Journal of Applied Mathematics and Physics
Vol.02 No.08(2014), Article ID:47579,11 pages
10.4236/jamp.2014.28086
Uniform Exponential Attractors for Non-Autonomous Strongly Damped Wave Equations
Hongyan Li
College of Management, Shanghai University of Engineering Science, Shanghai, China
Email: lihongyan580@sohu.com
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 3 May 2014; revised 3 June 2014; accepted 15 June 2014
ABSTRACT
In this paper, we study the existence of exponential attractors for strongly damped wave equations with a time-dependent driving force. To this end, the uniform Hölder continuity is established to the variation of the process in the phase apace. In a certain parameter region, the exponential attractor is a uniformly exponentially attracting time-dependent set in the phase apace, and is finite-dimensional no matter how complex the dependence of the external forces on time is. On this basis, we also obtain the existence of the infinite-dimensional uniform exponential attrac- tor for the system.
Keywords:
Exponential Attractor, Uniform Attractor, Strongly Damped Wave Equation
1. Introduction
In this paper, we study the following non-autonomous strongly damped wave equation on a bounded domain with smooth boundary
:
(1.1)
where is a real-valued function on
Let
we make the following assumptions on functions
(1.2)
(1.3)
(1.4)
where are positive constants. And we assume that the external force
belongs to the space
and satisfies
(1.5)
for some given (possibly large) constant.
Wave equations, describing a great variety of wave phenomena, occur in the extensive applications of mathe- matical physics. Equation (1.1) can be regarded as a perturbed equation of a continuous Josephson junction where, see [1] . There is a large literature on the asymptotic behavior of solutions for strongly damped wave equations (see, for instance, [1] - [9] ). In [9] , the author showed the uniform boundedness of the global attractor for large strongly damping and obtained an estimate of the upper bound of the Hausdorff dimen- sion of an attractor for strongly damped wave Equation (1.1) when
is independent of
. But when the equations depend explicitly on
, the case can be complex.
Recently, motivated by [6] , the authors have given a new explicit algorithm allowing to construct the expo- nential attractor, and this method makes it possible to consider more general processes in applications [10] [11] .
An exponential attractor is a compact semi-invariant set of the phase space whose fractal dimension is finite and which attracts exponentially the images of the bounded subsets of the phase space
. In non-autono- mous dynamical systems, instead of a semigroup, we have a so-called (dynamical) process
depending on two parameters
(or
for discrete times). The asymptotic behavior of non-autonomous dy- namical systems is essentially less understood and, to the best of our knowledge, the finite-dimensionality of the limit dynamics was established only for some special (e.g. quasiperiodic) dependence of the external forces on time. Indeed, there exists, at the present time, one of the different approaches for extending the concept of a glo- bal attractor to the non-autonomous case which is based on the embedding of the non-autonomous dynamical system into a larger autonomous one by using the skew-product flow. This approach naturally leads to the so- called uniform attractor
which remains time-independent in spite of the fact that the dynamical system now depends explicitly on the time, see [12] . We note that however the uniform attractor reduces to an autono- mous system via the skew-product flow. It seems natural to generalize the concept of an exponential attractor to the non-autonomous case, see [11] [13] [14] . But in all these articles, the uniform attractor’s approach was used in order to construct an exponential attractor for the non-autonomous system considered and, consequently, an (uniform) exponential attractor remained time-independent. Since, under this approach, an exponential attractor should contain the uniform attractor, all the drawbacks of uniform attractors (artificial infinite-dimensionality and high dynamical complexity) described above are preserved for exponential attractors.
In the present article, we study exponential attractors of the system (1.1) based on the concept of a non-au- tonomous (pullback) attractor. Thus, in the approach, an exponential attractor is also time-dependent. To be more precise, a family of compact semi-invariant (i.e.,
) sets of the dynami- cal process (1.1) is an (non-autonomous) exponential attractor if
1) The fractal dimension of all the sets is finite and uniformly bounded with respect to
2) There exist a positive constant and a monotonic function
such that, for every
and every bounded subset
of
,
(1.6)
We emphasize that the convergence in (1.6) is uniform with respect to and, consequently, under this approach, we indeed overcome the main drawback of global attractors [13] .
This article is organized as follows. In Section 2, we first provide some basic settings and show the absorbing and continuous properties in proper function space about Equation (1.1). In Section 3 and Section 4, we prove the existence of the uniform attractor and exponential attractor of Equation (1.1), respectively. Finally, we prove the existence of infinite-dimensional exponential attractor, and compare it with the non-autonomous exponential attractor in Section 5.
2. Preliminaries
We will use the following notations as that in Pata and Squassina [15] . Let be the (strictly) positive operator on
defined by
with domain
and consider the family of Hilbert spaces with the standard inner products and norms, respec- tively,
and
Then we have
and the compact, dense injections
In particular, naming the first eigenvalue of
, we get the inequlities
We recall the continuous embedding
and the interpolation results: given for any
, there exists
such that
and let
Equation (1.1) is equivalent to the following initial value problem in the Hilbert space E
(2.1)
where
and
It is well known (see, e.g., [3] , [9] ) that, under the above assumptions, Equation (2.1) possesses, for every and
a unique (mild) solution
Thus, Equation (1.1) defines a dynamical process
in the phase space
by
(2.2)
Define a new weighted inner product and norm in as
for any
where is chosen as
(2.3)
Obviously, the norm in (2.3) is equivalent to the usual norm
in
.
Let
where is chosen as
(2.4)
and then the system (1.1) can be written as
(2.5)
where
Lemma 2.1 For any we have
where
(2.6)
Proof. Since is dense in
, and
; we only need to prove lemma 2.1 for any
By (2.4) and (2.6), elementary computation shows
The proof is completed. ,
Lemma 2.2 Let assumptions (1.2)-(1.5) be satisfied. For any initial data, there exists a positive con- stant
depending only on the coefficients of (1.3) and (2.4) and
such that the following dissipative esti- mate holds:
where is a monotonic function and where the positive number
depends also on
(but is indepen- dent of the concrete choice of
).
Proof. Write Let
be the solution of the system (2.5) with the initial
value Taking the inner product
of (2.5) with
, we have
(2.7)
By (1.2), (1.3) and Poincaré inequality, there exist two positive constants such that
(2.8)
(2.9)
By (2.4) and (2.6), Let
By (2.8)-(2.9), we have
(2.10)
where By (2.7) and (2.10),
By Gronwall’s inequality, we have an absorbing property:
This completes the proof. ,
Theorem 2.1 Given any and for the solutions of (2.5) with any two initial data
such that
, we have the following Lipschitz continuity in E
for some.
The proof is similar to Theorem 2 in [15] . ,
Theorem 2.2 For the solutions of (2.5) with different external forces and
satisfying (1.5) and with the initial data
and
, the following contiuity holds:
where and
are independent of
and
The proof is similar to Lemma 4 in [5] .
3. Existence of the Uniform Attractor
The dissipativity property obtained in Lemma 2.2 yields the existence of an absorbing set for the process on
. In the following section, we assume that
holds, where
is specified in (3.11).
Theorem 3.1 The process possesses a uniform attractor
in
Proof. We consider such that
(3.1)
and we introduce the splitting where
satisfies
(3.2)
satisfies
(3.3)
and is the solution of
(3.4)
We now define the families of maps and
in
where
First step: We prove that is bounded in
that the solution of (2.5) is starting in bounded sets of initial data
. The system (3.2) can be written as
(3.5)
where,
(3.6)
Similar to Lemma 2.1, we have
where is as (2.6). Multiply (3.5) by
, so we get
(3.7)
Similar to Lemma 2.2, applying (3.7) and Young, Poincaré, Gronwall inequalities, we obtain
Now we multiply (3.5) by and integrate over
to obtain
(3.8)
with
Then from (3.8) we have
i.e.,
(3.9)
for the first term on the right-hand side of (3.9), we have
(3.10)
By (3.9), (3.10), and Lemma 2.2, there is such that for all
(3.11)
let using the Gronwall’s lemma, we have
(3.12)
By (1.2), (1.3) and (2.8), (2.9), from (3.12), we obtain
(3.13)
Lemma 2.2 and (3.13) imply that is bounded in
.
Second step: Let, we will prove that there exists
independence of
such that
Multiply (3.3) by, we thus obtain
due to Gronwall and Poincaré inequalities, then
(3.14)
Since the embedding is compact, (3.13), (3.14) and the following lemma imply that
is compact in
.
Lemma 3.1 (see [16] ) Let be a complete metric space and
be a subset in
, such that
with and
is compact in
, then
is compact in
.
Third step: Let, the same arguments in the Equation (3.4) lead to
(3.15)
Then from (3.15), Lemma 2.2, and the compactness of, the system (2.5) exists a uniform attractor
in
It is easy to see that the process
defined by (2.5) has the following relation with:
(3.16)
where is an isomorphism of
Since the process possesses a uniform attractor
by (3.16),
also possesses a uniform attractor
,
4. Existence of Exponential Attractors
The main result of this section is the following theorem.
Theorem 4.1 Let the function f and the external force g satisfy the above assumptions. Then, for every ex- ternal force g enjoying (1.5), there exists an exponential attractor of the dynamical process (1.1) which satisfies the following properties:
1) The sets are semi-invariant with respect to
and translation-invariant with respect to time-shifts:
(4.1)
where and
is the group of temporal shifts,
2) They satisfy a uniform exponential attraction property as follows: there exist a positive constant and a monotonic function
(both depending only on
) such that, for every bounded subset
of
, we have
(4.2)
3) The sets are compact finite-dimensional subsets of
(4.3)
where the constant is independent of
and
.
4) The map is Hölder continuous in the following sense:
(4.4)
where the positive constants and
are independent of
and
,
denotes the symme- tric Hausdorff distance. In particular, the function
is uniformly Hölder continuous in the Hausdorff metric:
(4.5)
where and
are also independent of
and
.
Proof. Firstly, we construct a family of exponential attractors for the discrete dynamical processes associated with Equation (2.5). According to Lemma 2.2, it only remains to construct the required exponential attractors for initial data belonging to the ball
where is a sufficiently large number depending only on
given in (1.5), is a uniform absorbing set for all the processes
generated by Equation (1.1). Moreover, from Theorem 2.1, Theorem 2.2 and Theo- rem 3.1, it follows Lipschitz continuity and smooth properties for the difference of two solutions
and
. Thus, by Theorem 2.1 in [13] , the family of discrete dynamical processes
possess exponential attractors
For obtain- ing exponential attractors of the family of dynamical processes
, we need the Hölder continuity of the processes
with respect to the time, see the following lemma.
Lemma 4.1 Let the above assumptions on Equation (1.1) hold. Then, for every we have
(4.6)
where the constant depends on
, and is independent of
Moreover, for every
we also have
(4.7)
where is a positive number and the positive constant
depends on
but is independent of
and
.
Proof. Note that there is a such that
and is uniformly bounded in
and Lemma 2.2, which imply the Hölder continuity (4.6). In order to verify (4.7), we note that, due to (4.6) and Theorem 2.2, for every
, we have
where all the constants depend on, but are independent of
and
Using the previously mentioned interpolation inequality in Section 2 finishes the proofs of estimate (4.7). ,
Now, we can define the exponential attractors for continuous time by the following formula
with respect to The proofs of the semi-invariance with respect to
and translation-invariance with respect to time-shifts is similar to [11] [13] . Estimate (4.2) follows in a standard way from Lemma 2.2, Theorem 3.1 for the processes
. Thus, it only remains to verify the finiteness of the fractal dimension of
. In order to prove this, we first note that, according to the Hölder continuities Theorem 2.1 in [13] and (4.7), we have
for all,
and
. Since the map
are uniformly Lipschitz conti- nuous, Theorem 3.1 in [13] implies that
for a given, and some constant
and
which are independent of
. The proof of Theorem 4.1 is completed. ,
5. Infinite-Dimensional (Uniform) Exponential Attractor and Non-Autonomous Exponential Attractor
Finally, we compare the non-autonomous exponential attractor obtained above with the so-called infinite-dimensional (uniform) exponential attractor constructed in [11] [13] . To the existence of the uniform at- tractor for strongly damped wave equations, we use the results in [4] and [5] as a model example.
Let be some external force. Let
be the hull of
in
, i.e.,
where denotes the closure in
. Evidently,
for any
.
Using the standard skew product flow in [4] and [5] , for every external forces satisfying (1.5), we can embed the dynamical process
into the autonomous dynamical system
acting on the extended phase space
via
It is known that is a semigroup. If this semigroup possesses the global attractor
, then, its projection
onto the first component of the Cartesian product is called the uniform at- tractor associated with problem (1.1).
It is also known that the uniform attractor exists under the relatively weak assumption that the hull
is compact in
, but, unfortunately, for more or less general external forces
its Haus- dorff and fractal dimensions are infinite. Instead, the following estimate for its Kolmogorov’s
-entropy holds, see [4] .
Proposition 5.1 Let the above assumptions hold and the hull of the initial external forces be compact. Then, Equation (1.1) possesses the uniform attractor
and its
-entropy can be estimated in terms of the
-entropy of the hull
as follows:
(5.1)
for some positive constants and
depending on
.
Definition 5.1 [11] [13] A set is an (uniform) exponential attractor of Equation (1.1) if the fol- lowing properties are satisfied:
1) Entropy estimate: is a compact subset of the phase space
which satisfies estimate (5.1) (possibly, for larger constants
and
).
2) Semi-invariance: for every there exists
such that
for all
3) Uniform exponential attraction property: there exists a positive constant and a monotonic function
such that, for every
and every bounded subset
, we have
(5.2)
[13] points out that a uniform exponential attractor can be constructed if the (non-autonomous) exponential attractor
has been constructed, so we have
Theorem 5.1 Let the assumptions of Theorem 4.1 hold and let, in addition, the hull of some external forces satisfying (1.5) be compact. Then, there exists a uniform exponential attractor
for problem (1.1) which can be constructed as follows:
Remark 1 When, Equation (1.1) reduces to the following damped wave equation on a bounded domain
with smooth boundary
:
(5.3)
Equation (1.1) reduces to the damped wave equation modeling the Josephson junction in superconduction which was studied by many authors (see [1] [6] [17] ). We assume that the function satisfy (1.2)-(1.4). The Equation (5.3) also possesses a finite dimensional exponential attractor.
Remark 2 When, Theorem 4.1 remains valid for the following strongly damped wave equation was studied by many authors (cf. [7] [18] ):
if we assume that the function satisfy (1.2)-(1.4).
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11101265) and Shanghai Edu- cation Research and Innovation Key Project of China (14ZZ157).
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