Finite Fractal Dimensionality of Compact Kernel Sections for Dissipative Non-Autonomous Klein-Gordon-Schrödinger Lattice Systems ()
1. Introduction
In recent years, great progress had been made in the study of non-autonomous infinite dimensional dynamical systems. See, e.g., [1] [2] [3] [4] [5] and the references therein. Lattice dynamical systems (Hereafter LDSs) are infinite dimensional ordinary differential equations, which were widely and deeply investigated in the past decades due to its wide application in many fields such as laser systems, material science, electrical engineering, biology, chemical reaction theory and etc. See, e.g., [6] - [17] and so on. Nowadays, the study of non-autonomous LDSs appealed to more and more researchers, but there are few papers for non-autonomous LDSs until now. See e.g., [18] - [23] and etc.
As to the dissipative autonomous Klein-Gordon-Schrödinger (Hereafter KGS) lattice systems, many authors have studied them. For example, Abdallah in [24], Abounouh, Goubet and Hakim in [25], Yin and Zhou et al. in [26] investigated the existence, regularity, upper semicontinuity, Kolmogorov entropy of global attractor and so forth. Meanwhile, the following dissipative non-autonomous KGS lattice system
(1.1)
was investigated by many researchers either. Specifically, the existence of uniform exponential attractors for the dissipative non-autonomous KGS lattice system (1.1) with quasi-periodic symbols is studied in weighted spaces of infinite sequences by Abdallah in [27], simultaneously, some main results that the solution semigroup associated with such a system is Lipschitz continuous, α-contraction and satisfies the squeezing property, are obtained under some premise. Huang et al. in [28] proved the existence of a compact uniform attractor and obtained an upper bound of the Kolmogorov entropy of the compact uniform attractor. In addition, an upper semicontinuity of the compact uniform attractor is established as well. Zhao and Zhou in [29] proved the existence of compact kernel sections and obtained an upper bound of the Kolmogorov entropy of the compact kernel sections, but they didn’t study the fractal dimension of the compact kernel sections. In Zhou and Han [30], some sufficient conditions for the existence of a uniform exponential attractor for a family of continuous processes on separable Hilbert spaces and the space of infinite sequences are presented at first, and then the existence of uniform exponential attractors for the dissipative non-autonomous KGS lattice system (1.1) and for the dissipative non-autonomous Zakharov lattice system driven by quasi-periodic external forces in the spaces of infinite sequences is studied. However, what’s more important, so far to our knowledge, this problem that the fractal dimension of the compact kernel sections was not studied in Zhao and Zhou [29] is still an open topic till today. In view of this point, this paper is to estimate the fractal dimension of the compact kernel sections for the dissipative non-autonomous KGS lattice system (1.1). For our purpose, we first mention that as we all know, if
is a compact set in a metric space such that the fractal dimension of
is less or equal to
for some
, then there exists an injective Lipschitz mapping
such that its inverse is Hölder continuous. In the sequel of this paper, we will present a criterion for estimating the fractal dimension of a family of compact subsets of a separable Hilbert space and then apply this criterion to obtain an upper bound of the fractal dimension of the compact kernel sections associated with the dissipative non-autonomous KGS lattice system (1.1).
The remaining of this paper is organized as below. We give the preliminaries in Section 2. In Sections 3, a criterion is used to estimate the fractal dimension of the compact kernel sections for the dissipative non-autonomous KGS lattice system, and an upper bound is obtained. Lastly, Section 4 presents the conclusions.
2. Preliminaries
To begin, we introduce
where
,
and
denote the integral, real and complex numbers, respectively.
Write
or
, and endow H with the inner product and norm as below
where
is the conjugate of
. Clearly, H is a Hilbert space.
Define linear operators A and B as follows
For any
, define a bilinear form by means of
where
as in the dissipative non-autonomous KGS lattice system (1.1) presented above. This bilinear form is obviously an inner product in Hilbert space H.
In the end, we express Hilbert spaces
,
and
as
Set
and equip it with the following norm and inner product
where
.
Define
and denote by
and
respectively the set of continuous and bounded functions from
into
and
.
Definition 2.1. A two-parameter family of mappings
is called to be a process in a Hilbert space
, if
1)
;
2)
;
3)
(identity operator of
),
.
Definition 2.2. A function
, is said to be a complete trajectory of the process
, if
. The kernel
of the process
consists of all bounded complete trajectories of
, i.e.,
and the kernel sections
of the kernel
at time
is
Definition 2.3. The fractal dimension
of a compact set
in a metric space
is defined as follows, namely
where
is the minimal number of closed sets of radius
which cover the set
.
The criterion below is directly cited from Zhou et al. [21].
Lemma 2.1. Let
be a continuous process on a Hilbert space
and
be a family of compact, negatively invariant (i.e.,
for all
) subsets of
. Assume that
1) there exists a uniform finite covering of closed subsets with diameter 2 of
for all
, that is, there exists
closed balls of
with diameter 2 covering
for all
, where
is independent of t;
2) for any
, there exists
and
, which are all independent of
such that for
,
a) there exists
yields
i.e.,
is Lipschitz on
;
b) there exists finite-dimensional orthoprojector P of
satisfies
then
3. Fractal Dimension of Compact Kernel Sections for Dissipative Non-Autonomous KGS Lattice System
Consider the dissipative non-autonomous KGS lattice system with the initial conditions as vector form
(3.1)
where
,
;
,
;
,
;
,
;
is the imaginary numbers’ unit;
,
,
and
are positive constants;
,
,
denotes
or
.
We set
Thus, (3.1) can be written as below
(3.2)
where
,
,
, and
From Zhao and Zhou [29], we can see, for given
with
,
with
, the solution mappings of (3.2), that is,
,
,
, generate a family of continuous processes
in
. Moreover, the family of processes
, possess a family of compact kernel sections
, where
is included in a uniformly bounded set
and satisfies
, here
(3.3)
(3.4)
In the sequel, we get an upper bound of the fractal dimension of the compact kernel sections
, which is generated by the process of the dissipative non-autonomous KGS lattice system (3.1).
Suppose
,
, then
for
,
. Set
, then by (3.2), we have
(3.5)
where
, and
,
,
.
Lemma 3.1. For any
,
is Lipschitz on
, i.e.,
(3.6)
where
(3.7)
and
as in (3.3) and (3.4), respectively.
For brief, we denote by
and
respectively the real part and imaginary part of inner product
.
Proof. Taking the real part of the inner product
of (3.5) with
, we have
(3.8)
By simple computation, we get
(3.9)
(3.10)
(3.11)
Actually
and
thus
(3.12)
Applying Young’s inequality to (3.12), it is obvious to know that (3.11) holds.
Taking (3.8)-(3.11) into account, we see
(3.13)
Set
,
, and then apply Gronwall’s inequality to (3.13), it is easy to see that (3.6) holds. The proof is completed.
Lemma 3.2. There exists a finite dimensional orthoprojector
of
and
such that
(3.14)
Proof. For this purpose, we choose an increasingly smooth function
, yielding
and at the same time, there exists a constant
such that
,
.
Let M be a fixed positive integer, set
,
. Taking the real part of the inner product
in (3.5) with
, we get
(3.15)
Similar to Zhou [14], we have
(3.16)
(3.17)
and analogous to (3.11), we obtain
(3.18)
Combining (3.15)-(3.18), we get
From Zhao and Zhou [29], we know that for
, there exist
and an integer
, which satisfies
where
such that
Thus, for any
and
, we have
(3.19)
By (3.13), it can easily obtain
(3.20)
From (3.19) and (3.20), we get
Furthermore, by Gronwall’s inequality, we have
Set
(3.21)
and define
, it is clear that
. Let
be the finite dimensional orthoprojector from
to
, then for
, (3.14) holds with
(3.22)
where
and
as in (3.3), (3.4), (3.7) and (3.21), respectively. The proof is completed.
As a straightforward consequence of Lemma 2.1, Lemma 3.1 and Lemma 3.2, we get the following Theorem 3.1.
Theorem 3.1. The compact kernel sections
has a finite fractal dimension
, which satisfies
(3.23)
where
and
as in (3.3), (3.4), (3.7), (3.21) and (3.22), respectively.
4. Conclusions
This paper studied the fractal dimension of the compact kernel sections which is generated by the process of the dissipative non-autonomous KGS lattice system described in (3.1) by applying a criterion given in Lemma 2.1 cited directly from Zhou et al. [21], and then an upper bound of the fractal dimension is obtained in (3.23) presented in Theorem 3.1.
Remark. We can use the argument in this paper to study the dissipative non-autonomous Klein-Gordon-Schrödinger lattice system defined on
with
,
. In this case, operator A possesses the following decomposition
meanwhile
where
means the norm in space H, K is a positive constant. Here, linear operator
and its adjoint operator
are defined by
where
and
.
Acknowledgements
The author would like to thank the anonymous referees for their helpful comments and thank the editors for their help.