G. G. Lin and L. J., Hu have studied the existence of a global attractor for coupled Kirchhoff type equations with strongly linear damping in  . In this paper, we are concerned with the finite dimensions of the global attractor as mentioned above:
is a bounded domain in
with smooth boundary
is real number,
is positive integer.
are given functions later.
In demonstrating the longtime behavior of evolutional equation, we currently aim to show that the dynamics of the equation is finite dimensional. To be precise, one possible way to express it is to say that the dynamical systems of equation exists a global attractor with finite Hausdorff and Fractal dimensions.
Concerning the wave equation with linear and semi-linear dissipative system, existence of the global attractor with finite Hausdorff and Fractal dimensions is proved in  , for the nonlinear wave equation, the existence of the global attractor with finite Hausdorff and Fractal dimensions is proved in    . When the equation is nonlinear, the process of dimension estimation is more complicated. The method of linearization works very well on it, and meanwhile we take fully consideration of assumptions on the nonlinearities of the equation.
Recently, Z. J, Yang  studied the longtime behavior of the Kirchhoff type equation with strong damping on
. It showed that the related continuous semi-group
possesses a global attractor which is connected and has finite Fractal and Hausdorff dimensions.
At the same time, Z. J. Yang  dealt with the global attractors and their Hausdorff dimensions for a class of Kirchhoff models, and got the existence, regularity, and Hausdorff dimensions of global attractors for a class of Kirchhoff models arising in elastoplastic flow.
Furthermore, X. M. Fan and S. F. Zhou  proved the existence of compact kernel sections for the process generated by strongly damped wave equations of non-degenerate Kirchhoff type modelling the nonlinear vibrations of an elastic string, and they obtained a precise estimate of upper bound of Hausdorff dimension of kernel sections.
In addition, G. G. Lin and Y. L. Gao  studied the longtime behavior of solution to initial boundary value problem for a class of strongly damped higher-order Kirchhoff type equation:
they got the existence and uniqueness of the solution by the Galerkin method and obtained the existence of the global attractor in
according to the attractor theorem, besides, the estimation of the upper bound of Hausdorff dimension for the attractor was established.
The paper is arranged as follows. In Section 2, some preliminaries and main results are stated. In Section 3, in order to acquire the result of the estimation, we show the differentiability of semigroup. Eventually, the Hausdorff and Fractal dimensions of the global attractor for the dynamics system associated with problem (1.1)-(1.5) are discussed in detail.
2. Preliminaries and Main Results
Throughout this paper, we need some notations for convenience. We consider a family of Hilbert spaces
, whose inner products and norms are given by
For our purpose, we define a weighted inner product and norm in
Next, we make the following assumptions for problem (1.1)-(1.5).
is not decreasing function and for positive constants
(A2) There exists
, and for every
, there exist
3. The Hausdorff and Fractal Dimensions of the Attractor
In order to obtain the result of the dimension estimation, we should prepare the following lemmas.
Lemma 3.1. (  ) Suppose that the assumptions of  hold, the constants
and initial value
, then for the problem (1.1)-(1.5), there exists a unique weak solution such that
Lemma 3.2. (  ) Suppose that
satisfy assumptions of  respectively. Then for
, the problem (1.1)-(1.5) possesses the global attractor
, which is compact among bounded absorbing set
, that is
this lemma is result on the existence of a global attractor of system (1.1)-(1.5) generated by semigroup
The first step will be to prove the differentiability of
. In what follows, we put
for simplicity. The first variation equations of (1.1)-(1.5) as the following
is the solution of Equations (1.1)-(1.5) with
, the solution
, by standard methods, we can prove that for any
, the linear initial boundary value problem (3.1)-(3.4) possesses a unique solution
Lemma 3.3. for any
, the mapping
is Frechet differentiable on
. Its differential at
is the linear operator on
is the solution of (3.1)-(3.4).
, we denote
. First, we can prove a Lipschitz property of
on the bounded sets on
, that is
We now consider the difference
the solution of (3.1)-(3.4), clearly,
Taking the scalar product of each side of (3.8)-(3.9) with
, and then we have
Taking the scalar product of right side of (3.16) with
, and then we obtain
which implies that
which means that
Taking (3.15)-(3.18) into (3.14), we have
applying the Gronwall inequality and (3.6) we deduce from (3.20) that
This is equivalent to
The differentiability of
The next step will be used in demonstrating the process of dimension estimation. It seems obvious that the equations (1.1)-(1.2) also can be written as
Lemma 3.4. For any
, we have
Proof. For any
, through the above definition, we get
By applying the Holder inequality, Young’s inequality and Poincare inequality, we deal with the terms in (3.26) by as follows
, and substituting (3.27)-(3.28) into (3.26), we obtain
The proof of lemma 3.4 is completed.
Consider the first variation equation of (3.23)
is a solution of (3.23),
Lemma 3.5. (  ) Let there be given
as above. Then for any
, and for any orthonormal family of elements of
, we have
is the eigenvalue of
Proof. This is a direct consequence for Lemma 6.3 of  .
Theorem 3.1. Let
be the global attractor of problem (1.1)-(1.5), then the Hausdorff dimension of global attractor
is less than or equal to
and its fractal dimension is less than or equal to
be settled. Consider
of (3.30), and we memorize that
we see that
is the orthogonal projection in
onto the space spanned by
. At a given time
, denote an orthonormal basis of
With respect to the scalar product
, we omit for the moment variable
By the Lemma 3.4, we have
By the assumption (H3) in  , the mean value theorem and the Sobolev embedding theorem
, we can easily obtain that
, we have
, there exist
, such that
is as in (3.36), then setting
The proof of theorem 3.1 is completed.
The authors express their sincere thanks to the anonymous referee for his/her careful reading of the paper, giving valuable suggestions and comments, which have greatly improved the presentation of this paper.