EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 1, No. 2, 2008, (3-10)
ISSN 1307-5543 – www.ejpam.com

Asymptotic Attractors of Benjamin-Bona-Mahony Equations

Chaosheng Zhu∗†

School of mathematics and statistics, Southwest University, Chongqing, 400715, P. R. China

Abstract. In this paper, we consider the long time behavior of solution for the Benjamin-Bona-Mahony
equations with periodic boundary conditions. By the method of orthogonal decomposition, we show
that the existence of asymptotic attractor which overcome difficulty come from the precision of approx-
imate inertial manifolds. Moreover, the dimensions estimate of the asymptotic attractor is obtained.
Key words: Benjamin-Bona-Mahony equation, Asymptotic attractor, Dimensions estimate, Orthogonal
decomposition.

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 infinite dimensional dynamical systems, see, for
example, [6, 8]. Inertial manifold is a finite dimensional invariant manifold in the phase
space H of the system which attracts exponentially all orbits. It is constructed as the graph of
a mapping from PH to (I − P)H, where P is a projection of finite 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 [7]
have been introduced.

But the precision of approximate inertial manifolds is inextricable difficulty at all times. To
overcome this difficulty, recently, new concept of asymptotic attractor has been introduced
[12].

Now let us recall the definition of asymptotic attractor. We consider the solution u(t) of a
differential equation

ut + Au= F(u), (1.1)

with initial data
u(0) = u0. (1.2)

The variable u(t) belongs to a linear space E called the phase space, and F is a mapping of E

into itself. The semigroup
n

S(t)
o

t≥0
associated to problems (1.1)-(1.2):

S(t) : u0 ∈ E→ u(t) ∈ E. (1.3)

∗Corresponding author. Email address: zhu.cauchy@yahoo.com.cn (Chaosheng Zhu)
†This work was supported by Doctor Fund of Southwest University (SWUB2008003)

http://www.ejpam.com 3 c© 2007 EJPAM All rights reserved.



Chaosheng Zhu / Eur. J. Pure Appl. Math, 1 (2008), (3-10) 4

If B is a bounded absorbing set, then

A =
⋂

s≥0

⋃

t≥s,u0∈B

S(t)u0. (1.4)

is global attractor for problems (1.1)-(1.2).

Definition 1.1. [12]. Let E be a finite-dimensional subspace of the phase space E, and let B be
a bounded absorbing set in E. Suppose there exists a number t∗(B) > 0 such that for all u0 ∈ B
and all t ≥ t∗(B), there exists a sequence {uk(t)}N ⊂ E such that

‖uk(t)− S(t)u0‖E→ 0, k→∞. (1.5)

Then the sequence of setsA k defined by

A k =
⋂

s≥0

⋃

t≥s,u0∈B

uk(t) (1.6)

is called an asymptotic attractor of the problem (1.1)-(1.2).

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

ut −δux x t −µux x + uux = f (x), (1.7)

u(x , 0) = u0(x), (1.8)

where u(x , t) = u(x + 2π, t), x ∈ R1,
∫ 2π

0
u(x , t)d x = 0 and δ, µ are positive constants.

The Benjamin-Bona-Mahony equation was proposed in [3] as a model for propagation of
long waves which incorporates nonlinear dispersive and dissipative effects. The existence and
uniqueness of solutions, as well as the decay rates of solutions for this equation were studied
by many authors, see, for example, [1, 2, 4]. On the other hand, the long-time behavior for
this equation were considered also by many authors, see, for example, [5,9–11,13–15].

Here, by the method of orthogonal decomposition, we show that the existence of asymp-
totic attractor for problems (1.7)-(1.8). Furthermore, the dimensions estimate of the asymp-

totic attractor is obtained. Throughout this paper, we set Ω=(0,2π), ‖u‖2 =
∫ 2π

0
|u|2d x and

Ḣ1
per(Ω) =:

¨

u

�

�

�

�

�

u ∈ L2(Ω), ux ∈ L2(Ω);

∫ 2π

0

u(x , t)d x = 0;

u(x , t) = u(x + 2π, t), x ∈ R1

«

.

Applying Faedo-Galerkin method, it is easy to prove that the problems (1.7)-(1.8) exists
a unique solution u(t) ∈ Ḣ1

per(Ω) if u0(x) ∈ Ḣ1
per(Ω) and f (x) ∈ L2(Ω). Moreover, there are

t0 > 0 and ρ0 > 0 such that

B =
n

u(t) ∈ Ḣ1
per(Ω) : ‖u(t)‖2+δ‖ux(t)‖2 ≤ ρ2

0 , t ≥ t0

o

is a bounded absorbing set. Now we are in position to state our main result:



Chaosheng Zhu / Eur. J. Pure Appl. Math, 1 (2008), (3-10) 5

Theorem 1.1. If u0(x) ∈ Ḣ1
per(Ω) and f (x) ∈ L2(Ω), the semigroup S(t) associated with prob-

lems (1.7)-(1.8) possesses an asymptotic attractor A k in Ḣ1
per(Ω). Moreover, the dimensions of

A k satisfies

NA k = min









N ∈ N

�

�

�

�

�

�

2(4δ−
3
4ρ2

0 + ‖ f ‖)
2

ρ2
0c1µ(N + 1)2

≤ 1 ,
2
�p

2ρ0δ
− 1

4 + cρ0δ
− 1

2

�2

c1µ(N + 1)2
< 1









,

where c1 = min
�

µ(N+1)2

2
, µ

2δ

�

.

2. Asymptotic Attractor

In this section, we show that the existence of asymptotic attractor for problems (1.7)-
(1.8) by the method of orthogonal decomposition. Let {sin kx , cos kx , k = 1, 2, · · ·} is an or-
thonormal basis of L̇2

per([0,2π]), denote

HN = Span{sin kx , cos kx , k = 1, 2, · · · , N}.

Let PN : L̇2
per([0,2π])→ HN , QN = I − PN . For any u(x , t) ∈ L̇2

per([0,2π]), we denote

p = PN u, q =QN u.

By projecting (1.7) on the HN , we have

pt −δpx x t −µpx x + PN (uux) = PN f , (2.1)

and
qt −δqx x t −µqx x +QN (uux) =QN f . (2.2)

For any u0(x) ∈ B, we set uk = p+ qk satisfies:






q0
t −δq0

x x t −µq0
x x +QN (ppx) =QN f ,

q0(x , t) = q0(x + 2π, t),
q0(x , 0) =QN u0.

(2.3)







qk
t −δqk

x x t −µqk
x x +QN (uk−1uk−1

x ) =QN f ,
qk(x , t) = qk(x + 2π, t),

qk(x , 0) =Qk
N u0.

(2.4)

where Qk
N =QN −Q2k+1N , k = 1,2, · · · .

Thus by (2.3)-(2.4), we can get a sequence {uk(t)} for problems (1.7)-(1.8). To prove
Theorem 1.2, we only to check the condition (1.5), that is, we only prove the following
Lemma 2.1 and Lemma 2.2.



Chaosheng Zhu / Eur. J. Pure Appl. Math, 1 (2008), (3-10) 6

Lemma 2.1. Assume that u(x , t) is solution for problems (1.7)-(1.8) with u0(x) ∈ B, and qk

(k = 0,1, 2, · · · ) satisfy (2.3)-(2.4), there are N0 ∈ N and t∗1(B) > 0 such that for N ≥ N0 we
have

‖uk‖2+δ‖uk
x‖

2 ≤ 2ρ2
0 , t ≥ t∗1(B), k = 0,1, 2, · · · . (2.5)

Proof : We only need to prove the following inequality:

‖qk‖2+δ‖qk
x‖

2 ≤ ρ2
0 . (2.6)

Here we verify (2.6) by the inductive method. Firstly, multiplying (2.3) by q0, we have

1

2

d

d t
(‖q0‖2+δ‖q0

x‖
2) +µ‖q0

x‖
2

≤ ‖p‖
1
2 ‖px‖

3
2 ‖q0‖+ ‖ f ‖‖q0‖

≤ ρ
1
2
0 δ
− 3

4ρ
3
2
0 ‖q

0‖+ ‖ f ‖‖q0‖

≤ (δ−
3
4ρ2

0 + ‖ f ‖)‖q
0‖

≤ (δ−
3
4ρ2

0 + ‖ f ‖)
1

N + 1
‖q0

x‖

≤
µ

2
‖q0

x‖
2+

1

2µ(N + 1)2
(ρ2

0δ
− 3

4 + ‖ f ‖)2.

It follows that

d

d t
(‖q0‖2+δ‖q0

x‖
2) +µ‖q0

x‖
2 ≤
(ρ2

0δ
− 3

4 + ‖ f ‖)2

µ(N + 1)2
.

Noting that

µ‖q0
x‖

2 =
µ

2
‖q0

x‖
2+
µ

2
‖q0

x‖
2

≥
�

µ(N + 1)2

2
‖q0‖2+

µ

2δ
δ‖q0

x‖
2

�

≥ c1(‖q0‖2+δ‖q0
x‖

2),

we have

d

d t
(‖q0‖2+δ‖q0

x‖
2) + c1(‖q0‖2+δ‖q0

x‖
2)≤

(ρ2
0δ
− 3

4 + ‖ f ‖)2

µ(N + 1)2
.

By Gronwall’s Lemma, we have

‖q0(t)‖2+δ‖q0
x(t)‖

2 ≤ (‖q0(0)‖2+δ‖q0
x(0)‖

2)e−c1 t

+
(ρ2

0δ
− 3

4 + ‖ f ‖)2

c1µ(N + 1)2
(1− e−c1 t).



Chaosheng Zhu / Eur. J. Pure Appl. Math, 1 (2008), (3-10) 7

There exists a t∗11(B)> 0, such that for ∀t ≥ t∗11(B), we have

‖q0(t)‖2+δ‖q0
x(t)‖

2 ≤
2(ρ2

0δ
− 3

4 + ‖ f ‖)2

c1µ(N + 1)2
.

Let N is large enough, such that

2(ρ2
0δ
− 3

4 + ‖ f ‖)2

ρ2
0c1µ(N + 1)2

≤ 1, (2.7)

we have
‖q0(t)‖2+δ‖q0

x(t)‖
2 ≤ ρ2

0 , t ≥ t∗11(B). (2.8)

Now assume that ‖qk−1‖2+δ‖qk−1
x ‖

2 ≤ ρ2
0 holds, we shall prove that for ∀k (2.6) is holds.

Multiplying (2.4) by qk, we have

1

2

d

d t
(‖qk‖2+δ‖qk

x‖
2) +µ‖qk

x‖
2 ≤ (4δ−

3
4ρ2

0 + ‖ f ‖)‖q
k‖.

By using similar argument as above, we can obtain

d

d t
(‖qk‖2+δ‖qk

x‖
2) + c1(‖qk‖2+δ‖qk

x‖
2)≤

(4ρ2
0δ
− 3

4 + ‖ f ‖)2

µ(N + 1)2
.

By Gronwall’s Lemma, there exists a t∗12(B)> 0 such that for ∀t ≥ t∗12(B) we have

‖qk(t)‖2+δ‖qk
x(t)‖

2 ≤
2(4δ−

3
4ρ2

0 + ‖ f ‖)
2

c1µ(N + 1)2
.

Let N is large enough, such that

2(4δ−
3
4ρ2

0 + ‖ f ‖)
2

ρ2
0c1µ(N + 1)2

≤ 1, (2.9)

we have
‖qk(t)‖2+δ‖qk

x(t)‖
2 ≤ ρ2

0 , t ≥ t∗12(B). (2.10)

Let t∗1(B) = max(t∗11(B), t∗12(B)), then (2.6) follows from (2.8) and (2.10). The proof of
Lemma 2.1 is completed.

Lemma 2.2. Under the hypotheses of Lemma 2.1, there are N1 ∈ N and t∗2(B) > 0 such that for
N ≥ N1 we have

‖qk − q‖2+δ‖qk
x − qx‖2→ 0, k→∞, t ≥ t∗2(B). (2.11)



Chaosheng Zhu / Eur. J. Pure Appl. Math, 1 (2008), (3-10) 8

Proof : Here we verify (2.11) by the inductive method. Firstly, set w0 = q0 − q, by (2.2)
and (2.3) we have

w0
t −δw0

x x t −µw0
x x +QN (ppx − uux) = 0. (2.12)

Multiplying (2.12) by w0 we obtain

1

2

d

d t
(‖w0‖2+δ‖w0

x‖
2) +µ‖w0

x‖
2

≤ 2‖u‖
1
2 ‖ux‖

3
2 ‖w0‖ ≤ 2ρ2

0δ
− 3

4 ‖w0‖.

By using similar argument as above, we can obtain

d

d t
(‖w0‖2+δ‖w0

x‖
2) + c1(‖w0‖2+δ‖w0

x‖
2)≤

4ρ4
0

δ
3
2µ(N + 1)2

. (2.13)

By Gronwall’s Lemma, there exists a t∗20(B)> 0, such that

‖w0(t)‖2+δ‖w0
x(t)‖

2 ≤
8ρ4

0

c1δ
3
2µ(N + 1)2

, t ≥ t∗20(B). (2.14)

Denote wk = qk − q, by (2.2) and (2.4), we have

wk
t −δwk

x x t −µwk
x x +QN (u

k−1uk−1
x − uux) = 0, (2.15)

where k = 1,2, · · · . Here we note that

uk−1uk−1
x − uux = uk−1wk−1

x +wk−1ux .

Multiplying (2.15) by wk we have

1

2

d

d t
(‖wk‖2+δ‖wk

x‖
2) +µ‖wk

x‖
2

≤
p

2δ−
1
4ρ0‖wk−1

x ‖‖w
k‖+δ−

1
2ρ0‖wk−1‖L∞‖wk‖

≤
�p

2ρ0δ
− 1

4 + cρ0δ
− 1

2

�

‖wk−1
x ‖‖w

k‖

≤
1

N + 1

�p
2ρ0δ

− 1
4 + cρ0δ

− 1
2

�

‖wk−1
x ‖‖w

k
x‖

≤
1

2µ(N + 1)2
�p

2ρ0δ
− 1

4 + cρ0δ
− 1

2

�2
‖wk−1

x ‖
2+
µ

2
‖wk

x‖
2.

It follows that

d

d t
(‖wk‖2+δ‖wk

x‖
2) + c1(‖wk‖2+δ‖wk

x‖
2)

≤
1

µ(N + 1)2
�p

2ρ0δ
− 1

4 + cρ0δ
− 1

2

�2
‖wk−1

x ‖
2. (2.16)



REFERENCES 9

where k = 1, 2, · · · . Let k = 1 in (2.16), we have

d

d t
(‖w1‖2+δ‖w1

x‖
2) + c1(‖wk‖2+δ‖wk

x‖
2)

≤
1

µ(N + 1)2
�p

2ρ0δ
− 1

4 + cρ0δ
− 1

2

�2
‖w0

x‖
2. (2.17)

By Gronwall’s lemma, there is a t∗21(B)> 0 such that

‖w1(t)‖2+δ‖w1
x(t)‖

2

≤
2

c1µ(N + 1)2
�p

2ρ0δ
− 1

4 + cρ0δ
− 1

2

�2
‖w0

x(t)‖
2, t ≥ t∗21(B). (2.18)

By the inductive method, there is a t∗2k(B)> 0 such that

‖wk‖2+δ‖wk
x‖

2

≤
2k

ck
1µ

k(N + 1)2k

�p
2ρ0δ

− 1
4 + cρ0δ

− 1
2

�2k
‖w0

x(t)‖
2, t ≥ t∗2k(B) (2.19)

where k = 1, 2, · · · . If N is large enough, such that

2
�p

2ρ0δ
− 1

4 + cρ0δ
− 1

2

�2

c1µ(N + 1)2
< 1, (2.20)

then (2.11) follows from (2.14) and (2.19). The proof of Lemma 2.2 is completed.

References

[1] J. Albert, Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation,
J. Diff. Equ., 63(1):117-134, 1986.

[2] J. Avrin, The generalized Benjamin-Bona-Mahony equation in Rn with singular initial data, Non-
lin. Anal., 11(1): 139-147, 1987.

[3] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear disper-
sive systems, Philos. Trans. Roy. Soc. Londan , 272(1): 47-78, 1972.

[4] P. Biler, Long time behaviour of solutions of the generalized Benjamin-Bona-Mahony equation in
two-space dimensions, Diff. Integ. Equ., 5(4): 891-901, 1992.

[5] A. O. Celebi, V. K. Kalantarov and M. Polat, Attractors for the generalized Benjamin-Bona-Mahony
equation, J. Diff. Equ., 157(2): 439-451, 1999.

[6] S. N. Chow and K. Lu, Inertial manifolds for flows in Banach spaces, J. Diff. Eqns., 74(1): 285-
317, 1988.

[7] C. Foias, O. Manley and R. Temam, Sur l’interaction des petits et grands tourbillons dans les
ecoulements turbulents, C. R. Acad. Sci. Paris, Serie I, 305: 497-500, 1987.

[8] C. Foias, G. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Diff.
Eqns., 73(2): 309-353, 1988.

[9] M. Stanislavova, On the global attractor for the damped Benjamin-Bona-Mahony equation, Pro-
ceedings of the fifth international conference on dynamical systems and differential equations, June
16-19, 2004, Pomona, CA, USA



REFERENCES 10

[10] M. Stanislavova, A. Stefanov and B. Wang, Asymptotic Smoothing and attractors for the gener-
alized Benjamin-Bona-Mahony equation on R3, J. Diff. Equ., 219(2): 451-483, 2005.

[11] B. Wang, Strong Attracrors for the Benjamin-Bona-Mahony equation, Appl. Math. Lett., 10(1):
23-28, 1997.

[12] G. X. Wang, Z. R. Liu, The asymptotic attractors of Kuramoto-Sivashinsky equation, Acta Math.
Appl. Sinica(in Chinese), 23(2): 329-336, 2000.

[13] B. Wang, W. Yang, Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J.
Phys. A, Math. Gen., 30(12):4877-4885, 1997.

[14] C. S. Zhu, Global attractor for the damped Benjamin-Bona-Mahony equations on R1, Appl. Anal.,
86(1):59-65, 2007.

[15] C. S. Zhu, C. L. Mu, Exponential decay estimates for time-delayed Benjamin-Bona-Mahony equa-
tions, Appl. Anal., 87(4):401-07, 2008.