Exponential Attractors of the Nonclassical Diffusion Equations with Lower Regular Forcing Term ()
![](https://www.scirp.org/html/htmlimages\3-2340114x\8898c970-54e4-4831-94c7-ec0770e16e37.png)
1. Introduction
We consider the asymptotic behavior of solutions to be the following nonclassical diffusion equation:
(1.1)
where
is a bounded domain with smooth boundary
, and the external forcing term
, non-linear function
with
and satisfies the following conditions:
(1.2)
and
(1.3)
where
is a positive constant and
is the first eigenvalue of
on
. The number
is called the critical exponent; since the nonlinearity
is not compact in this case, this is one of the essential difficulties in studying the asymptotic behavior.
This equation appears as a nonclassical diffusion equation in fluid mechanics, solid mechanics and heat conduction theory, see for instance [1] -[3] and the references therein.
Since Equation (1.1) contains the term
, it is different from the usual reaction diffusion equation essentially. For example, the reaction diffusion equations has some smoothing effect, that is, although the initial data only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity. However, for Equation (1.1), if the initial data
belongs to
, the solution
with
is always in
and has no higher regularity because of
, which is similar to the hyperbolic equation. Consequently, its dynamics would be more complex and interesting.
The long-time behavior of the solutions of (1.1) has been considered by many researchers; see, e.g. [4] -[9] , and the references therein. For instance, for the case
, the existence of a global attractor of (1.1) in
was obtained in [4] under the assumptions that
satisfies (1.2) and (1.3) corresponding to ![](https://www.scirp.org/html/htmlimages\3-2340114x\6d9a5fd8-aca6-4676-8bfc-d6702c0197b0.png)
and the additional condition
with
, which essentially requires that the nonlinearity is subcritical. In [7] the authors investigated the existence of the global attractors for
, and proved the asymptotic regularity and existence of exponential attractors for
only under the conditions (1.2)-(1.3). Recently, the authors in [9] showed the asymptotic regularity of solutions of Equation (1.1) in
for any
and for
,
only under the assumptions (1.2)-(1.3).
For the limit of our knowledge, the existence of exponential attractors of Equation (1.1) has not been achieved by predecessors for
. On the other hand, we note that in [10] the authors scrutinized the asymptotic regularity of the solutions for a semilinear second order evolution equation when
, and based on this regularity, they constructed a family of finite dimensional exponential attractors. However, they require the following additional technical assumptions besides (1.2) and (1.3):
![](https://www.scirp.org/html/htmlimages\3-2340114x\683bac8d-8466-41a6-b5da-043142ab70bb.png)
and
![](https://www.scirp.org/html/htmlimages\3-2340114x\fd1c8502-faac-42c8-b08b-cf5de523c8fb.png)
In this article, motivated by the work in [10] -[12] , based on the asymptotic regularity in [9] , we construct a finite dimensional exponential attractor of (1.1) only under the conditions (1.2) and (1.3).
Our main result is Theorem 1.1 Assume
and satisfies (1.2)-(1.3),
. Then the semigroup
associated with problem (1.1) has an exponential attractor
in
.
Remark 1.1 If
is a global attractor of (1.1) in
, we know that
, then Theorem 1.1 implies that fractal dimension of the global attractor
is finite.
2. Notations and Preliminaries
In this section, for convenience, we introduce some notations about the functions space which will be used later throughout this article.
•
with domain
, and consider the family of Hilbert space
with the standard inner products and norms, respectively,
![](https://www.scirp.org/html/htmlimages\3-2340114x\3d778351-ff1b-42ed-9aed-244b7178c4b0.png)
Especially,
means the
inner product and norm, respectively.
•
with the usual norm
. Especially, we denote
and
.
•
are continuous increasing functions.
•
denote the general positive constants,
, which will be different from line to line.
We also need the following the transitivity property of exponential attraction, e.g., see [[12] , Theorem 5.1]:
Lemma 2.1 ([13] ) Let
be subsets of
such that
![](https://www.scirp.org/html/htmlimages\3-2340114x\5ecc261e-bec7-4db0-b461-07966bcb9e7f.png)
for some
and
. Assume also that for all
there holds
![](https://www.scirp.org/html/htmlimages\3-2340114x\e334e3d2-4af1-41ad-a645-97c8fe297222.png)
for some
and
. Then it follows that
![](https://www.scirp.org/html/htmlimages\3-2340114x\f23d6310-de3d-45e7-b7ab-b9cc3c129a3f.png)
where
and
.
3. Exponential Attractor
In this subsection, based on the asymptotic regularity obtained in [9] , we will construct an exponential attractor by the methods and techniques devised in [10] -[12] . We first need the following Lemmas:
Lemma 3.1 ([7] ) Let
satisfies (1.2)-(1.3) and
. Then for any
and any
, there is a unique solution
of (1.1) such that
![](https://www.scirp.org/html/htmlimages\3-2340114x\c7cb8a7d-85e2-485b-9ca7-f6f4dc0476ce.png)
Moreover,the solution continuously depends on the initial data in
.
In the remainder of this section, we denote by
the semigroup associated with the solutions of (1.1)-(1.3).
Lemma 3.2 ([7] ) Under conditions of above Lemma, There is a positive constant
such that for any bounded subset
, there exists
such that
(3.1)
From this Lemma, we know that the semigroup of operators
generalized by (1.1) possesses a bounded absorbing set
in
.
Lemma 3.3 Under conditions of
, and
be two solutions of (1.1) with
, respectively, it follows that
(3.2)
Proof Let
satisfies the following equation
(3.3)
Taking the scalar product of (3.3) with
, we find,
(3.4)
From the condition (1.2), by using the Hölder inequality, and noting the embedding
, we have
![](https://www.scirp.org/html/htmlimages\3-2340114x\91d373e2-79d9-4d5f-a0aa-24ff0680575f.png)
And then, by means of (3.1), we obtain
(3.5)
So, combining with Equation (3.4), (3.5), we get
![](https://www.scirp.org/html/htmlimages\3-2340114x\fa0f9dac-bab6-44d6-81e5-30ab99934028.png)
then using the Gronwall lemma to above inequality, we can conclude our lemma immediately.
Lemma 3.4 ([9] ) Let
and satisfies (1.2), (1.3),
. Then, for any
, there exists a subset
, a positive constant
and a monotone increasing function
such that for any bounded set
,
(3.6)
where
and
depend on
but
is independent of
;
satisfying
(3.7)
for some positive constant
; And
is the unique solution of the following elliptic equation
(3.8)
where the constant
such that
. Furthermore, we know that the solution
only belongs to
when
satisfies (1.2)-(1.3).
Lemma 3.5 ([9] ) Under the assumption of Lemma 3.4, for any bounded subset
, if the initial data
, then the solution
of (1.1) has the following estimates similar to (3.7) in Lemma 3.4, more precisely, we have
(3.9)
where the constant
depends only on
and the
-bound of
.
Lemma 3.6 There exists
such that
(3.10)
Proof For the solution
of (1.1), we now decompose
as follows
(3.11)
where
is a fixed solution of (3.8), and
satisfies the following equation :
(3.12)
At the same time, noticing the embedding
, and from Lemma 3.5 we yield
(3.13)
Taking the inner product of (3.12) with
, we get
(3.14)
By means of (3.1) and (3.13) and together with H
lder, Young inequalities, it follows that
(3.15)
![](https://www.scirp.org/html/htmlimages\3-2340114x\7fb4096f-f233-4c96-8606-65e329a77cab.png)
![](https://www.scirp.org/html/htmlimages\3-2340114x\63c403f7-d8e7-4341-9d31-a0ce581335d3.png)
Thus, combining with (3.14), there holds
![](https://www.scirp.org/html/htmlimages\3-2340114x\b2794348-f4e9-46cc-89a0-0a12e91e1aec.png)
Integrating the above inequality on
and noting
, the proof completes.
Next, we will prepared for constructing an exponential attractor of
in
by applying the abstract results devised in [10] -[12] [14] .
Firstly, for each fixed
, we define
(3.16)
where
is the set obtained in Lemma 3.4. Then, from Lemma 3.5 we know that
(3.17)
Secondly, let us establish some properties of this set.
•
is a compact set in
, due to Lemma 3.4.
•
is positive invariant. In fact, from the continuity of
, we have
(3.18)
• There holds
(3.19)
Indeed, it is apparent that
(3.20)
Hence, (3.19) follows from Lemma 2.1.
• There is
such that
(3.21)
This is a direct consequence of Lemma 3.6.
Therefore such a set
is a promising candidate for our purpose.
Finally, we need the following two lemmas.
Lemma 3.7 For every
, the mapping
is Lipschitz continuous on
.
Proof For
and
we have
(3.22)
The first term of the above inequality is handled by estimate (3.2). Concerning the second one,
(3.23)
Hence, there exists a constant
, such that
(3.24)
On the other hands, for each initial data
, we can decompose the solution
of (1.1) as
(3.25)
where
and
solve the following equations respectively:
(3.26)
and
(3.27)
Therefore, we will have the following lemma:
Lemma 3.8 The following two estimates hold:
(3.28)
and
(3.29)
where the constant
depends only on
and
.
Proof Given two solutions
of Equation (1.1) origination from
, respectively.
Set
![](https://www.scirp.org/html/htmlimages\3-2340114x\f2ebb578-a2e8-4ce4-9416-27b19899a218.png)
where
and
solve the following equations respectively:
(3.30)
and
(3.31)
It is apparent that
and ![](https://www.scirp.org/html/htmlimages\3-2340114x\96d4e17f-87d7-4beb-b681-d39965f7c174.png)
Taking the product of (3.30) with
in
, we get
(3.32)
So
(3.33)
Hence, setting
(3.34)
we have
![](https://www.scirp.org/html/htmlimages\3-2340114x\87488877-5fe9-4110-acd1-0e21fffa7e33.png)
So, we obtain the result (3.28).
On the other hands, taking the product of (3.31) with
in
, we ge
(3.35)
Since
, we have
.
So, from (1.2) and using H
lder inequality, we have
(3.36)
where the constant
comes from the embedding
,
.
From Lemma 3.3, we obtain the inequality
![](https://www.scirp.org/html/htmlimages\3-2340114x\56cd623e-69e6-42ca-aac5-f13a84ea2a20.png)
and an integration on
, we can get the estimate (3.29).
Proof of Theorem 1.1 Applying the abstract results devised in [10] -[12] , from Lemma 3.7 and Lemma 3.8, we can prove the existence of an exponential attractor
for
in
immediately.
Remark 3.9 As a direct consequence of Theorem 1.1 and the a priori estimates given in [[9] , Lemma 3.5] and Lemma 3.8, we decompose
as
, where
is bounded in
for any
and
is the unique solution of (3.8).
Acknowledgements
The authors thank the referee for his/her comments and suggestions, which have improved the original version of this article essentially. This work was partly supported by the NSFC (11061030,11101334) and the NSF of Gansu Province(1107RJZA223), in part by the Fundamental Research Funds for the Gansu Universities.
NOTES
*Corresponding author.