The Estimates of the Upper Bounds of Hausdorff Dimensions for the Global Attractor for a Class of Nonlinear Coupled Kirchhoff-Type Equations ()
1. Introduction
Guohuang Lin, Ming Zhang [1] studied the initial boundary value problem for a class of Kirchhoff-type coupled equations and obtained the existence of the global attractor. Next, in this paper, we consider the Hausdorff dimensions for the global attractor for the following Kirchhoff-type equations:
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
where
is a bounded domain in
with the smooth boundary
,
is a constant.
is a nonnegative
function,
and
are strong- ly damping terms,
and
are nonlinear source terms,
and
are given forcing function.
Jingzhu Wu, Guoguang Lin [2] consider a class of damped Bossinesq equation:
(1.6)
(1.7)
(1.8)
where
,
, then they obtain the existence of the global attractor and the limited of Hausdorff dimension and the limited of Fractal dimension.
Xiaoming Fan, Shengfan Zhou [3] consider the following non-autonomous strongly damped wave equation of non-degenerate Kirchhoff-type:
(1.9)
(1.10)
(1.11)
where
is a real-valued function on
,
,
is an open bounded set of
,
with a smooth boundary
,
is called the strong damping,
,
,
.
,
,
,
is the set of continuous bounded functions from
into
. And then, they obtained a precise estimate of upper bound of Hausdorff dimension of kernel sections, which decreases as the strong damping grows for large strong damping under some conditions, particularly in the autonomous case.
Guoguang Lin, Yunlong Gao [4] concerned the following nonlinear Higher-order Kirchhoff-type equations:
(1.12)
(1.13)
(1.14)
where
is an integer constant,
are constants and
is a real number.
is a bounded domain of
with a smooth boundary
and
is the unit outward normal on
.
is a nonlinear function specified later. And they obtained the existence of the global attractor. In this case, they considered that the estimation of the upper bounds of Hausdorff for the global attractors is obtained.
2. Hausdorff Dimensions of the Global Attractor
In this paper, some inner product, norms, abbreviations and some assumptions (H1) - (H4) and notations needs in the proof of our results in refer to [1] .
2.1. Differentiability of the Semigroup
In order to estimate dimensions, we suppose:
(H5) For every
, there exist
, such that:
(2.1)
(2.2)
where
;
;
;
;
.
(H6) There exists constant
, such that
(2.3)
We define
,
. The inner product and norm in
space are defined as follows:
we have
(2.4)
(2.5)
Setting
,
,
, the Equations (1.1)-(1.5) is equivalent to
(2.6)
where
(2.7)
(2.8)
Lemma 2.1 For any
, we have
(2.9)
Proof.
By (2.3), we get
(2.10)
By using holder inequality and Young’s inequality and Poincare inequality, we deal with the terms in (2.9) by as follows:
(2.11)
(2.12)
(2.13)
(2.14)
By
and substituting (2.11)-(2.14) into (2.10), we obtain
(2.15)
Proof finished.
The linearized equations of (1.1)-(1.5), the above equations as follows:
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
where
,
is the solution of with
Given
and
, the solution
, by stand methods we can show that for any
, the linear initial boundary value problem (2.16)-(2.20) possess a unique solution
.
Theorem 2.1 For any
,
, the mapping
is Frechet differentiable on. Its differential at
is the linear operator on
where
,
is the solution of (2.16)-(2.20).
Proof.
Let
,
with
,
, we denote
,
. We can get the Lipchitz property of
on the bounded sets of
, that is
(2.21)
Let
,
is the solution of problem
(2.22)
(2.23)
(2.24)
(2.25)
where
(2.26)
(2.27)
Let
,
, so we can get
(2.28)
And
(2.29)
Then, we have
(2.30)
(2.31)
(2.32)
(2.33)
By using (2.30)-(2.33), we have
(2.34)
Similarly
(2.35)
And by using (H5)
(2.36)
Similar
(2.37)
So, we can get
(2.38)
Then, by using Gronwall’s inequality and (2.21), we obtain
(2.39)
Then, we get
(2.40)
as
in
. The proof is competed.
2.2. The Upper Bounds of Hausdorff Dimensions for the Global Attractor
Consider the first variation of (2.6) with initial condition:
(2.41)
where
,
,
and
is a solution of (2.41),
(2.42)
(2.43)
(2.44)
It is easy to show form Theorem 2.1 that (2.41) is a well-posed problem in
, the mapping
:
®
is frechet differentiable on
for any
, its differential at
is the linear operator on
where
is the solution of (2.40).
Lemma 2.2 [5] For any orthonormal family of elements of
,
,
, we have
(2.45)
Proof. This is a direct consequence of Lemma VI. 6.3 of [5] .
Theorem 2.2 If (H1)-(H6) hold, satisfying, then there exists
, such that the Hausdorff dimension of global attractor
in
satisfies
(2.46)
where
is as in Lemma 2.6 in [1] .
Proof. Let
be fixed. Consider
solutions
of (2.41). At a given time
, let
denote the orthogonal projection in
onto span
.
Let
,
, be an orthonormal basis of
(2.47)
with respect to the inner product
and norm
.
Suppose
(2.48)
then
,
. By
and Lemma 2.1, we have
(2.49)
(2.50)
Then, by the Sobolev embedding theorem:
(2.51)
Therefore
(2.52)
By Young’s inequality, we have
(2.53)
So exist
satisfying
(2.54)
(2.55)
We obtain
(2.56)
If
, then
(2.57)
Proof finish.