A Family of Exponential Attractors and Inertial Manifolds for a Class of Higher Order Kirchhoff Equations ()
1. Introduction
In the 1990s, C. Foiao proposed the concept of an exponential attractor. The exponential attractor has a compact positive invariant set with finite fractal dimension and is exponentially attractive to the solution orbit. The exponential attractor has deeper and more practical properties. Compared with the global attractor, the exponential attractor has a uniform exponential convergence rate on the invariant absorption set of its solution. The exponential attractor is more robust under numerical approximation and perturbation. The family of inertial manifolds is concerned with the long-time behavior of the solution of a dissipative evolution equation, which is a finite-dimensional invariant Lipschitz manifold and attracts all solution orbits in the phase space with an exponential rate. The family of inertial manifolds is an important link between finite-dimensional and infinite-dimensional dynamical systems. In this paper, we study the family of exponential attractors and inertial manifolds of the following nonlinear Kirchhoff equations
(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.
In reference [1], Fan Xiaoming constructed the spatial discretization based on the wave equation on
, and studied the exponential attractor of the second-order lattice dynamical system with nonlinear damping:
In reference [2], Yang Zhijian et al. studied the exponential attractor of Kirchhoff equation with strong damping of nonlinear term and supercritical nonlinear term:
They prove the existence of the exponential attractor by using the weak quasistability estimate; in reference [3], Xu Guigui, Wang Libo and Lin Guoguang studied a class of second-order nonlinear wave equations with time delay under the assumption that the delay time is small:
The existence of inertial manifolds. Need to know more references [4] - [17].
2. Basic Assumption
For convenience, space and symbols are defined as follows:
,
,
,
,
.
and
represent the inner product and the norm on H space, then
,
.
Nonlinear function
meets the following conditions:
(H1)
(H2)
(H3)
(H4)
3. Exponential Attractors
In order to prove the need of later, so the inner product and norm of
are defined
(3.1)
(3.2)
Let
,
,
then the Equation (1.1) is equivalent to
(3.3)
where
By definition, we know that
are two Hilbert spaces,
is dense and compact in
, let
is the mapping of
to
,
.
Definition 3.1. [4] If there is a compact set
,
attracts all bounded sets in
, and it is an invariant set under
,
,
. Then we say that a semigroup
has a family of
-compact attractors
.
Definition 3.2. [5] If
and
1)
,
;
2)
has finite fractal dimension,
;
3) There exist universal constants
, such that
where,
,
is the positive invariant set of
in
. The compact set
is called a family of
exponential attractors for the system
.
Definition 3.3. [5] if there exists limited function
, such that
(3.4)
Then the semigroup
is Lipschitz continuous in
.
Definition 3.4. [5] If
and exists an orthogonal projection
of rank N, such that for
,
(3.5)
or
(3.6)
Then
is said to satisfy the discrete squeezing property in
, where
.
Theorem 3.1. [5] Assume that
1)
possesses a family of
-compact attractors
;
2) In
, there exists a compact set
with positive invariance to the action of
;
3)
is Lipschitz continuous and is squeezed in
.
possesses a family of
-compact attractors
,
,
. The fractal dimension of
satisfies
, where,
is the smallest N which makes the discrete squeezing property established.
Proposition 3.2. [6] After making reasonable assumptions about
and
, the initial boundary value problem (1.1)-(1.3) has a unique smooth solution and the solution has the following properties:
The solution
of the equation is expressed by Theorem 3.1, then
is a semigroup of continuous operators in
, we have the ball
(3.7)
(3.8)
are absorbing sets of
in
and
respectively.
Lemma 3.1. For any
, there is
.
Proof. By
and
then
(3.9)
where
.
From the Young’s inequality and the Poincare’s inequality, we can get
(3.10)
(3.11)
where
is the first eigenvalue of the operator
.
Derived from Formula (3.9) to Formula (3.11)
Due to
then
thus
where
Let
, where
;
, where
;
Let
; where
, then
satisfies
(3.12)
(3.13)
To verify that the Formulas (1.1)-(1.3) has an exponential attractor, it is necessary to prove that the dynamical system
is Lipschitz continuous on
.
Lemma 3.2. (Lipschitz property). For any
, there is
Proof. We can get the inner product of
and Formula (3.4) in
space,
(3.14)
A proof similar to lemma 3.1, obtained
(3.15)
From Young’s inequality and Poincare’s inequality and the mean value Theorem, we can get
(3.16)
let
then
(3.17)
(3.18)
then
(3.19)
where
By Gronwall’s inequality, we can get
(3.20)
where
.
Then
Apparently,
is an unbounded self-adjoint closed positive operator, and
is compact, we know that there is an Orthonormal basis of H through the basic theory of spectral spacing, it is made up of the eigenvector
of
, so that
is an orthogonal projection in
.
.
Next, were going to use
Lemma 3.3. For any
,
, then
Proof. Apply
to Equation (3.4), we get
(3.21)
The sum of (3.11) and
is the inner product of
, we get
(3.22)
Known by Young’s inequality and hypothesis condition
(3.23)
(3.24)
(3.25)
Replace (3.13) with (3.12) to get
(3.26)
By Gronwall’s inequality, we can get
(3.27)
Thus lemma 3.3 is proved.
Lemma 3.4. (Discrete squeezing property). For any
, if
then
Proof. If
then
(3.28)
Let
be big enough
(3.29)
and let
be big enough
(3.30)
Replace the Formulas (3.17) and (3.18) into the Formula (3.16), we get
(3.31)
Theorem 3.3. Under the appropriate assumptions above,
,
,
,
,
then the solution semigroup of the initial boundary value problem (1.1)-(1.3) has a family of
exponential attractors on
,
and the fractal dimension is satisfied
.
Proof: According to Theorem 3.1, Lemma 3.2, Lemma 3.4, Theorem 3.3 is easy to prove.
4. Inertial Manifolds
Definition 4.1. [18] Assum
is a solution semigroup of Banach space
,
, a subset
satisfies the following three properties:
1)
is finite dimensional Lipschitz manifold;
2)
is positively invariant,
;
3)
attracts exponentially all the orbits of the solution, and
, there are constants
, then
It is said that
is an inertial manifold of
.
Definition 4.2. [19] Let
be an operator and assume that
satisfies the Lipschitz condition
(4.1)
If the point spectrum of the operator
can be divided into two parts
and
, where
is finite,
(4.2)
(4.3)
and satisfies the condition
(4.4)
and the orthogonal decomposition
(4.5)
set
and
are both continuous orthogonal projections, then the operator
is said to satisfy the spectral interval condition.
Lemma 4.1. [18] Let the eigenvalues
be non-decreasing, and
, there exists
, such that
and
are consecutive adjacent values.
Equation (1.1) is equivalent to the following first order evolution equation
(4.6)
where
a graph defined on
by the quantity product:
(4.7)
where
,
respectively represents the conjugation of y and z
,
.
, there is
(4.8)
Therefore, the operator
is monotonically increasing, and
is a nonnegative real number.
The characteristic equation
,
is equivalent to
(4.9)
(4.10)
Therefore,
satisfies the following eigenvalue problem
(4.11)
Given by Formulas (4.11) and (4.12), the corresponding eigenvectors take the form
(4.12)
where
is the eigenvecroot of
in
.
For
, there is
(4.13)
Take the position of (4.12) u in
and use
to get the inner product
(4.14)
Consider the Formula (4.16) as the quadratic equation of
, as follows:
(4.15)
Theorem 4.1. If
is large enough, when
, the following inequality holds
(4.16)
then the operator
is said to satisfy the spectral interval condition.
Proof. Because all the eigenvalues of
are positive real numbers, and the known sequence
and
is incremented.
This theorem is then proved in four steps.
Step 1: Known non-subtractive sequence of
, according to lemma 4.1, for
,
makes
and
adjacent, the eigenvalues of the operator
can be decomposed into
(4.17)
(4.18)
Step 2: The corresponding
can be decomposed into
(4.19)
(4.20)
In order to make the two subspace orthogonal and satisfy the interspectral Formula (4.4).
. further decomposition
, i.e.
(4.21)
(4.22)
and let
.
Next, we specify the quantity product of the eigenvalues over
, makes
and
orthogonal, here are two functions
,
.
(4.23)
(4.24)
where
,
respectively represents the conjugation of y and z.
Set
, then
(4.25)
Since
is sufficiently large, can Be obtained
,
, thus
is positive definite.
In the same way,
, there is
(4.26)
,
are positive definite.
Specifies the inner product of
:
(4.27)
where
and
are maps of
and
respectively.
The above formula can be abbreviated to
(4.28)
In the inner product of
,
and
orthogonal. If
and
orthogonal, just have to prove it
(4.29)
According to (4.13), and
(4.30)
(4.31)
Step 3: Further estimating the Lipschitz constant
of F, where
,
. If
,
,
, then
(4.32)
Step 4: Now verify that the spectral interval condition
holds.
Then
(4.33)
where
.
There exists
, such that for
,
. We can get
(4.34)
According to assumption (H3), we can easily see that
(4.35)
Then according to (4.16) and (4.32)-(4.35), we have
(4.36)
So the spectral interval condition holds.
Theorem 4.2. Under the assumption of Theorem 4.1, the initial boundary value problem (1.1)-(1.3) has the family of inertial manifolds
on the space
, and the form is
(4.37)
where
is defined in Formulas (4.19)-(4.20),
is a Lipschitz continuous function.