Time-periodic solution for a

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Electronic Journal of Qualitative Theory of Differential Equations 2013, No. 7, 1-15;http://www.math.u-szeged.hu/ejqtde/

Time-periodic solution for a

fourth-order parabolic equation describing crystal surface growth

^∗

Xiaopeng Zhao^†, Bo Liu, Ning Duan

College of Mathematics, Jilin University, Changchun, 130012, P. R. China

Abstract. In this paper, by using the Galerkin method, the existence and uniqueness of time-periodic generalized solutions to a fourth-order parabolic equation describing crystal surface growth are proved.

Keywords. Time-periodic solution, fourth-order parabolic equation, Galerkin method.

1 Introduction

In the study of crystal surface growth, there arises the following diffusion equation

ut=−jx+f(x, t),

where u(x, t) denotes the variation of height from the average, j is the atom current parallel to the surface, andf(x, t) is a noise term caused by shot noise in the incoming flux. Takingj=uxxx+_1+|u^u^x_x_|², we obtain the well-known BCF model (see [4, 5, 7, 8, 11, 13])

ut+uxxxx+

ux

1 +|ux|²

x

=f(x, t), in (0,1)×R. (1.1) During the past years, many authors have paid much attention to the equation (1.1). It was Rost and Krug [13] who studied the unstable epitaxy on singular surfaces using equation (1.1) with a prescribed slopedependent surface current.

In their paper, they derived scaling relations for the late stage of growth, where power law coarsening of the mound morphology is observed. In [11], in the limit of weak desorption, O. Pierre-Louis et al. derived the equation (1.1) for a vicinal surface growing in the step flow mode. This limit turned out to be singular, and nonlinearities of arbitrary order need to be taken into account.

Recently, H. Fujimura and A. Yagi [4] considered the equation of (1.1). In their paper, the uniqueness local solutions and the global solutions are obtained, a dynamical system determined from the initial-boundary value problem of the model equation was also constructed. In [5], H. Fujimura and A. Yagi continued a study on the model equation (1.1). They considered the asymptotic behavior of trajectories of the dynamical system by constructing exponential attractors and a Lyapunov function. There is much literature concerned with the Eq.(1.1), for more results we refer the reader to [6, 14] and the references therein.

∗This work was supported by Graduate Innovation Fund of Jilin University (Project 20121059).

†Corresponding author, E-mail address: zxp032@gmail.com.

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Furthermore, several authors have paid attention to the time-periodic problems [1, 19, 20]. But, to the best of our knowledge, only a few papers deal with time periodic solutions of fourth-order diffusion equations. In [10, 17], the existence of time periodic solutions for the Cahn-Hilliard type equation and viscous Cahn-Hilliard equation with periodic concentration dependent potentials and sources has been investigated. In [15, 16], Wang et. al. considered the existence and uniqueness of time-periodic generalized solutions and time-periodic classi- cal solutions to the generalized Ginzburg-Landau model equation in 1D and 2D case. In [3], by using the Galerkin method and the Leray-Schauder fixed point theorem, Fu and Guo studied the existence and uniqueness of a time periodic solution for the viscous Camassa-Holm equation. There are also many papers were denoted to the periodic problems, for example [9, 12, 18] and so on.

Here, we investigate the existence and uniqueness of time-periodic generalized solutions to the equation (1.1) in one spatial dimension together with the condition

u(x+ 1, t) =u(x, t), t∈R, (1.2) and the time-periodic condition

u(x, t+ω) =u(x, t), x∈(0,1), t∈R, (1.3) whereω >0 is a constant andf(x, t) isω-periodic functions with respect to the timet, which also satisfies

Z

Ω

f(x, t)dx= 0.

Throughout this paper, we use the following notations.

LetX be a Banach space,C_ω^k(R;X) denotes the set ofX-valuedω-periodic functions onRwith continuous derivatives up to orderk. The norm inC_ω^k(R;X) is defined as

kuk_Ck

ω(R;X)= sup

0≤t≤ω

{

k

X

i=0

kD_tⁱukX},

whereDt=_∂t^∂,k · kX is the norm inX. We also defineL^p_ω(R;X) (1≤p≤ ∞) as the set ofω-periodicX-valued measurable functions onRsuch that

kuk_L^p_ω_(R;X)= Z ω

0

kuk^p_Xdt ¹_p

<∞, where 1≤p <∞, kuk_L^p_ω_(R;X)= ess sup

0≤t≤ω

kukX<∞, wherep=∞.

LetW_ω^k,p(R;X) denote the set of functions which belong toL^p_ω(R;X) together with their partial derivatives with respect totup to the orderk.

In the following, we frequently use the Poincar´e inequality (see [2]):

kuk²≤1

2kuxk², where Z 1

0

u(x, t)dx= 0.

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Denotek ·k_L²(0,1)byk ·k,k ·k_L^∞(0,1)byk ·k∞,k ·k_L^p(0,1)byk ·kpandk ·k_H^m(0,1)

byk · kH^m, respectively.

2 Integration estimations and existence of the approximate solutions for problem (1.1)-(1.3)

Let{yj(x)} (j= 1,2,· · ·) be the orthonormal base in L²(0,1) being composed of the eigenfunctions of the eigenvalue problem

y^′′+λy= 0, y^′(0) =y^′(1) = 0, corresponding to eigenvaluesλj (j= 1,2,· · ·).

Suppose that uN(x, t) = PN

j=1uN j(t)yj(x) is the Galerkin approximate solution to the problem (1.1)-(1.3), where a group of function uN j(t) (j = 1,2,· · ·, N)∈C¹(ω,R),N is a natural number.

Performing the Galerkin procedure for the equation (1.1), we obtain u_{N t}+u_{N xxxx}+

u_{N x} 1 +|uN x|²

x

=f, in (0,1)×R. (2.1) with

uN(x+ 1, t) =uN(x, t), t∈R, (2.2) and the time-periodic condition

uN(x, t+ω) =uN(x, t), x∈(0,1), t∈R, (2.3) Lemma 2.1 Suppose thatf ∈C_ω(R;L²(0,1)),M1= sup_0≤t≤ωkf(·, t)k. Then, there exists a approximate solutionu_N for problem (2.1)-(2.3), which satisfies

sup

0≤t≤ω

kuN(·, t)k²≤c0M₁², wherec0 is a positive constant independent of N,M1.

Proof. Using Poincar´e’s inequality and H¨older’s inequality, we have kuNk²≤1

2kuN xk²=−1 2

Z 1 0

uNuN xxdx≤ 1

4kuNk²+1

4kuN xxk², that is

kuNk²≤ 1

3kuN xxk². (2.4)

Multiplying both sides of (2.1) byu_N, and integrating it over (0,1), making use of H¨older’s inequality and (2.4), we obtain

1 2

d

dtkuNk²+kuN xxk²

= Z 1

0

u²_{N x}

1 +|uN x|²dx+ Z 1

0

f uNdx≤ kuN xk²+1

8kuNk²+ 2kfk²

≤ 1

2kuN xxk²+5

8kuNk²+ 2kfk²≤17

24kuN xxk²+ 2M₁².

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Then,

d

dtkuNk²+ 7

12kuN xxk²≤4M₁². (2.5) Integrating (2.5) over [0, ω], we get

Z ω 0

kuN xxk²≤48

7 M₁²ω. (2.6)

It then follows from (2.6) that there is at1∈(0, ω) such that kuN xx(·, t1)k²≤48

7 M₁². (2.7)

Adding (2.4) and (2.7) together gives kuN(·, t1)k²≤ 16

7 M₁². (2.8)

Integrating (2.5) again over [t1, t+ω] (∀t∈[0, ω]), using (2.8), we get sup

0≤t≤ω

kuN(·, t)k²≤ kuN(·, t1)k²+ 8ωM₁²= (16

7 + 8ω)M₁². (2.9) Settingc0= ¹⁶₇ + 8ω, we complete the proof.

Employing the Leray-Schauder fixed-point argument, we can prove that there exists at least one solutionuN(t) =PN

j=1uN jyj(x) for the problem (2.1)- (2.3).

Lemma 2.2 Suppose that the assumptions of Lemma 2.1 hold and f ∈Cω(R;H²(0,1)), ft∈Cω(R;L²(0,1)).

Then

sup

0≤t≤ω

(kuN t(·, t)k²_H²+kuN(·, t)k²_H⁴)≤c1(M2),

where M2 = sup_0≤t≤ω(kfkH² +kftk). Here and in the sequel, c_i(M2)(i = 1,2,· · ·) is nondecreasing with respect to M2 andlimM2→0ci(M2) =ci(0) = 0, ci(M2)is independent of N.

Proof. Multiplying both sides of (2.1) by−uN xx, and integrating it over (0,1), we obtain

1 2

d

dtkuN xk²+kuN xxxk²+ Z 1

0

u_{N x}u_{N xxx} 1 +|uN x|²dx+

Z 1 0

f u_{N xx}dx= 0, Then, using H¨older’s inequality, we get

1 2

d

dtkuN xk²+kuN xxxk²

≤ 1

4kuN xxxk²+ Z 1

0

u²_{N x}

(1 +|uN x|²)²dx+1

4kfk²+kuN xxk²

≤ 1

4kuN xxxk²+kuN xk²+1

4kfk²+kuN xxk².

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By Nirenberg’s inequality, we have kuN xk²≤

c^′₁kuN xxxk³¹kuNk²³ +c^′₂kuNk2

≤1

8kuN xxxk²+c2(M2).

and

kuN xxk²≤

c^′₁kuN xxxk²³kuNk¹³ +c^′₂kuNk2

≤1

8kuN xxxk²+c3(M2).

Summing up, we get d

dtkuN xk²+kuN xxxk²≤c4(M2), (2.10) wherec4(M2) = 2c2(M2) + 2c3(M2) +¹₂M₂². Integrating (2.10) over [0, ω], we have

Z ω 0

kuN xxx(·, t)k²dt≤c4(M2)ω. (2.11) It then follows from (2.11) that there exists a timet2∈(0, ω) such that

kuN xxx(·, t2)k²≤c4(M2). (2.12) Then

kuN x(·, t2)k²≤1

8kuN xxx(·, t2)k²+c2(M2)≤ 1

8c4(M2) +c2(M2). (2.13) Integrating (2.10) again over [t2, t+ω] (∀t∈[0, ω]), using (2.13), we obtain

sup

0≤t≤ω

kuN x(·, t)k²≤ kuN x(·, t2)k²+ 2c4(M2)ω=c5(M2). (2.14) Based on Sobolev’s embedding theorem, we have

kuN(·, t)kC[0,1] ≤ckuNkH¹ ≤c6(M2), t∈[0, ω]. (2.15) Multiplying both sides of (2.1) by uN xxxx, and integrating it over (0,1), we obtain

1 2

d

dtkuN xxk²+kuN xxxxk²= Z 1

0

uN x

1 +|uN x|²

x

uN xxxxdx+ Z 1

0

f uN xxxxdx.

Then, using H¨older’s inequality, noticing thats≤¹₂(1 +s²), we get 1

2 d

dtkuN xxk²+kuN xxxxk²

≤ 1

4kuN xxxxk²+ Z 1

0

uN x

1 +|uN x|²

2

dx+1

4kuN xxxxk²+ Z 1

0

f²dx

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≤ 1

2kuN xxxxk²+ Z 1

0

uN xx

1 +|uN x|²− 2|uN x|²uN xx

(1 +|uN x|²)² ²

dx+ Z 1

0

f²dx

≤ 1

2kuN xxxxk²+ 2 Z 1

0

uN xx

1 +|uN x|² 2

dx +2

Z 1 0

2|uN x|²u_{N xx} (1 +|uN x|²)²

2

dx+ Z 1

0

f²dx

= 1

2kuN xxxxk²+ 2 Z 1

0

u²_{N xx} 1

(1 +|uN x|²)²dx +8

Z 1 0

u²_{N xx} |uN x|⁴

(1 +|uN x|²)⁴dx+ Z 1

0

f²dx

≤ 1

2kuN xxxxk²+ 2kuN xxk²+1

2kuN xxk²+M². Using Nirenberg’s inequality, we derive that

5

2kuN xxk²≤5

2(c^′1kuN xxxxk¹³kuN xk²³ +c^′₂kuN xk)²≤1

4kuN xxxxk²+c7(M2), Summing up, we deduce that

d

dtkuN xxk²+1

2kuN xxxxk²≤c8(M2), (2.16) wherec8(M2) = 2c7(M2) + 2M2². Integrating (2.16) over [0, ω], we have

Z ω 0

kuN xxxx(·, t)k²dt≤2c8(M2)ω. (2.17) It then follows from (2.17) that there exists a timet3∈(0, ω) such that

kuN xxxx(·, t3)k²≤2c8(M2). (2.18) Then

kuN xx(·, t3)k²≤ 1

10kuN xxxx(·, t3)k²+2

5c7(M2)≤1

5c8(M2) +2

5c7(M2).(2.19) Integrating (2.16) again over [t3, t+ω] (∀t∈[0, ω]), using (2.19), we obtain

sup

0≤t≤ω

kuN xx(·, t)k²≤ kuN xx(·, t3)k²+ 2c8(M2)ω=c9(M2). (2.20) Based on Sobolev’s embedding theorem, we have

kuN x(·, t)kC[0,1]≤ckuNk_H²≤c10(M2), t∈[0, ω]. (2.21) Multiplying both sides of (2.1) by uN xxxxxx, and integrating it over (0,1), we obtain

1 2

d

dtkuN xxxk²+kuN xxxxxk²+ Z 1

0

uN x

1 +|uN x|²

xx

u_{N xxxxx}dx= Z 1

0

f_xu_{N xxxxx}dx.

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Using H¨older’s inequality, we get 1

2 d

dtkuN xxxk²+kuN xxxxxk²

≤ Z 1

0

uN x

1 +|uN x|² ²

xx

dx+ Z 1

0

f_x²dx+1

2kuN xxxxxk²

≤ 4 Z 1

0

u²_{N xxx}

(1 +|uN x|²)²dx+ 144 Z 1

0

u²_{N x}u⁴_{N xx}

(1 +|uN x|²)⁴dx+ 16 Z 1

0

u⁴_{N x}u²_{N xxx} (1 +|uN x|²)⁴dx +256

Z 1 0

u⁶_{N x}u⁴_{N xx} (1 +|uN x|²)⁶dx+

Z 1 0

f_x²dx+1

2kuN xxxxxk²

≤ 5kuN xxxk²+ 40kuN xxk⁴₄+M₂²+1

2kuN xxxxxk². By Nirenberg’s inequality, we have

5kuN xxxk²≤5(c^′₁kuN xxxxxk¹³kuN xxk²³+c^′₂kuN xxk)²≤ 1

8kuN xxxxxk²+c11(M2), and

40kuN xxk⁴₄≤40(c^′₁kuN xxxxxk¹²¹kuN xxk¹¹¹²+c^′₂kuN xxk)⁴≤1

8kuN xxxxxk²+c12(M2).

Summing up, we deduce that d

dtkuN xxxk²+1

2kuN xxxxxk²≤c13(M2), (2.22) wherec13(M2) = 2c11(M2) + 2c12(M2) + 2M₂². Integrating (2.22) over [0, ω], we have

Z ω 0

kuN xxxxx(·, t)k²dt≤2c13(M2)ω. (2.23) It then follows from (2.23) that there exists a timet4∈(0, ω) such that

kuN xxxxx(·, t4)k²≤2c13(M2). (2.24) Then

kuN xxx(·, t4)k² ≤ 1

40kuN xxxxx(·, t4)k²+1

5c11(M2)

≤ 1

20c13(M2) +1

5c11(M2). (2.25) Integrating (2.22) again over [t4, t+ω] (∀t∈[0, ω]), using (2.25), we obtain

sup

0≤t≤ω

kuN xxx(·, t)k²≤ kuN xxx(·, t4)k²+ 2c13(M2)ω=c14(M2). (2.26) Based on Sobolev’s embedding theorem, we have

kuN xx(·, t)kC[0,1]≤ckuNkH³ ≤c15(M2), t∈[0, ω]. (2.27)

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Multiplying both sides of (2.1) byuN xxxxxxxx, and integrating it over (0,1), we obtain

1 2

d

dtkuN xxxxk²+kuN xxxxxxk²+ Z 1

0

uN x

1 +|uN x 2

xxx

uN xxxxxxdx

= Z 1

0

f_xxu_{N xxxxxx}dx.

Using H¨older’s inequality, we get 1

2 d

dtkuN xxxxk²+kuN xxxxxxk²

≤ 1

2kuN xxxxxxk²+ Z 1

0

uN x

1 +|uN x 22

xxx

dx+ Z 1

0

f_xx² dx

≤ 1

2kuN xxxxxxk²+ Z 1

0

f_xx² dx+c Z 1

0

u_{N xxxx} 1 +|uN x|²

² dx +c

Z 1 0

u_{N x}u_{N xx}u_{N xxx} (1 +|uN x|²)²

2

dx+c Z 1

0

|uN xx|³ (1 +|uN x|²)²

dx +c

Z 1 0

|uN x|²uN xx

(1 +|uN x|²)²

dx+c Z 1

0

|uN x|²|uN xx|³ (1 +|uN x|²)³

dx

+c Z 1

0

|uN x|³uN xxuN xxx

(1 +|uN x|²)³

dx+c Z 1

0

|uN x|⁴|uN xx|³ (1 +|uN x|²)⁴

dx

≤ 1

2kuN xxxxxxk²+ Z 1

0

f_xx² dx+c(kuN xxxxk²+kuN xxk⁴₄ +kuN xxk⁶₆+kuN xxxk⁴₄+kuN xxk²).

By Nirenberg’s inequality, we get

ckuN xxxxk² ≤ c(c^′₁kuN xxxxxxk³¹kuN xxxk²³ +c^′₂kuN xxxk)²

≤ 1

16kuN xxxxxxk²+c16(M2),

ckuN xxk⁴₄ ≤ c(c^′₁kuN xxxxxxk¹⁶¹kuN xxk¹⁵¹⁶ +c^′₂kuN xxk)⁴

≤ 1

16kuN xxxxxxk²+c17(M2),

ckuN xxk⁶₆ ≤ c(c^′₁kuN xxxxxxk¹²¹kuN xxk¹¹¹² +c^′₂kuN xxk)⁶

≤ 1

16kuN xxxxxxk²+c18(M2), and

ckuN xxxk⁴₄ ≤ c(c^′₁kuN xxxxxxk¹²¹ kuN xxxk¹¹¹² +c^′₂kuN xxxk)²

≤ 1

16kuN xxxxxxk²+c19(M2).

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Summing up, we deduce that d

dtkuN xxxxk²+1

2kuN xxxxxxk²≤c21(M2), (2.28) wherec21(M2) = 2(c16(M2) +c17(M2) +c18(M2) +c19(M2) +M₂²+cc9(M2)).

Integrating (2.28) over [0, ω], we have Z ω

0

kuN xxxxxx(·, t)k²dt≤2c21(M2)ω. (2.29) It then follows from (2.23) that there exists a timet5∈(0, ω) such that

kuN xxxxxx(·, t5)k²≤2c21(M2). (2.30) Then

kuN xxxx(·, t5)k² ≤ 1

16ckuN xxxxxx(·, t5)k²+1

cc16(M2)

≤ 1

8cc21(M2) +c16(M2)

c . (2.31)

Integrating (2.28) again over [t5, t+ω] (∀t∈[0, ω]), using (2.31), we obtain sup

0≤t≤ω

kuN xxxx(·, t)k²≤ kuN xxxx(·, t5)k²+ 2c21(M2)ω=c22(M2). (2.32) Based on Sobolev’s embedding theorem, we have

kuN xxx(·, t)kC[0,1]≤ckuNkH⁴ ≤c22(M2), t∈[0, ω]. (2.33) Multiplying both sides of (2.1) byuN t, and integrating it over (0,1), we obtain

kuN tk²

= (−uN xxxx−( uN x

1 +|uN x|²)x+f, uN t)

≤ 1

2kuN tk²+3

2kuN xxxxk²+3

2kfk²+3 2

Z 1 0

uN x

1 +|uN x|² 2

x

dx

≤ 1

2kuN tk²+3

2kuN xxxxk²+3 2kfk² +3

Z 1 0

uN xx

1 +|uN x|² 2

dx+ 3 Z 1

0

2u²_{N x}uN xx

(1 +|uN x|²)² ²

dx

≤ 1

2kuN tk²+3

2kuN xxxxk²+3

2kfk²+ckuN xxk²

≤ 1

2kuN tk²+3

2c22(M2) +3

2M₂²+cc9(M2).

Hence

kuN tk²≤c23(M2)≡3c22(M2) + 3M₂²+ 2cc9(M2). (2.34)

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Differentiating (2.1) with respect tot, we get uN tt+uN xxxxt+ ( uN x

1 +|uN x|²)xt=ft, (x, t)∈(0,1)×R. (2.35) Multiplying both sides of (2.35) by u_{N xxt}, and integrating it over (0,1), we obtain

1 2

d

dtkuN xtk²+kuN xxxtk²

= Z 1

0

u_{N x} 1 +|uN x|²

t

uN xxxtdx− Z 1

0

ftuN xxtdx.

≤ 1

4kuN xxxtk²+ 2 Z 1

0

uN t

1 +|uN x|² 2

dx +2

Z 1 0

2u²_{N x}uN xt

(1 +|uN x|²)² 2

dx+1

4kuN xxtk²+M₂²

≤ 1

4kuN xxxtk²+5

2kuN xtk²+1

4kuN xxtk²+M₂². Using Nirenberg’s inequality, we obtain

5

2kuN xtk²≤ 5

2(c^′₁kuN xxxtk¹³kuN tk²³ +c^′₂kuN tk)²≤ 1

8kuN xxxtk²+c24(M2), and

1

4kuN xxtk²≤1

4(c^′₁kuN xxxtk²³kuN tk¹³+c^′₂kuN tk)²≤1

8kuN xxxtk²+c25(M2).

Summing up, we have d

dtkuN xtk²+kuN xxxtk²≤c26(M2), (2.36) wherec26(M2) = 2c24(M2) + 2c25(M2) + 2M₂². Integrating (2.36) over [0, ω], we have

Z ω 0

kuN xxxt(·, t)k²dt≤c26(M2)ω. (2.37) It then follows from (2.37) that there exists a timet6∈(0, ω) such that

kuN xxxt(·, t6)k²≤c26(M2). (2.38) Then

kuN xt(·, t6)k²≤ 1

20kuN xxxt(·, t6)k²+2

5c24(M2)≤ 1

20c26(M2) +2

5c24(M2).(2.39) Integrating (2.36) again over [t6, t+ω] (∀t∈[0, ω]), using (2.39), we obtain

sup

0≤t≤ω

kuN xt(·, t)k²≤ kuN xt(·, t6)k²+ 2c26(M2)ω=c27(M2). (2.40)

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Based on Sobolev’s embedding theorem, we have

kuN t(·, t)kC[0,1]≤ckuN tkH¹≤cc27(M2), t∈[0, ω]. (2.41) Multiplying both sides of (2.35) byuN xxxxt, and integrating it over (0,1), we obtain

1 2

d

dtkuN xxtk²+kuN xxxxtk²=− Z 1

0

u_{N x} 1 +|uN x|²

xt

u_{N xxxxt}dx+ Z 1

0

f_tu_{N xxxxt}dx.

Therefore 1 2

d

dtkuN xxtk²+kuN xxxxtk²

≤ 1

4kuN xxxxtk²+ 2 Z 1

0

u_{N x} 1 +|uN x|²

2 xt

dx+ 2 Z 1

0

f_t²dx

≤ 1

4kuN xxxxtk²+ 8 Z 1

0

uN xxt

1 +|uN x|² 2

dx+ 8 Z 1

0

6uN xuN xxuN xt

(1 +|uN x|²)² 2

dx +8

Z 1 0

2u²_{N x}u_{N xxt} (1 +|u²_{N x})²

dx+ 8

Z 1 0

4u³_{N x}u_{N xx}u_{N xt} (1 +|uN x|²)³

2

dx+ 2 Z 1

0

f_t²dx

≤ 1

4kuN xxxxtk²+ckuN xxtk²+ckuN xtk²+ 2M₂². Nirenberg’s inequality gives

ckuN xxtk²≤c(c^′₁kuN xxxxtk³¹kuN xtk²³ +c^′₂kuN xtk)²≤1

4kuN xxxxtk²+c28(M2).

Summing up, we get d

dtkuN xxtk²+kuN xxxxtk²≤c29(M2), (2.42) wherec29(M2) = 2cc27(M2) + 2cc28(M2) + 4M₂². Integrating (2.42) over [0, ω], we have

Z ω 0

kuN xxxxt(·, t)k²dt≤c30(M2)ω. (2.43) It then follows from (2.43) that there exists a timet7∈(0, ω) such that

kuN xxxxt(·, t7)k²≤c30(M2). (2.44) Then

kuN xxt(·, t7)k²≤ 1

4ckuN xxxxt(·, t7)k²+c28(M2)

c ≤ 1

4cc30(M2) +c28(M2) c (2.45). Integrating (2.42) again over [t7, t+ω] (∀t∈[0, ω]), using (2.45), we obtain

sup

0≤t≤ω

kuN xxt(·, t)k²≤ kuN xxxxt(·, t7)k²+ 2c30(M2)ω=c31(M2). (2.46)

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Based on Sobolev’s embedding theorem, we have

kuN xt(·, t)kC[0,1]≤ckuN tkH² ≤cc21(M2), t∈[0, ω]. (2.47) Combining (2.9), (2.16), (2.21), (2.27), (2.33), (2.35), (2.40) and (2.46) together, we complete the proof of Lemma 2.2.

3 Existence and uniqueness of solutions for the problem (1.1)-(1.3)

Theorem 3.1 Suppose that the assumptions in Lemma 2.2 is satisfied, then there exists a generalized time-periodic solution

u(x, t)∈L²_ω(R;H⁴(0,1)), ut(x, t)∈L²_ω(R;H²(0,1)), (3.1) for problem (1.1)-(1.3), which satisfies

Z ω 0

Z 1 0

ut+uxxxx+ ( ux

1 +|ux|²)x−f

ϕdxdt= 0, ∀ϕ∈L²_ω(R;L²(0,1)).(3.2) Especially, ifM2 is sufficiently small, the solution is unique.

Proof. Based on Lemma 2.2 and Sobolev’s embedding theorem, we obtain the following estimate

sup

0≤t≤ω

(kuN t(·, t)kC¹[0,1]+kuNkC³[0,1])≤c32(M2). (3.3) It then follows from (3.3) and Ascoli-Arzel´a’s theorem that there exists a func- tionu(x, t) and a subsequence of {uN(x, t)}, still denoted by{uN(x, t)}, such that, whenN →+∞,{uN(x, t)}, uN x(x, t)} uniformly converge tou(x, t) and ux(x, t) on [0, ω]×(0,1). Based on the result of Lemma 2.2, whenN →+∞, the subsequences{uN xx},{uN xxx},{uN xxxx}, {uN t},{uN xt}and {uN xxt} weakly converge touxx,uxxx,uxxxx,ut,uxt anduxxtin L²_ω(R;L²(0,1)). Set

W ={u|u∈L²_ω(R;H⁴(0,1)), ut∈L²_ω(R;H²(0,1))}.

Aubin’s compact lemma implies that the embeddingW ֒→L²_ω(R;H²(0,1)) is compact. Owing to the assumptions, we know that there exists a subsequence of{uN(x, t)} still denoted by{uN(x, t)} such that, whenN→+∞,{uN(x, t)}

is convergent inL²_ω(R;H³(0,1)).

Setting F(s) = _1+|s|^s ², according to the previous subsequences {uN(x, t)}, we conclude that{[F(uN x)]x}={

uN x

1+|u_{N x}|²

x}weakly converges to [F(ux)]x= ux

1+|ux|²

x in L²_ω(R;L²(0,1)). In fact, for any ω ∈ L²_ω(R;L²(0,1)), by (3.3),

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we have

| Z ω

0

([F(uN x)]x−[F(ux)]x, w)dt|

≤ Z ω

0

Z 1 0

(|F^′(uN x)−F^′(ux)||uN xx|+|F^′(ux)||uN xx−u_xx|)|ω|dxdt

≤ Z ω

0

Z 1 0

(|F^′′(θuN x+ (1−θ)ux)||uN x−u_x||uN xx||ω|)dxdt +

Z ω 0

Z 1 0

(|F^′(ux)||uN xx−u_xx||ω|)dxdt

≤ c32(M2) Z ω

0

Z 1 0

(|uN x−u_x|+|uN xx−u_xx|)|ω|dxdt

≤ c32(M2)[kuN x−uxkL²((0,ω)×(0,1))+kuN xx−uxxkL²((0,ω)×(0,1))]

·kωkL²((0,ω)×(0,1)), (3.4)

whereθ∈(0,1). By (3.4), we know that there exists a subsequence{uN(x, t)}

such that{[F(uN x)]x} weakly converges to [F(ux)]x in L²(R;L²(0,1)). Then, problem (1.1)-(1.3) admits a generalized time-periodic solution u(x, t), which satisfies (3.1) and (3.2).

Now, we are going to prove the uniqueness of the solution. Suppose that u(x, t) andv(x, t) are two solutions of (1.1)-(1.3). Letξ(x, t) =u(x, t)−v(x, t), thenξ(x, t) satisfies the following problem







ξt+ξxxxx+ [F(ux)]x−[F(vx)]x= 0, x∈(0,1), t∈R, ξx(0, t) =ξx(1, t) =ξxxx(0, t) =ξxxx(1, t), t∈R, ξ(t+ω) =ξ(x, t), t∈R.

(3.5) Multiplying both sides of the equation of (3.5) byξ, integrating the products over (0,1) and using the mean value theorem, we obtain that

1 2

d

dtkξk²+kξxxk²= Z 1

0

(F(ux)−F(vx))ξxdx

= Z 1

0

(F^′(θux+ (1−θ)vx)ξxx)ξxdx≤c33(M2)kξxxkkξxk

≤ 1

4kξxxk²+ [c33(M2)]²kξxk²= 1

4kξxxk²−[c33(M2)]²(ξ, ξxx)

≤ 1

2kξxxk²+ [c33(M2)]⁴kξk². (3.6) Using Poincar´e’s inequality, we get

kξk²≤ 1

2kξxk²=−1

2(ξ, ξxx)≤ 1

4kξxxk²+1 4kξk², which means

kξk²≤ 1 3kξxxk².

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It then follows form (3.6) and the above inequality that d

dtkξk²+ (3−2[c32(M2)]⁴)kξk²≤0. (3.7) Taking M2 sufficiently small such that 3−2[c32(M2)]⁴ > 0, using Gronwall’s inequality, we have

kξ(·, t)k²≤ kξ(·,0)k²e^−(3−2[c³²^(M²^)]

4)t

, ∀t >0.

Sinceξ(x, t) is time-periodic, for anyt∈R, there exists a natural number N0

such thatt+N0ξ >0 and

kξ(·, t)k²=kξ(·, t+N0ω)k²≤ kξ(·,0)k²e^−(3−2[c³²^(M²^)]

4)N ω

, ∀N > N0, that is

kξ(·, t)k²= 0, ∀t∈R. Then, Theorem 3.1 is proved.

Acknowledgment

The authors thank the referees for their valuable comments and suggestions about this article.

References

[1] L. Biasco, L. Di Gregorio,Time periodic solutions for the nonlinear wave equation with long minimal period, SIAM J. Math. Anal., 38(2006), 1090–

1125.

[2] C. M. Elliott, S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96(1986), 339-357.

[3] Y. Fu, B. Guo,Time periodic solution of the viscous Camassa-Holm equation, J. Math. Anal. Appl., 313(2006), 311-321.

[4] H. Fujimura, A. Yagi,Dynamical system for BCF model describing crystal surface growth, Vestnik Chelyab. Univ. Ser. 3 Mat. Mekh. Inform., 10(2008), 75–88.

[5] H. Fujimura, A. Yagi,Asymptotic behavior of solutions for dynamical system for BCF model describing crystal surface growth, Int. Math. Forum., 3(2008), 1803–1812.

[6] H. Fujimura, A. Yagi, Homogeneous stationary solution for BCF model describing crystal surface growth, Sci. Math. Jpn., 69(2009), 295-302.

[7] A. W. Hunt, C. Orme, D. R. M. Williams, B. G. Orr, L. M. Sander,Insta- bilities in MBE growth, Europhys. Lett., 27(1994), 611–616.

[8] M. D. Johnson, C. Orme, A. W. Hunt, D. Graff, J. Sudijion, L. M. Sauder, B. G. Orr,Stable and unstable growth in molecular beam epitaxy, Pys. Rev.

Lett., 72(1994), 116-119.

(15)

[9] Y. Li, J. Yin, C. Jin,A Cahn-Hilliard type equation with periodic gradient- dependent potentials and sources, Math. Methods Appl. Sci., 33(2010), 263- 272.

[10] Y. Li, J. Yin, The viscous Cahn-Hilliard equation with periodic potentials and sources, J. Fixed Point Theory Appl., 9(2011), 63-84.

[11] O. Pierre-Louis, C. Misbah, Y. Saito,New nonlinear evolution equation for steps during molecular beam epitaxy on vicinal surfaces, Phys. Rev. Lett., 80(19), 4221-4224.

[12] C. V. Pao, Numerical methods for time-periodic solutions of nonlinear parabolic boundary value problems, SIAM J. Numer. Anal., 39 (201), 647–

667.

[13] M. Rost, J. Krug, Coarsening of surface structures in unstable epitaxial growth, Phys. Rev. E, 55(1997), 3952–3957.

[14] M. Rost, P. Smilauer, J. Krug,Unstable epitaxy on vicinal surfaces, Surface Science, 369(1996), 393-402.

[15] Y. Wang, G. Chen, Time-periodic solution to a Ginzburg-Landau model equation in population problems, Acta Math. Sci. Ser. A Chin. Ed., 26(2006), 258-266.

[16] Y. Wang, Y. Zhang, Time-periodic solution to a nonlinear parabolic type equation of higher order, Acta Math. Appl. Sin. Engl. Ser., 24(2008), 129- 140.

[17] L. Yin, Y. Li, R. Huang, J. Yin, Time periodic solutions for a Cahn-Hilliard type equation, Math. Comput. Modelling, 48(2008), 11-18.

[18] J. Yin, Y. Li, R. Huang, The Cahn-Hilliard type equations with periodic potentials and sources, Appl. Math. Comput., 211(2009), 211-221.

[19] S. Ukai, Time-periodic solutions of the Boltzmann equation, Discrete Con- tin. Dyn. Syst., 14(2006), 579–596.

[20] F. Zaouch,Time-periodic solutions of the time-dependent Ginzburg-Landau equations of superconductivity, Z. Angew. Math. Phys., 54(2003), 905–918.

(Received July 8, 2012)