Journal of Applied Mathematics and Physics
Vol.05 No.12(2017), Article ID:81503,14 pages
10.4236/jamp.2017.512197
Hausdorff Dimension and Fractal Dimension of the Global Attractor for the Higher-Order Coupled Kirchhoff-Type Equations
Guoguang Lin, Sanmei Yang
Department of Mathematics, Yunnan University, Kunming, China

Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/



Received: November 28, 2017; Accepted: December 26, 2017; Published: December 29, 2017
ABSTRACT
This paper mainly deals with the higher-order coupled Kirchhoff-type equations with nonlinear strong damped and source terms in a bounded domain. We obtain some results that are estimation of the upper bounds of Hausdorff dimension and Fractal dimension of the global attractor.
Keywords:
Higher-Order Coupled Kirchhoff-Type Equations, Source Term, Hausdorff Dimension, Fractal Dimension, Nonlinear Dissipation

1. Introduction
Guoguang Lin and Sanmei Yang [1] had studied the existence and uniqueness of the solution and global attractors for the higher-order coupled Kirchhoff-type equations. Furthermore, we consider the Hausdorff dimension and Fractal dimension of the global attractor for the following Hinder-order coupled Kirchhoff equations:
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
where
is an integer constant and
is a bounded domain of
with a smooth Dirichlet boundary
and initial value.
and
are the unit outward normal on
,
is a nonnegative
function,
and
are strongly damping,
and
are nonlinear source terms,
and
are given forcing function.
When considering single Higher-order Kirchhoff-type equation,
,
and
in
, becomes following Higher-order Kirchhoff-type wave equation with nonlinear strongly damping:
(1.7)
(1.8)
(1.9)
This equation had been studied some main results that are existence and uniqueness of the solution in
and global attractors by Yuting Sun, Yunlong Gao and Guoguang Lin, see [2] .
In case of
and
in
, the Equation (1.1) becomes a Higher-order Kirchhoff-type equation with nonlinear strongly dissipation and source term:
(1.10)
(1.11)
(1.12)
This equation had been investigated the existence and uniqueness of the solution, global attractors and estimation Hausdorff and fractal dimensions of the global attractor by Chen, Wei Wang and Guoguang Lin, see [3] . As for the study of estimation Hausdorff dimension of the global attractor, we applied different method from theirs.
Under the situation of
and
, the problem (1.1) becomes a class of strongly damped Higher-order Kirchhoff-type equation:
(1.13)
(1.14)
(1.15)
This equation had been studied the existence and uniqueness of the solution, global attractors and estimation of the upper bounds of Hausdorff for the global attractors and the existence of a fractal exponential attractor with non-supercritical and critical cases by Guoguang Lin and Yunlong Gao, see [4] . Their novelty is that it overcomes
by using generalized Gronwall’s inequality in Lemma 2.
Next, the main purpose of this paper is to study a precise estimation of upper bounds of Hausdorff dimension and Fractal dimension of the global attractor.
2. Preliminaries
To better carry out our work, We denote the some simple symbol,
represents norm, (,) stands for inner product and
,
,
,
,
,
,
,
.
,
are constants.
is the first eigenvalue of the operator
.
Next, we give some assumptions needed for problem (1.1)-(1.6).
(H1)
(2.1)
(H2)
,
(2.2)
(H3)
, there exists
, such that
(2.3)
(2.4)
(2.5)
Lemma 2.1. (Young’s inequality [5] ) For any
and
, then
(2.6)
where
Lemma 2.2. (Gronwall’s inequality [5] ) If
and
, such that
(2.7)
where
are constants.
Lemma 2.3. (Sobolev-Poincare inequality [6] ) Let’s be a number with
and
. Then there is a constant k depending on
and s such that
(2.8)
3. Hausdorff Dimensions and Fractal Dimension for the Global Attractor
3.1. Differentiability of the Semigroup
We denote
. The inner product and the norm in
space are defined as follows:
, we can get
(3.1)
(3.2)
Setting
, the equations (1.1) and (1.2) are equivalent to
(3.3)
where
(3.4)
(3.5)
Lemma 3.1.1.
, we can get
(3.6)
Proof. According to (3.1)-(3.5), Holder inequality, Young’s inequality and Poincare inequality, we can obtain
(3.7)
The proof of Lemma 3.1.1. is completed.
The linearized equations of (1.1)-(1.6), the above equations as follows:
(3.8)
(3.9)
(3.10)
(3.11)
where
is the solution of (1.1)-(1.6) with
.
Given
and
, the solution
by stand methods we can show that for any
, the linear initial boundary value problem (3.8)-(3.11) possess a unique solution:
(3.12)
Lemma 3.1.2. For any
, the mapping
is Frechet differentiable on. It is differential at
is the linear operator on
, where
and
are solutions of (3.8)-(3.11)
Proof. Let
,
with
, we define
. We can obtain the Lipchitz property of
on the bounded sets of
, that is
(3.13)
Let
and
are solutions of problem
(3.14)
(3.15)
(3.16)
(3.17)
with
(3.18)
(3.19)
where
.
Taking the scalar product of each side of (3.14) with
. Because of
(3.20)
(3.21)
So
(3.22)
(3.23)
(3.24)
(3.25)
(3.26)
where
.
By (H3) and Young’s inequality, we can get
(3.27)
By using Young inequality, we can get
(3.28)
From (3.23)-(3.28), we have
(3.29)
Take the scalar product of each side of (3.15) with
. Because of
(3.30)
Summing up (3.29) and (3.30), we have
(3.31)
By using Gronwall inequality, we can get
(3.32)
where
.
From (3.32), we can get
(3.33)
here
(3.34)
As
in
. The lemma 3.1.2 is completed.
3.2. The Upper Bounds of Hausdorff Dimension and Fractal Dimension for the Global Attractor
Consider the first variation of (3.3) with initial condition;
(3.35)
where
and
are solutions of (3.35),
(3.36)
(3. 37)
(3.38)
where
It is easy to show from lemma 3.1.2. that (3.35) is a well-posed problem in
, the mapping
, where
,
,
,
.
is Frechet differentiable on
for any
, its differential at
is the linear operator on
,
, where
is the solution of (3.35).
Lemma 3.2.1. For any orthonormal family of elements of
,
,
, we have
(3.39)
(3.40)
where
is the eigenvalue of
.
Proof. This is a direct consequence of lemma VI 6.3 of [7]
Theorem 3.2.2. If we take proper
satisfy
and
(H1)-(H3) hold, then there exists
, such that the Hausdorff dimension and Fractal dimension of global attractor
in
satisfies
(3.41)
(3.42)
where
is as in [1] , and
(3.43)
here
.
Proof. Let
be fixed. Consider
solutions
of (3.35). At given time
, let
define the orthogonal projection in
onto
. Let
be an Orthonormal basis of
(3.44)
with respect to the inner product
and norm
.
Suppose
(3.45)
then
. By
and Lemma 4.1.1, we can get
(3.46)
(3.47)
where
.
By the hypothesis (H4) in [1] , the mean value theorem and Sobolev embedding theorem:
(3.48)
where
.
Thus, by Lemma 2.4. in [1] and (3.47), for
,
(3.49)
There exists
, such that
(3.50)
(3.51)
(3.52)
(3.53)
For
, by
, there exists
, such that
(3.54)
(3.55)
(3.56)
(3.57)
For
, by (H4) in [1] there exists
such that
(3.58)
(3.59)
(3.60)
(3.61)
From above, we have
(3.62)
where
,
,
,
,
.
According to Lemma 2.4. in [1] , we can get
(3.63)
where
.
If
, then
,
.
By Lemma VI 6.3 of [8] , Young’s inequality and existing
satisfying
, we obtain
if
, then
(3.64)
where
.
If
hold, then
(3.65)
Thus, by Lemma 4 of (S. Zhou, 1999 [9] ), we obtain (3.39) and (3.40). The proof is completed.
4. Conclusion
In this paper, we study estimation of the upper bounds of Hausdorff dimension and Fractal dimension of the global attractor for a class of Higher-order coupled Kirchhoff-type equations. In the process of research, we have avoided further restriction by taking the overall treatment of
item, thus making the application of this model more extensive. Although the theoretical derivation of the beam vibration model is not combined with the application in real life, it is necessary to combine with practical application to further study.
Acknowledgements
We express our heartful thanks to the anonymous reader for his/her careful reading of this paper. We hope that we can obtain valuable comments and advices. These contributions vastly improved the paper and making the paper better.
Cite this paper
Lin, G.G. and Yang, S.M. (2017) Hausdorff Dimension and Fractal Dimension of the Global Attractor for the Higher-Order Coupled Kirchhoff-Type Equations. Journal of Applied Mathematics and Physics, 5, 2411-2424. https://doi.org/10.4236/jamp.2017.512197
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