∈ L²(Q)insensitizing the functionalΦalong the solution of (1.1), where M is same as in Theorem1.2.

Remark 1.4. In view of Theorem1.3, we can obtain the null controllability of (1.3)–(1.4) with the nonlinearities f(s) =o(s(log(|s|)))for |s| → ∞. For the scalar Cahn–Hilliard type equa-

tion 













yt+yxxxx+F(y) =hχω in Q, y(0,t) =0=y(1,t) in (0,T), y_x(0,t) =0=y_x(1,t) in (0,T), y(x, 0) =y⁰(x) in I

(1.11)

with nonlinearities such that F(s) = o(s(log⁷²(|s|)))for |s| → ∞, it seems possible to obtain the null controllability of (1.11). Indeed, following the same idea as in [18], we can choose a small timeT^∗< Tand find a controlhthat drives the solution to zero atT^∗, then extendhby zero to the rest interval[T^∗,T].

However, for the system (1.3)–(1.4), since the existence of the nonhomogeneous termξ, the above method does not work. More precisely, even though we can obtain the null control- lability of (1.3)–(1.4) at a small time T^∗, the zero control in [T^∗,T]cannot guarantee the null controllability of (1.3)–(1.4) atTowing toξ. According to the existent methods, the best result for the nonlinearities in (1.3)–(1.4) we can obtain is f(s) = o(s(log(|s|))) for |s| → _{∞. The} key point is the estimate (4.7) in Section 4. The same reason can also be found in [3] which considers the insensitizing controls for a heat equation.

The paper is organized as follows. In Section 2, we present some well-posedness results by the classical semigroup theory, multipliers method and suitable energy estimates. Then, we establish a Carleman estimate for the fourth order parabolic operator. The observability estimate is established in Section 3. In Section 4, by means of the variational approach, the observability estimate in the above section and Kakutani’s fixed point theorem, we establish the existence of insensitizing controls for the Cahn–Hilliard equation.

2 Some preliminaries

In order to prove Theorem 1.2, we should establish a global Carleman estimate for a fourth order parabolic operator.

Letψ∈ C^∞(_Ω)satisfy thatψ>0 in Ω,ψ(0) = ψ(1) =0,kψk_C(Ω) =1,|ψx |>0 in Ω\ω₀, ψ_x(0)>0 andψ_x(1)<0. For any given positive constantsλandµ, we setd(x,t) = ^{eµ(ψ(x)+3)−e5µ}

t(T−t) ,

θ(x,t) =e^λd(x,t)and ϕ(x,t) = ^eµ_t⁽₍^ψ_T^(x₋⁾⁺_t)³⁾,∀(x,t)∈ Q. LetPbe an operator Py:=y_t+y_xxxx,

defined on U := y ∈ L²(0,T;H⁴(I)) | y(t, 0) = y(t, 1) = yx(t, 0) = yx(t, 1) = 0, t ∈ (0,T), Py∈ L²(0,T;L²(I)) .

Proposition 2.1. There exist four constantsµ0>1, C0 >0,C₁ >0and C2 >0such that forµ=µ0

and for everyλ≥C₀(T+T²)and y ∈ U,we have Z

Q

1

λϕθ²(y²_t+y²_xxxx) +λϕθ²y²_xxx+λ³ϕ³θ²y²_xx+λ⁵ϕ⁵θ²y²_x+λ⁷ϕ⁷θ²y² dx dt

≤C_{1 Z}

Qθ²|Py|²dx dt+

Z

Q^ωλ⁷ϕ⁷θ²y²dx dt

.

(2.1)

Moreover, Z

Q

λ⁻¹t(T−t)θ²(y²_t +y²_xxxx) +λt⁻¹(T−t)⁻¹θ²y²_xxx+λ³t⁻³(T−t)⁻³θ²y²_xx +λ⁵t⁻⁵(T−t)⁻⁵θ²y²_x+λ⁷t⁻⁷(T−t)⁻⁷θ²y²

dx dt

≤C_{2 Z}

ω×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷y²dxdt+

Z

Qθ²|Py|²dx dt

.

(2.2)

Remark 2.2. A Carleman estimate for the fourth order parabolic operator was previously ob- tained in [28]. Our Carleman estimate is a generalization to the result in [28]. More precisely, we can also obtain the estimate forR

Q 1

λϕθ²(y²_t +y²_xxxx) +λϕθ²y²_xxx

dx dt. We only sketch the proof in the Appendix.

Now, we present a regularity result for the following system















y_t+y_xxxx+ay= g inQ, y(0,t) =0=y(1,t) in(0,T), y_x(_0,t) =₀=y_x(_1,t) _in(_0,T)_, y(x, 0) =y⁰(x) in I.

(2.3)

Proposition 2.3.

(i) If g∈ L²(0,T;L²(I)),a∈ L^∞(Q)and y⁰∈ L²(I),system(2.3)has a unique mild solution y in C([0,T];L²(I))∩L²(0,T;H₀²(I)).Moreover, there exists a constant C=C(T),such that

kyk_C([0,T];L2(I))∩L²(0,T;H²₀(I)) ≤Ce^{C(kakL∞(Q)+1)}

kgk_L2(0,T;L²(I))+ky⁰k_L2(I)

.

(ii) If g ∈L²(0,T;L²(I)),a ∈L^∞(Q)and y⁰ ∈H₀²(I),system(2.3)has a unique mild solution y in C([0,T];H²₀(I))∩L²(0,T;H⁴(I)).Moreover, there exists a constant C=C(T),such that

kyk_C([0,T];H2

0(I))∩L²(0,T;H⁴(I)) ≤Ce^{C(kakL∞(Q)+1)}

kgk_L2(0,T;L²(I))+ky⁰k_H2 0(I)

.

Remark 2.4. By the classical semigroup theory, multipliers method and suitable energy esti- mates [19,24], we can obtain Proposition2.3.

3 Proof of Theorem 1.2

Applying the classical estimates for the parabolic equation to the system (1.7)–(1.8), we can obtain the following lemma.

Lemma 3.1. System(1.7)–(1.8)has the following energy estimates (i)

Z

Ip²(t2)dx≤e^{2kbkL∞(Q)(t2−t1)} Z

Ip²(t₁)dx, ∀t₁ <t2; (ii)

kz(t)k²_L2(I) ≤

Z _T

t e^{(2kakL∞(Q)+1)(s−t)}kp(s)k_L2(O)ds, ∀t ∈[0,T]. In particular, we have

Z

Ip²

t+ ^T 4

dx ≤e^kbkL∞(Q)T2 Z

Ip²(t)dx, ∀t ∈ T

4,3T 4

, and hence,

Z

I×(^T₂,T)p²dx dt≤e^kbkL∞(Q)T2 Z

I×(^T₄,^3T₄ )p²dx dt. (3.1) On the other hand,

Z _T

t

kz(s)k²_L2(I)ds≤ (T−t)e^{(2kakL∞(Q)+1)T2} Z _T

t

kp(s)k_L2(O)ds, ∀t ∈ T

2,T

,

thus Z

I×(^T₂,T)z²dxds≤ e^{(kakL∞(Q)+1)T} Z

O×(^T₂,T)p²dxds. (3.2) By the same method as in [5, Lemma 2.4], a simple calculation yields

Lemma 3.2. Set m₀ =_min

x∈_Ω

e^5µ−e^µ(ψ(x)+3)

and M₀ =_max

x∈_Ω

e^5µ−e^µ(ψ(x)+3) . (i) Whenλ> _2M^7T2

0,the function e⁻

2λM0

T(T−t)(T−t)⁻⁷ is decreasing in(0,T). (ii) Whenλ> ^15T_8m²

0,we haveλ⁸θ²t⁻¹⁵(T−t)⁻¹⁵≤2³⁰T⁻¹⁴m⁻₀⁸e⁻⁸. In particular, we have that for anyt∈ (_0, ^T₂)_,

e⁻

2λM0

t(T−t)t⁻⁷(T−t)⁻⁷ =e^−2λMTt0t⁻⁷·e⁻

2λM0

T(T−t)(T−t)⁻⁷

≥e^−2λMTt0 T

2 −7

·e⁻

2λM0 T T2

T 2

−7

=e^−2λMTt0 ·C(T). Proof of Theorem1.2

We first assume that p⁰ ∈ H₀²(I)_.

According to Proposition 2.1, we obtain that there exists a positive λ₀, such that when λ≥ λ₀

Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷p²dxdt+

Z

Qλ⁵θ²t⁻⁵(T−t)⁻⁵p²_xdx dt +

Z

Qλ³θ²t⁻³(T−t)⁻³p²_xxdx dt+

Z

Qλ¹θ²t⁻¹(T−t)⁻¹p²_xxxdx dt

≤C _Z

ω×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt+

Z

Qθ²|bp|²dx dt

≤C Z

ω×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt+¹ 2

Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt

and Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt+

Z

Qλ⁵θ²t⁻⁵(T−t)⁻⁵z²_xdxdt +

Z

Q

λ³θ²t⁻³(_T−t)⁻³_z²_{xxdx dt}+

Z

Q

λθ²t⁻¹(_T−t)⁻¹_z²_{xxxdx dt}

≤C Z

ω×(0,T)

λ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt+

Z

Q

θ²|pχ_O−az|²dx dt

≤C _Z

ω×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt+¹ 2

Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt+

Z

Qθ²p²χ_Odx dt

.

Then we have Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt+

Z

Qλ⁵θ²t⁻⁵(T−t)⁻⁵p²_xdx dt +

Z

Qλ³θ²t⁻³(T−t)⁻³p²_xxdx dt+

Z

Qλθ²t⁻¹(T−t)⁻¹p²_xxxdx dt

≤C Z

ω×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt

(3.3)

and

Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt+

Z

Qλ⁵θ²t⁻⁵(T−t)⁻⁵z²_xdx dt +

Z

Qλ³θ²t⁻³(T−t)⁻³z²_xxdx dtdt+

Z

Qλθ²t⁻¹(T−t)⁻¹z²_xxxdx dt

≤C _Z

ω×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt+

Z

Qθ²p²χ_Odx dt

.

(3.4)

Let us consider two open sets B₁ and B2 such that B₁ ⊂ B2 ⊂ ω∩ O, and let us set u=λ⁷θ²t⁻⁷(T−t)⁻⁷. Consider a functionξ₁ ∈C₀^∞(I)such that 0≤ξ₁ ≤1 in I,ξ₁ =1 in B₁, suppξ₁⊂ B₂⊂ω∩ O, ^|ξ1x|

ξ^1/2₁ ,^|ξ1xx|

ξ^1/2₁ ,^|ξ1xxx|

ξ₁^1/2 ,^|ξ1xxxx|

ξ^1/2₁ ∈ L^∞(Q).

From (3.3), we can deduce that Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt+

Z

Qλ⁵θ²t⁻⁵(T−t)⁻⁵p²_xdx dt +

Z

Qλ³θ²t⁻³(T−t)⁻³p²_xxdx dt+

Z

Qλθ²t⁻¹(T−t)⁻¹p²_xxxdx dt

≤ C Z

B1×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt

≤ C Z

O×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt

≤ C Z

Q

ξ₁up·pχOdx dt

= C Z

Qξ₁up·(−z_t+z_xxxx+a(x,t)z)dx dt

= C Z

Q

zp·(ξ₁u)_t+ (a−b)zp·ξ₁u+4zp_xxx·(ξ₁u)_x+6zp_xx·(ξ₁u)_xx +4zpx·(ξ₁u)_xxx+zp·(ξ₁u)_xxxxdx dt

=:C(I₁+I₂+I₃+I₄+I₅+I₆). Since

|ξ₁u|=λ⁷e^2λat⁻⁷(T−t)⁻⁷ξ₁

|(ξ₁u)_t| ≤Cλ⁸e^2λat⁻⁹(T−t)⁻⁹ξ₁

|(ξ₁u)_x| ≤Cλ⁷e^2λat⁻⁷(T−t)⁻⁷|ξ_1x|+Cλ⁸e^2λat⁻⁸(T−t)⁻⁸ξ₁

|(ξ₁u)_xx| ≤Cλ⁷e^2λat⁻⁷(T−t)⁻⁷|ξ_1xx|+Cλ⁸e^2λat⁻⁸(T−t)⁻⁸|ξ_1x| +Cλ⁸e^2λat⁻⁸(T−t)⁻⁸ξ₁+Cλ⁹e^2λat⁻⁹(T−t)⁻⁹ξ₁

|(ξ₁u)_xxx| ≤Cλ⁷e^2λat⁻⁷(T−t)⁻⁷(|ξ_1xxx|+λt⁻¹(T−t)⁻¹|ξ_1xx|

+λ²t⁻²(T−t)⁻²|ξ_1x|+λ²t⁻²(T−t)⁻²|ξ₁|+λt⁻¹(T−t)⁻¹|ξ_1x| +λ³t⁻³(T−t)⁻³|ξ₁|+λt⁻¹(T−t)⁻¹|ξ₁|)

|(ξ₁u)_xxxx| ≤Ce^2λa(λ⁹t⁻⁹(T−t)⁻⁹|ξ_1xx|+λ¹⁰t⁻¹⁰(T−t)⁻¹⁰|ξ_1x|

+λ¹¹t⁻¹¹(T−t)⁻¹¹|ξ₁|+λ⁸t⁻⁸(T−t)⁻⁸|ξ_1xxx|+|ξ_1xxxx|),

by the Cauchy–Schwartz inequality and following the ideas in [5], it holds that for sufficiently largeλ₀

|I1|= Z

Qzp·(ξ₁u)_{tdx dt}

≤δ Z

Q

λ⁷θ²t⁻⁷(T−t)⁻⁷p²ξ₁dx dt+C(δ)

Z

Q

λ⁹θ²t⁻¹¹(T−t)⁻¹¹z²ξ₁dx dt,

|I₂|= Z

Q

(a−b)zp·ξ₁u dx dt

≤δ Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷p²ξ₁dx dt+C(δ)

Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷z²ξ₁dx dt,

|I₃|= Z

Qzp_xxx·(ξ₁u)_xdx dt

≤δ Z

Qλθ²t⁻¹(T−t)⁻¹p²_xxxξ₁dx dt+C(δ)

Z

Qλ¹⁵θ²t⁻¹⁵(T−t)⁻¹⁵z²χ_B2dx dt,

|I₄|= Z

Qzp_xx·(ξ₁u)_xxdx dt

≤ δ Z

Qλ³θ²t⁻³(T−t)⁻³p²_xxξ₁dx dt+C(δ)

Z

Qλ¹⁵θ²t⁻¹⁵(T−t)⁻¹⁵z²χ_B2dx dt,

|I₅|= Z

Qzp_x·(ξ₁u)_xxxdx dt

≤ δ Z

Qλ⁵θ²t⁻⁵(T−t)⁻⁵p²_xξ₁dx dt+C(δ)

Z

Qλ¹⁵θ²t⁻¹⁵(T−t)⁻¹⁵z²χ_B2dx dt,

|I₆|= Z

Qzp·(ξ₁u)_xxxxdx dt

≤ δ Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷p²ξ₁dx dt+C(δ)

Z

Qλ¹³θ²t⁻¹³(T−t)⁻¹³z²χ_B2dx dt with λ≥ λ₀. Thus

Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt+

Z

Qλ⁵θ²t⁻⁵(T−t)⁻⁵p²_xdx dt +

Z

Qλ³θ²t⁻³(T−t)⁻³p²_xxdx dt+

Z

Qλ¹θ²t⁻¹(T−t)⁻¹p²_xxxdx dt

≤ C(I₁+I₂+I₃+I₄+I₅+I₆)

≤ δ _Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷p²ξ₁dx dt+λ⁵θ²t⁻⁵(T−t)⁻⁵p²_xξ₁dx dt +

Z

Qλ³θ²t⁻³(T−t)⁻³p²_xxξ₁dx dt+

Z

Qλθ²t⁻¹(T−t)⁻¹p²_xxxξ₁dx dt

+C(δ)

Z

Q

λ¹⁵θ²t⁻¹⁵(T−t)⁻¹⁵z²χ_B2dx dt.

Then we have Z

Q

λ⁷θ²t⁻⁷(T−t)⁻⁷p²dx dt+

Z

Q

λ⁵θ²t⁻⁵(T−t)⁻⁵p²_xdx dt +

Z

Qλ³θ²t⁻³(T−t)⁻³p²_xxdx dt+

Z

Qλθ²t⁻¹(T−t)⁻¹p²_xxxdx dt

≤C Z

Qλ¹⁵θ²t⁻¹⁵(T−t)⁻¹⁵z²χ_B2dx dt, namely,

Z

Qθ²t⁻⁷(T−t)⁻⁷p²dx dt ≤C Z

Qλ⁸θ²t⁻¹⁵(T−t)⁻¹⁵z²χB₂dx dt. (3.5) From (3.4) and (3.5), we can deduce that for sufficient large λ₀

Z

Qλ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt

≤C Z

B2×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt+

Z

Qθ²pχ²_Odx dt

≤C _Z

B₂×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt+λ⁷ Z

Qθ²t⁻⁷(T−t)⁻⁷pχ²_Odx dt

≤C _Z

B2×(0,T)λ⁷θ²t⁻⁷(T−t)⁻⁷z²dx dt+

Z

Qλ¹⁵θ²t⁻¹⁵(T−t)⁻¹⁵z²χ_B2dx dt

≤C(T)

Z

Qλ¹⁵θ²t⁻¹⁵(T−t)⁻¹⁵z²χ_B2dx dt

withλ≥λ₀. Namely Z

Qθ²t⁻⁷(T−t)⁻⁷z²dx dt≤ C(T)

Z

Qλ⁸θ²t⁻¹⁵(T−t)⁻¹⁵z²χB₂dx dt. (3.6) Setλ≥λ₁:=max_7T2

2M0,^15T_8m²

0,λ₀ andM_λ = ^2λM_T ⁰. On the one hand, according to Lemma 3.2, (3.6) and the definition ofθ, we have

Z

I×(0,^T₂)e^−Mtλz²dx dt

≤C(T)

Z

I×(0,^T₂)θ²t⁻⁷(T−t)⁻⁷z²dx dt

≤C(T)

Z

Qθ²t⁻⁷(T−t)⁻⁷z²dx dt

≤C(T)

Z

B2×(0,T)λ⁸θ²t⁻¹⁵(T−t)⁻¹⁵z²dx dt.

On the other hand, from (3.1), (3.2) and (3.5), it holds that Z

I×(^T₂,T)

e^−Mtλz²dx dt

≤

Z

I×(^T₂,T)z²dx dt

≤e^{C(T)(kakL∞(Q)+1)} Z

O×(^T₂,T)p²dx dt

≤e^{C(T)(kakL∞(Q)+1)} Z

I×(^T₂,T)p²dx dt

≤e^{C(T)(kakL∞(Q)+kbkL∞(Q)+1)} Z

I×(^T₄,^3T₄ )p²dx dt

≤e^{C(T)(kakL∞(Q)+kbkL∞(Q)+1)} Z

I×(^T₄,^3T₄ )θ²t⁻⁷(T−t)⁻⁷p²dx dt

≤e^{C(T)(kakL∞(Q)+kbkL∞(Q)+1)} Z

I×(0,T)θ²t⁻⁷(T−t)⁻⁷p²dx dt

≤e^{C(T)(kakL∞(Q)+kbkL∞(Q)+1)} Z

B2×(0,T)λ⁸θ²t⁻¹⁵(T−t)⁻¹⁵z²dx dt.

Thus, in view of Lemma3.2, we have Z

Qe^−Mtλz²dx dt

=

Z

I×(0,^T₂)e^−Mtλz²dx dt+

Z

I×(^T₂,T)e^−Mtλz²dx dt

≤e^{C(T)(kakL∞(Q)+kbkL∞(Q)+1)} Z

B2×(0,T)λ⁸θ²t⁻¹⁵(T−t)⁻¹⁵z²dx dt

≤e^{C(T)(kakL∞(Q)+kbkL∞(Q)+1)} Z

B2×(0,T)z²dx dt.

(3.7)

Finally, settingλ= λ₁in (3.7) and we define M = M_λ1.

By a density argument, (3.7) holds for the solution (p,z) of (1.7)–(1.8) if the initial data p⁰ ∈ L²(I). Indeed, we can choose a sequence {p⁰_n} ⊂ H₀²(I)such that p⁰_n → p⁰ in L²(I). By i) of Proposotion2.3, we obtain that R

Qe^−Mt z²_ndx dt → R

Qe^−Mt z²dx dt and R

ω×(0,T)z²_ndx dt →