The Family of Exponential Attractors and Inertial Manifolds for a Generalized Nonlinear Kirchhoff Equations ()
1. Introduction
Exponential attractor is a compact positive invariant set with finite fractal dimension and exponentially attracts every orbit, which is an important feature to describe the long-term behavior of nonlinear partial differential equations. In reference [1], since Foias and others put forward this concept in 1994, many mathematicians have made in-depth research on exponential attractors. Inertial manifold refers to the positive invariant Lipschitz manifold of finite dimension, which includes the global attractor attracting all solution orbits at exponential speed, and it is an important bridge between infinite dimensional dynamical system and finite dimensional dynamical system.
In reference [2], the author studied the exponential attractors of the following nonlinear wave equations by using operator decomposition and finite covering methods.
Contrary to the global attractor, the exponential attractor has a uniform exponential convergence rate on the invariant absorption set of its solution. Because of this, the exponential attractor has deeper and more practical properties, and under the perturbation and numerical approximation, the exponential attractor is more robust than the whole attractor.
In reference [3], Perikles G. Papadopoulos, Nikos M. Stavrakakis studied the global existence and blow-up of the following equations
Initial condition
.
Li et al. [4]. studied the global existence and blow-up of solutions for the following high-order Kirchhoff type equations with nonlinear dissipation terms
where
is a bounded open region with smooth boundary,
is an outward normal vector,
is a positive integer and
is a normal number. In this paper, using the concavity method, it is obtained that the solution has global existence when
, but when
, for any initial value with negative initial energy, the solution explodes in a finite time with the norm in
. Salim [5] not only improves the results in reference [4] by modifying the proof method, but also proves that when the positive initial energy has an upper bound, the solution explodes in a finite time. Inspired by reference [4] [5], Ye et al. [6] studied the following hyperbolic equations of Kirchhoff type with damping term and source term:
where
,
is a positive integer,
is a bounded region with smooth boundary,
is an outward normal vector, and
and
are normal numbers. The author not only obtains the global existence of the solution by constructing a stable set in
, but also proves the estimation of energy attenuation by using Komornik lemma.
For more research on exponential attractors and inertial manifolds, we can read the literature [7] - [16].
Inspired by the above research, this paper will discuss a family of the existence of exponential attractors and inertial manifolds of a generalized Kirchhoff equation with damping term:
(1)
where
,
is a bounded domain with smooth boundary
,
is an external force term,
is the stress term of Kirchhoff equation,
,
is a strong dissipative term,
is a nonlinear source term.
In this paper, our main difficulty is the handling of
and nonlinear terms
. In order to overcome the difficulties, certain assumptions are needed to solve them. The algorithm of proof process has been used by predecessors. The previous algorithms are combined and extended to solve the difficulty of nonlinear term in the paper. This paper is organized as follows. Section 2 is some basic assumptions. Section 3 proves the existence a family of exponential attractors. Section 4 proves the existence of a family of the inertial manifolds by using the Hadamard graph transformation method.
2. Preliminaries
For brevity, we used the follow abbreviation:
,
,
,
,
and
denotes positive constant,
is the first eigenvalue of
with homogeneous Dirichlet boundary condition on
.
The notation
for the H inner product and norm,that is
,
.
(H1) assume that Kirchhoff type stress term
satisfies:
where
is a constant.
(H2)
.
3. Exponential Attractors
We denote the inner product and norm in
as following:
,
we have
(1)
(2)
Setting
, then Equation (1.1) can be converted into the following first-order evolution equation
(3)
where
(4)
(5)
In order to accomplish the proof, we need to construct a map. Let
are two Hilbert spaces with
is dense and continuous injection, and
is compact. Let
is a solution semigroup generated by Equation (3.3).
In the following definitions,
.
Definition 3.1 [17]
compact set
is called an exponential attractor for
if
and
1)
,
2)
has finite fractal dimension,
,
3) There exist universal constans
such that
(6)
where
,
is a positively invariant set for
in
.
Definition 3.2 [17] If for every
, there exist a time
, an integer
, and an orthogonal projection
of rank equal to
such that for every U and V in
, either
(7)
or
(8)
then we call
is squeezing in
, where
.
Theorem 3.1 [1] Assume that
1)
possesses a family of
-compact attractors
,
2)
exists a positive invariant compact set
,
3)
is a Lipschitz continuous map with a Lipschitz continuous function
on
, such that
, and satisfied the discrete squeezing property on
.
Then
has a family of
-compact exponential attractors
and
(9)
where
(10)
Moreover, the fractal dimension of
satisfies
, where
is the smallest N which make the discrete squeezing property established,
.
Proposition 3.1 [1] There exist
such that
is the positive invariant set of
in
, and
attracts all bounded subsets of
, where
is a closed bounded absorbing set for
in
.
Proposition 3.2 [1] Let
respectively are closed bounded absorbing set of Equation (3.3) in
, then
possesses a family of
-compact attractors
.
Under of the appropriate hypothesized, the initial boundary value problem Equation (1.1) exists unique smooth. This solution possesses the following properties:
(11)
(12)
We denote the solution in Theorem 3.1 by
, the
is a continuous semigroup in
, There exist the balls:
(13)
(14)
respectively is a absorbing set of
in
and
.
Lemma 3.1 For
, when we can obtain
(15)
Proof. By (3.1), (3.4) we get
(16)
By employing Hölder’s inequality, Young’s inequality and Poincaré’s inequality, we process the terms in (3.16), we have
(17)
(18)
By the value of
,and substituting (3.17)-(3.18), we have
(19)
because of
, so
,
,
.
Let
,
, we can get
The proof is completed.
Let
, where
,
, where
,
Next set
, where
, then
satisfies:
(20)
(21)
In order to certify Equation (1.1) exists a family of exponential attractors,we first show the semigroup
of system (1.1) is Lipschitz continuous on
.
Lemma 3.2 (Lipschitz property) For
, where
is the initial values of problem (1.1), and
, we have
(22)
Proof. Taking the inner product of the Equation (3.20) with
in
, we have
(23)
Next, we deal with the following items one by one. Similar to Lemma 3.1, we easily obtain
(24)
For convenience, let’s call
, then by (H1) and using the mean value theorem, Young’s inequality, we have
(25)
For the last term, we apply the mean value theorem, by (H2), we have
By the interpolation inequality
In the same way with
where
.
Therefore
(26)
Integrating (3.24) - (3.26) into (3.23), we have
(27)
where
.
By using Gronwall’s inequality, we have
(28)
where
, so we have
(29)
The proved is completed.
Now,we introduce the operator
, Obviously,
is an unbounded self-adjoin positive operator and
is compact. So, there is an orthonormal basis
of H consisting of eigenvectors
of
such that
,
.
denote by
the projector,
is an orthogonal projection,
.
As follows,we will need
Lemma 3.3 For
, where
is the initial values of problem (1.1). Let
then we have
(30)
Proof. Applying
to (3.20), we have
(31)
Taking the inner product of (3.31) with
in
, we have
(32)
Next, we deal with the following items one by one
(33)
For the last term, we apply the mean value theorem, by (H2), we have
By the interpolation inequality
In the same way with
where
.
Therefore
(34)
Integrating (3.33) - (3.34) into (3.32), we have
(35)
where
.
Using Gronwall’s inequality, we have
(36)
The proved is completed.
Lemma 3.4 (squeezing property) For
, if
(37)
then we have
(38)
Proof. If
, then
(39)
Let
be large enough
(40)
Also let
be large enough
(41)
Subsituting (3.39) - (3.41) into (3.38), we have
(42)
The proved is completed.
Theorem 3.2 Under the above assumptions,
,
,
.Then the initial boundary value problem (1.1) the solution semigroup has a family of
-compact exponential attractors
on
,
, and the fractal dimension is satisfied
.
Proof. According to Theorem 3.1, Lemma 3.2, Theorem 3.2 is easily proven.
4. A Family of Inertial Manifolds
Next, we will prove the existence of a family of inertial manifolds when N is large enough by using graph norm transformation method.
Definition 4.1 [18] Let
be the solution semigroup on Banach space
, and there is a subset
:
1)
is a finite-dimensional Lipschitz manifold;
2)
is the positive invariant set, that is
,
;
3)
attracts exponentially all orbits of solutions, that is, there are constants
,
, Such that
It is said that
is an inertial manifold about
.
Definition 4.2 [18] Let the operator
have several eigenvalues of positive real parts, and its eigenfunction
expands into the corresponding orthogonal space in
, and
satisfies the Lipschitz condition
(1)
If the point spectrum of the operator can be divided into two parts
and
, where
is finite,
(2)
(3)
Then
(4)
(5)
hold with continuous orthogonal projection
,
, So it is said that the operator
satisfies the spectral interval condition, P is orthogonal projection.
Lemma 4.1 Let the eigenvalues
is non-decreasing and for every
, when
, such that
and
are consecutive adjacent values.
Equation (1.1) are equivalent to the following first-order evolution equation:
(6)
with
(7)
(8)
we consider the graph norm on
, which induced by the scale product
(9)
where
;
represent the conjugation of
respectively;
. Obviously, the operator
defined in (4.2) is monotone. Indeed, for
,
(10)
Therefore,
is a non-negative real number.
In order to determine the characteristic value of
, we consider the following characteristic equation
(11)
that is
(12)
Substituting the first Equation of (4.12) into the second equation can be obtained
(13)
Taking the inner product of
, on both sides of the Equations of (4.13) respectively, we acquire
(14)
Regarding (4.14) as a quadratic equation of one variable with respect to
, for
, and let
,
, the corresponding eigenvalues of Equation (4.11) are as follows:
(15)
where
,
is the eigenvalue of
in
, then
. If
, then
, that is all the eigenvalues of
are positive real numbers, and the corresponding eigenvectors are in the form of
. For convenience, we note that for any
,
(16)
Theorem 4.1 Assue
,
is large enough, when
, the following inequality holds
(17)
Then the operator
satisfies the spectral gap condition
.
Proof: It is known that all the eigenvalues of
are positive real numbers,
, and the sequence
and
are monotonically increasing.
The following four steps to prove Theorem 4.1.
step 1: Because
is a non-decreasing sequence. According to Lemma 4.1, given N such that
and
are consecutive adjacent eigenvalues, the eigenvalues of the operator
are decomposed into
and
, where
is the finite parts, which are expressed as follows.
(18)
(19)
step 2: Consider the corresponding decomposition of
.
(20)
(21)
The purpose is to make these two orthogonal subspaces of
and satisfy the spectral gap Equation (4.4) is true when
,
. Further decomposition
, then
,
(22)
(23)
and set
. Note that
and
are finite dimensional, that
,
, and that the reason why
is not orthogonal to
is that, while it is orthogonal to
is not orthogonal to
Now we introduce two functions
,
, defined by
(24)
(25)
with
,
represents the conjugate of y and g respectively.
For
, then
(26)
For any k, there is
, and according to the initial hypothesis
, that is
,
is positive definite. Similarly, for
, then
(27)
that is
,
is positive definite.
Thus
and
define a scalar product, respectively on
and
, and we can define an equivalent scalar product in
, by
(28)
where
and
are projections of
to
and
respectively, for brief, (4.28) can be abbreviated as the following
We proved then to show that the subspaces
and
defined in (4.20), (4.21) are orthogonal with respect to the scalar product (4.28). In fact, it is sufficient to show that
is orthogonal to
, in turn, this reduces to showing that
(29)
Recalling (4.26) and (4.27),
(30)
according to (4.15)
thus, (4.30) is equivalent to
step 3: Further, we estimate the Lipschitz constant
of
then
Give
, we get
By the interpolation inequality
where
.
Therefore
thus
(31)
step 4: Now we need to verify that the spectral interval condition
is established.
and
, we can get
(32)
with
.
and
(33)
For formula (4.32). There,
, such that for
,
we can get
(34)
From the condition, it can be determined that
such that for all
, and with (4.32)
(35)
under the latter assumption, Theorem 4.1 is proved completely.
Theorem 4.2 In the conclusions of Theorem 4.1, initial boundary value problems admits an inertial manifold
in
of the form
(36)
where
is Lipschitz continuous with the Lipschitz constant
,and
represents the diagram of
.
Proof: According to Theorem 4.1, Lemma 4.1 and Definition 4.1 is easily proven.