A Family of the Global Attractor for Higher Order Nonlinear Kirchhoff Equation ()
1. Introduction
This paper intends to study the initial-boundary value problem of higher-order Kirchhoff-type equation
(1.1)
(1.2)
(1.3)
where
, and
,
is a bounded domain,
denotes the boundary of
,
is a nonlinear source term,
is a strongly dissipative term,
,
is an external force term.
Kirchhoff-type equation model is one of the hot topics in mathematical physics equation research in recent years, which shows the importance of its position and influence. There have been many achievements in the study of the long-term behavior of the solution of Kirchhoff-type equation, for details, refer to references ( [1] [2] [3] [4] [5] ). Cheng Jianling and Yang Zhijian studied the asymptotic behavior of the solution of Kirchhoff-type equation in reference [6]:
The existence of the global attractor of the corresponding operator semigroup
in phase space is proved.
Recently, Lin Guoguang et al. studied the existence of global attractors for higher order Kirchhoff type equations with nonlinear strong damping terms in reference [7]:
They proved the existence and uniqueness of the solution of the equation by using prior estimation and Galerkin’s method, and then obtained that the attractor exists in space
.
Guoguang Lin and Changqing Zhu studied asymptotic state of solutions for a class of nonlinear higher order Kirchhoff type equations in reference [8]:
For more results, please refer to references ( [9] - [15] ).
2. Basic Assumptions
For convenience, space and notations are defined as follows:
,
,
,
,
,
. Remember that
is a family of global attractors from
to
,
is a bounded absorption set in
. In which
,
is a constant;
,
represent the inner
product and norm on space H, namely
,
.
Kirchhoff type stress term
satisfies the following conditions:
(H1)
;
(H2)
where
are constant,
.
The nonlinear term
satisfies the following conditions:
(H3)
;
(H4) There is
,
,
;
(H5)
.
3. The Existence of the Family of Global Attractor
Lemma 3.1. set
satisfy assumption (H1),
,
, and
,
, then the smooth solution of problems (1)-(3)
and
satisfy
where
,
, where constants
and
, so that
Proof. It is proved that the inner product of
and Equation (1.1) can be obtained
(3.1)
(3.2)
(3.3)
By using the Poincare’s inequality, we obtain
(3.4)
By using the hypothesis (H3) and Young’s inequality, we obtain the following estimation
(3.5)
(3.6)
Substitute Inequality (3.2)-(3.6) into Equation (3.1), therefore,
(3.7)
let
,
, and let
,
, then
(3.8)
where
(3.9)
By using the Gronwall’s inequality, we get
(3.10)
where
(3.11)
Then
(3.12)
and
(3.13)
There are constants
and
, we, get
(3.14)
Lemma 3.1 is proved.
Lemma 3.2. If lemma 3.1 holds, and the condition is (H4), set
,
.
, then the smooth solution of Problems (1.1)-(1.3)
and
satisfy
where constants
and
, then
Proof. Set
. It is obtained by inner product of
and Formula (1.1).
(3.15)
(3.16)
(3.17)
(3.18)
From hypothesis (H4), we get
Then
(3.19)
(3.20)
Substitute Inequality (3.16)-(3.20) into Equation (3.15), therefore,
(3.21)
Let
,
,
, and let
,
,
then
(3.22)
where
(3.23)
By using the Gronwall’s inequality, we get
(3.24)
where
(3.25)
Then
(3.26)
and
(3.27)
There are constants
and
, then
(3.28)
Lemma 3.2 is proved.
Theorem 3.1. Assume that the nonlinear function
satisfies (H1), (H2),
,
, then the Problems (1.1)-(1.3) have a unique global smooth solution
, and
.
Proof: The proof of existence is divided into the following three steps by Galekin’s method:
Step 1: Approximate solution
Suppose the eigenvector
of
generates an orthonormal basis for
, where
is the eigenvalue of
with homogeneous Dirichlet boundary on
, define k order approximation
:
(3.29)
where
,
with in H,
(3.30)
This system of ordinary differential equations about
can determine
in the interval
; need to prove
.
Step 2: Prior estimation
According to the conclusion and proof method of lemma 3.1,
is uniformly bounded on
, then
(3.31)
(3.32)
(3.33)
thus it can be seen
, Inequality (3.32)-(3.33) shows
is bounded in
, and
is bounded in
.
And it’s actually available
in
and
in
.
Step 3: Limit process
According to Danford-Pttes theorem, Space
conjugate to space
; Space
conjugate to space
, select the subsequence
from the sequence
, such that
weakly * converges in
,
weakly * converges in
,
weakly converges in
.
According to Pellich-Kohdarachov theorem,
is compact embedded in
and
,
is strong convergence almost everywhere in
and
.
converges in
.
is weak * convergence in
.
is weak * convergence in
.
converges in
.
is weak * convergence in
.
From the Formula (3.30), we get
For
, according to the density of
.
(3.34)
and
weakly converges in
, and in
.
then
is satisfied for all j, so that existence can be proved.
Then prove the uniqueness of the solution.
Set
be two solutions of the Problem (1.1)-(1.3), let
, then
(3.35)
(3.36)
Take the inner product with
and (3.34).
(3.37)
(3.38)
(3.39)
By using hypothesis (H5), there are
, then
(3.40)
By using hypothesis (H1), Lemma 3.1 and Differential Mean Value Theorem
(3.41)
To sum up, we obtain
From Gronwall’s inequality, we get
(3.42)
Therefore
, the uniqueness is proved.
Theorem 3.2. [9] Let E be a Banach space,
Semigroups satisfy the following conditions
1) Semigroup
is uniformly bounded in E, then
, there is constant
, so that when
, there is
;
2) There is a bounded absorption set
in E;
3)
is a fully continuous operator. That is, semigroups
have compact global attractors
.
The Banach space E in theorem 3.2 is changed into Hilbert space
, there are the following existence theorems of global attractor’s families.
Theorem 3.3. Under the hypothesis of lemma 3.1 and lemma 3.2, Then there is a family of global attractor
for Problems (1.1)-(1.3). That is, there is a compact set
, make:
1)
;
2)
(
is a bounded set), among
,
is the solution semigroup
generated by Problems (1.1)-(1.3).
Proof: It is necessary to verify the conditions (1), (2) and (3) of Theorem 3.2. Solution Semigroup
generated by Theorem 3.1 and Lemma 3.2 to know Problems (1.1)-(1.3).
1) Knowing the arbitrary bounded set
by Lemma 3.2, have
and
(3.43)
where
,
, this shows that
is uniformly bounded with in
.
2) Further, about
, there is
(3.44)
Then
is a bounded absorbing set of semigroup
.
3) According to Rellich-Kondrachov compact embedding theorem
, then the bounded set in
is the compact set in
, therefore, the solution semigroup
is a fully continuous operator. Therefore, the family of global attractor
of the solution semigroup
can be obtained, where
(3.45)
The theorem is completed.
4. Dimension Estimation of the Family of Global Attractor
First consider the linearization of Problems (1.1)-(1.3)
(4.1)
(4.2)
(4.3)
where
,
is the solution of Problems (1.1)-(1.3), known
,
, can prove
, the linearization Problems (4.1)-(4.3) have unique solutions
.
Theorem 4.1.
, the mapping
is Frechet differentiable on
, and differentiate the linear operator of
, where
is the solution of problems (4.1)-(4.3).
Proof: Set
,
, then
, from this we can get the Lipchitz property of
on
the bounded set of
, that is,
(4.4)
Set
, then
(4.5)
(4.6)
let
,
then
(4.7)
Take the inner product of
and both sides of Formula (4.5)
(4.8)
Let
, according to lemma 3.2, differential mean value theorem and poincare’s inequality
(4.9)
(4.10)
(4.11)
(4.12)
It can be obtained from hypothesis (H1), Holder’s inequality, Young’s inequality, Poincare’s inequality and differential mean value theorem, let
(4.13)
Substitute (4.9)-(4.13) into Formula (4.8), then it can be obtained by Young’s inequality and Poincare’s inequality
According to Gronwall’s inequality
(4.14)
When
, there is
(4.15)
The theorem is proved.
Theorem 4.2. Under the condition of theorem 3.3, the family of global attractor
of Problems (1.1)-(1.3) have finite Hansdorff dimension and Fractal
dimension, and
.
Proof: assume
,
then
is an isomorphic mapping.
Linearize Equation (4.1)
(4.16)
(4.17)
where
(4.18)
(4.19)
(4.20)
(4.21)
where
. U is the solution of (4.16).
For a fixed
, Let
be n elements in
. letting
is several solutions of linear Equation (4.1) with initial value of
. Available by direct calculation
(4.22)
where
represents the outer product, tr stands for trace,
represents the orthogonal projection from
to the subspace generated by
.
For a given moment
, let
be the standard orthogonal basis of space
.
Define the inner product over
(4.23)
To sum up, there are
(4.24)
where
(4.25)
(4.26)
Present hypothesis
, according to theorem 3.3, A is a bounded absorption set in
.
;
, there is
with mapping
(4.27)
(4.28)
where
satisfy
(4.29)
Comprehensive the above contents are as follows
(4.30)
where
(4.31)
(4.32)
Almost any t has
(4.33)
So
(4.34)
Set
(4.35)
(4.36)
It can be known from (4.32)
(4.37)
Therefore, the Lyapunov exponent
of
is uniformly bounded
(4.38)
From the above knowledge, there are
and
,
(4.39)
(4.40)
So
(4.41)
Thus, we can get the conclusion
(4.42)