2 Asymptotic Attractor

Asymptotic Attractors Of Two-dimensional Generalized Benjamin-Bona-Mahony Equations

Sheng-Ping Zhang

, Chao-Sheng Zhu

Received 13 May 2016

Abstract

In this paper, we consider the long time behavior of solutions for two-dimensional generalized Benjamin-Bona-Mahony equations with periodic boundary conditions.

By the method of orthogonal decomposition, we show that the existence of asymp- totic attractor from the precision of approximate inertial manifolds. Moreover, the dimensions estimate of the asymptotic attractor is obtained.

1 Introduction

It is well known that the concept of an inertial manifold plays an important role in the investigation of the long-time behavior of in…nite dimensional dynamical systems, see, for example, [5, 6, 8, 10]. Inertial manifold is a …nite dimensional invariant manifold in the phase space H of the system which attracts exponentially all orbits. It is con- structed as the graph of a mapping from PH to (I-P)H, where P is a projection of

…nite dimension N. However, the existence usually holds under a restrictive spectral gap condition. To investigate the case when the spectral gap condition does not hold, the concepts of approximate inertial manifolds have been introduced in [7].

But the precision of approximate inertial manifolds is inextricably di¢ cult at all times. To overcome this di¢ culty the concept of asymptotic attractor has been intro- duced [14].

Now let us recall the de…nition of the asymptotic attractor. We consider the solution u(t)of a di¤erential equation

ut+Au=F(u); (1)

with initial data

u(0) =u0: (2)

The variableu(t)belongs to a linear spaceEcalled the phase space, andF is a mapping ofEinto itself. The semigroupS(t)_{t 0}denotes the solution map associated to problem (1)–(2):

S(t) :u02E!u(t)2E: (3)

Mathematics Sub ject Classi…cations: 20F05, 20F10, 20F55, 68Q42.

ySchool of Mathematics and statistics, Southwest University, Chongqing 400715, P. R. China

zCorresponding author, School of Mathematics and statistics, Southwest University, Chongqing 400715, P. R. China

36

IfB is a bounded absorbing set, then A= \

s 0

[

t s;u02B

S(t)u₀ (4)

is a global attractor for problem (1)–(2).

DEFINITION 1.1 ([14]). LetE be a …nite-dimensional subspaceE, and letB be a bounded absorbing set in E. Suppose there exists a number t (B)>0 such that for allu02B and allt > t (B), there exists a sequencefu^k(t)g^N E such that

ku^k(t) S(t)u0k^E!0; k! 1: (5) Then the sequence of setsA^k de…ned by

A^k = \

s 0

[

t s;u02B

u^k(t) (6)

is called anasymptotic attractor of the problem (1)–(2).

In an asymptotic attractor A^k, we know k kE is the module of phase space E, u^k(t) depends on the initial value of u₀, and t (B) only depends on the radius of absorbing sets. In other words, t is consistent with u0 of B. When k ! +1, if the limit value exists, then the limit value is a global attractor; otherwise, there is no global attractor. we can discuss the structure of A^k, since u^k is the solution of …nite dimensional dynamical systems. (5) guarantees the asymptotic approximation ofu^k(t) to the real solution of u(t), and not only the approximation. In the next section, we construct a …nite dimensional asymptotic solution to the generalized Benjamin-Bona- Mahony equations, and then prove the asymptotic solution to the real solution. Then we give the asymptotic attractor of the generalized Benjamin-Bon-Mahony equations.

In this paper, we will show the existence of the asymptotic attractor for the following generalized Benjamin-Bona-Mahony equations with periodic boundary conditions

ut ut u+r F(u) =h(x); (7)

@^ju(x; t)

@x^j_i =@^ju(x+ 2 ei; t)

@x^j_i ; j= 0;1;2; i= 1;2: (8) u(x;0) =u0(x); u0(x) =u0(x+ 2 ei); (9) Z

u(x; t)dx= 0; (10)

where x = (x₁; x₂) 2 , = [0;2 ] [0;2 ], e₁ = (1;0), e₂ = (0;1); and are positive constants, F = (F₁(s); F₂(s)) is a given vector …led satisfying the following properties:

(i) Fk(0) = 0,k= 1;2;

(ii) the functionFk,k= 1;2 are twice continuously di¤erentiable inR¹;

(iii) the functionsf_k(s) = _ds^dF_k(s),k= 1;2, satisfy the growth conditions jf_k(s)j C(1 +jsj^m); k= 1;2; 0 m <2:

The existence and uniqueness of solutions, as well as the decay rates of solutions for this equation was studied by many authors, see, for example, [1, 2, 3]. On the other hand, the long-time behavior for this equation were considered also by many authors, see, for example, [4, 9, 11, 12, 13, 15, 16, 17].

Here, by the method of orthogonal decomposition, we show the existence of as- ymptotic attractor for problem (7)–(10). Furthermore, the dimensions estimate of the asymptotic attractor is obtained. Throughout this paper, we setkuk²=R

juj²dxand H__per² ( ) =:

(

u:D u2L²( ); 80 j j 2;

Z

u(x; t)dx= 0; u(x; t) =u(x+ 2 ei; t); x2R² )

:

Applying Faedo-Galerkin method similar to [4], it is easy to prove that the problem (7)–(10) has a unique solution u(t)2H__per² ( ) ifu0(x)2H__per² ( ) andh(x)2 L²( ).

Moreover, there aret0>0 and ₀>0 such that B=n

u(t)2H__per² ( ) :ku(x; t)k²+ kru(x; t)k^{2 2}0; t t₀o is a bounded absorbing set. Now we are in a position to state our main result:

THEOREM 1.2. If u0(x) 2 H__per² ( ) and h(x) 2 L²( ), the semigroup S(t) as- sociated with problems (7)–(10) possesses an asymptotic attractor A^k in H__per² ( ).

Moreover, the dimension of A^k satis…es N_Ak= min

(

N 2N: (khk+ 2C₁ ¹² ₀(1 + ^m₀ ))² C2 (N+ 1)^{2 2}₀ 1;

2 C2

p2C₁(1 + ^m₀)

N+ 1 + C_{4 0} 2(N+ 1)^{2 12}

!2

<1 )

;

where C2= minf (N+ 1)²;₂ g.

2 Asymptotic Attractor

In this section, we show the existence of asymptotic attractor for problem (7)–(10) by the method of the orthogonal decomposition. Let

fcosk1x1cosk2x2; cosk1x1sink2x2; sink1x1cosk2x2; sink₁x₁sink₂x₂; k₁; k₂= 1;2; :g

be an orthogonal basis ofL_²_per( )and denote

HN = spanfcosk1x1cosk2x2; cosk1x1sink2x2; sink1x1cosk2x2; sink1x1sink2x2; k1; k2= 1;2; :::; Ng:

LetPN: L_²_per( )!HN andQN =I PN. For anyu(x; t)2L_²_per( ), we denote p=PNuandq=QNu:

By projecting (7) on the HN, we have

p_t p_t p+P_N(r F(u)) =P_Nh (11) and

qt qt q+QN(r F(u)) =QNh: (12) For anyu0(x)2B, we set u^k =p+q^k satisfying:

8<

:

q⁰_t q⁰_t q⁰+Q_N(r F(p)) =Q_Nh;

q⁰(x; t) =q⁰(x+ 2 e_i; t); i= 1;2;

q⁰(x;0) =Q_Nu₀

(13)

and 8

<

:

q_t^k q^k_t q^k+Q_N(r F(u^{k 1})) =Q_Nh;

q^k(x; t) =q^k(x+ 2 e_i; t); i= 1;2;

q^k(x;0) =Q^k_Nu₀: where Q^k_N =Q_N Q₂^k+1_N; k= 1;2; :

Thus by (12)–(13), we can get a sequence fu^k(t)g for problem (7)–(10). To prove Theorem 1, it su¢ ces to check the condition (5), that is, to prove the following Lemmas 2.2 and 2.3.

LEMMA 2.1. Under the hypotheses ofq=Q_NuforN2N, we can get krqk² 2(N+ 1)²kqk²:

PROOF. Here we have u =

X1 k;k1;k2=1

u¹_kcosk1x1cosk2x2+u²_kcosk1x1sink2x2

+u³_ksink₁x₁cosk₂x₂+u⁴_ksink₁x₁sink₂x₂ ; where u¹_k, u²_k,u³_k,u⁴_k are constants. Noting that

q=QNuforN 2N, it follows that

q =

X1 k;k1;k2=N+1

u¹_kcosk₁x₁cosk₂x₂+u²_kcosk₁x₁sink₂x₂ +u³_ksink1x1cosk2x2+u⁴_ksink1x1sink2x2

and

kqk²= X1 k=N+1

ju¹_kj²+ju²_kj²+ju³_kj²+ju⁴_kj² :

Therefore

rq = @q

@x1

; @q

@x2

=

X1 k;k1;k2=N+1

h

k₁u¹_ksink₁x₁cosk₂x₂ k₁u²_ksink₁x₁sink₂x₂ +k1u³_kcosk1x1cosk2x2+k1u⁴_ksink1x1sink2x2

i

; X1

k;k1;k2=N+1

h

k₂u¹_kcosk₁x₁sink₂x₂+k₂u²_kcosk₁x₁sink₂x₂

k2u³_ksink1x1sink2x2+k4u⁴_ksink1x1cosk2x2

i!

;

and

krqk² = @q

@x₁

2

+ @q

@x₂

2

=

X1 k;k1;k2=N+1

(k²₁+k²₂)(ju¹_kj²+ju²_kj²+ju³_kj²+ju⁴_kj²)

2(N+ 1)² X1 k=N+1

(ju¹_kj²+ju²_kj²+ju³_kj²+ju⁴_kj²)

= 2(N+ 1)²kqk²:

LEMMA 2.2. Assume thatu(x; t)is a solution for problem (7)–(10) withu₀(x)2B, and q^k(k = 0;1;2; )satisfy (12)–(13). Then there areN₀ 2Nand t₁(B)>0 such that for N N₀;

ku^kk²+ kru^kk² 2 ²₀; t t₁(B); k= 0;1;2; : (14)

PROOF. We only need to prove the following inequality:

kq^kk²+ krq^kk^{2 2}0: (15)

Here we verify (15) by the inductive method. Firstly, multiplying (12) byq⁰, we have 1

2 d

dt(kq⁰k²+ krq⁰k²) + krq⁰k² kq⁰k(khk+kr F(p)k) krq⁰k

p2(N+ 1)(khk+kf₁(p)p_x1+f₂(p)p_x2k) krq⁰k

p2(N+ 1)(khk+ 2C1(1 +jpj^m)krpk) krq⁰k

p2(N+ 1) khk+ 2C1(1 + ^m₀ ) ¹² ₀

2krq⁰k²+ 1

4 (N+ 1)² khk+ 2C1

1

2 0(1 + ^m₀)

2

: It follows that

d

dt(kq⁰k²+ krq⁰k²) + krq⁰k² 1

2 (N+ 1)² khk+ 2C₁ ¹² ₀(1 + ^m₀) ²: Note that

krq⁰k²=

2krq⁰k²+

2krq⁰k² (N+ 1)²kq⁰k²+

2 krq⁰k² C2(kq⁰k²+ krq⁰k²);

where C₂= minf (N+ 1)²;₂ g. Then we have

d

dt(kq⁰k²+ krq⁰k²) +C₂(kq⁰k²+ krq⁰k²) khk+ 2C1

1

2 0(1 + ^m₀ ) ² 2 (N+ 1)² : By Gronwall’s Lemma, we have

kq⁰(t)k²+ krq⁰(t)k²

(kq⁰(0)k²+ krq⁰(0)k²)e ^C2t+ khk+ 2C1

1

2 0(1 + ^m₀)

2

2C2 (N+ 1)² (1 e ^C2t):

There exists at₁₁(B)>0, such that for8t t₁₁(B), we have

kq⁰(t)k²+ krq⁰k² khk+ 2C₁ ¹² ₀(1 + ^m₀ ) ² C2 (N+ 1)² : LetN be so large that

khk+ 2C1

1

2 0(1 + ^m₀)

2

C2 (N+ 1)^{2 2}₀ 1; (16)

we have

kq⁰k²+ krq⁰k^{2 2}0; t t₁₁(B): (17) Now assume that kq^{k 1}k²+ krq^{k 1}k^{2 2}0 holds, we shall prove that for 8k (15) holds. Multiplying (13) byq^k, we have

1 2

d

dt(kq^kk²+ krq^kk²) + krq^kk² khk+ 2C1

1

2 0(1 + ^m₀ )² kq^kk: By using similar argument as above, we can obtain

d

dt(kq^kk²+ krq^kk²) +C2(kq^kk²+ krq^kk²) khk+ 2C₁ ¹² ₀(1 + ^m₀ ) ² 2 (N+ 1)² : By Gronwall’s Lemma, there exists at₁₂(B)>0, such that for8t t₁₂(B), we have

kq^k(t)k²+ krq^kk² khk+ 2C1

1

2 0(1 + ^m₀) ² C2 (N+ 1)² : LetN be so large that

khk+ 2C1

1

2 0(1 + ^m₀) ²

C2 (N+ 1)^{2 2}₀ 1; (18)

we have

kq^kk²+ krq^kk^{2 2}0; t t₁₁(B): (19) Lett₁(B) = maxft₁₁(B); t₁₂(B)g. Then (15) follows from (17) and (19). The proof of Lemma 2.2 is completed.

LEMMA 2.3. Under the hypotheses of Lemma 2.2, there areN₁2Nandt₂(B)>0 such that forN > N₁ we have

kq^k qk²+ krq^k rqk²!0; k! 1; t t₂(B): (20) PROOF. Here we verify (20) by the inductive method. Firstly, setw⁰=q⁰ q, by (11) and (12) we have

w_t⁰ w⁰_t w⁰+Q_N(r F(p) r F(u)) = 0: (21) Multiplying (21) byw⁰ we obtain

1 2

d

dt(kw⁰k²+ krw⁰k²) + krw⁰k² kr F(p) r F(u)kkw⁰k 4C1

1

2 0(1 + ^m₀)kw⁰k: By using similar arguement as above, we can obtain

d

dt(kw⁰k²+ krw⁰k²) +C2(kw⁰k²+ krw⁰k²) 8C₁^{2 2}₀(1 + ^m₀)² (N+ 1)² :

By Gronwall’s Lemma, there exists at₂₀(B)>0such that kw⁰(t)k²+ krw⁰(t)k² 16C₁^{2 2}₀(1 + ^m₀)²

C2 (N+ 1)² ; t t₂₀(B): (22) Denote w^k=q^k q, by (11) and (13), we have

w^k_t w_t^k w^k+QN r F(u^{k 1}) r F(u) = 0; (23) where k= 1;2; : Multiplying (23) byw^k, we have

1 2

d

dt(kw^kk²+ krw^kk²) + krw^kk²+ QN(r F(u^{k 1}) r F(u)); w^k = 0;

Now let us consider the last term

QN(r F(u^{k 1}) r F(u)); w^k

= X2 i=1

[fi(u^{k 1})u^k_xi¹ fi(u)uxi]; QNw^k

!

= X2 i=1

fi(u^{k 1})w_x^k_i ¹+ fi(u^{k 1}) fi(u) uxi ; QNw^k

!

= X2 i=1

fi(u^{k 1})w^k_xi¹; QNw^k

!

+ X2 i=1

Z 1 0

f_i⁰( u^{k 1}+ (1 )u)d w^{k 1}uxi; QNw^k

! :

So we have 1 2

d

dt(kw^kk²+ krw^kk²) + krw^kk²

2C1(1 +ju^{k 1}j^m)krw^{k 1}kkw^kk+C3kw^{k 1}kkrukkw^kk 2C₁(1 + ^m)krw^{k 1}k 1

p2(N+ 1)krw^kk +C3

1 2 0

1

2(N+ 1)²krw^{k 1}kkrw^kk

2krw^kk²+ 1 2

p2C1(1 + ^m₀ )

(N+ 1) + C_{3 0} 2(N+ 1)^{2 12}

!2

krw^{k 1}k²: It follows that

d

dt(kw^kk²+ krw^kk) +C₂(kw^kk²+ krw^kk²)

1 p

2C1(1 + ^m₀)

N+ 1 + C_{3 0} 2(N+ 1)^{2 12}

!2

krw^{k 1}k²; (24)

where(k= 1;2; :). Letk= 1in (24), we have d

dt(kw¹k²+ krw¹k²) +C₂(kw¹k²+ krw¹k²)

1 p

2C1(1 + ^m₀ )

N+ 1 + C_{3 0} 2(N+ 1)^{2 12}

!2

krw⁰(t)k²: (25) By Gronwall’s lemma, there is at₂₁(B)>0such that

kw¹k²+ krw¹k² 2

C₂

p2C1(1 + ^m₀)

(N+ 1) + C_{3 0} 2(N+ 1)^{2 12}

!2

krw⁰(t)k²; t t₂₁(B): (26) By the inductive method, there is a t_2k(B)>0such that

kw^kk²+ krw^kk² 2^k

C₂^{k k}

p2C1(1 + ^m₀)

N+ 1 + C_{3 0} 2(N+ 1)^{2 12}

!2k

krw⁰(t)k²; t t_2k(B); (27) where k= 1;2 :IfN is large enough, such that

2 C2

p2C₁(1 + ^m₀ )

N+ 1 + C_{4 0} 2(N+ 1)^{2 12}

!2

<1; (28)

then (20) follows from (22) and (27). The proof of Lemma 2.3 is completed.

In the above lemma, the asymptotic approximation of the real solution is proved, and the dimension of the asymptotic solution is estimated. Now we explain the com- pactness ofA^k inH__per² ( ). It is indispensable. First we use the characterisation

A^k = \

s 0

[

t s;u02B

u^k(t):

Since for t s, the sets S

t s;u02Bu^k(t) form a sequence of nonempty compact sets decreasing as t increases, their intersection A^k is nonempty and compact. Next, to show invariance, suppose that

x2 A^k =fy:9tn! 1; S^k(tn)u0!yg;

we …nd that there exist sequencesft_ngwith t_n! 1such thatS^k(t_n)u₀!xand S^k(t)S^k(tn)u0=S^k(t+tn)u0!S^k(t)u0

sinceS(t)is continuous. SoS(t)A^k A^k. To show equality, fort_n t+s, the sequence S^k(t_n t)u₀ is in the set

[

t s;u02B

u^k(t)

and so possesses a convergent subsequence S^k(t_nj t)u₀ ! y, and so y 2 A^k. But sinceS(t)is continuous,

x= lim

j!1S(t)S(t_nj t)x_nj =S(t)y;

and so A^K S(t)A^k. Thus S(t)A^k = A^k for all t s. The proof of invariance is completed. Therefore,A^k is the asymptotic attractor.

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