A Family of the Inertial Manifolds for a Class of Generalized Kirchhoff-Type Coupled Equations ()
1. Introduction
This paper investigates the following primal value problems of a system of generalized Kirchhoff-type coupled equations:
where
is a bounded region with a smooth boundary in
,
represents the boundary of
,
and
are known functions, where
are nonlinear terms and external interference terms, respectively, and are known functions on
,
is the normal number,
is a non-negative first-order continuous derivative function, and
is the normal number,
.
In order to overcome the research difficulties, G. Foias, G. R. Sell and R. Temam [1] proposed the concept of inertial manifolds, which greatly promoted the study of infinite-dimensional dynamical systems. Where the inertial manifold is a positive, finite-dimensional Lipschitz manifold, and the existence of an inertial manifold depends on the establishment of a spectral interval condition. Therefore, the research on a family of inertial manifolds is of great significance from both theoretical and practical aspects, and the relevant theoretical achievements can be referred to [2] - [9].
Guoguang Lin, Lingjuan Hu [10] studied a system of coupled wave equations of higher-order Kirchhoff type with strong damping terms
where
is a bounded region with a smooth boundary in
,
represents the boundary of
,
is a nonlinear source term,
is an external force interference term, and
,
is a strong dissipation terms. Using the Hadamard graph transformation method, the Lipschitz constant
of F is further estimated, and the inertial manifolds that satisfies the spectral interval condition is obtained.
Lin Guoguang, Liu Xiaomei [11] studied a family of inertial manifolds for a class of generalized higher-order Kirchhoff equations with strong dissipation terms
where
,
is a bounded domain with a smooth boundary in
,
is an external force term,
is the stress term of Kirchhoff equation,
is a strong dissipative term,
is a nonlinear source term. Based on appropriate assumptions and the Hadamard graph transformation method, the spectral interval condition is verified, and the existence of a family of the inertial manifolds of the equation is obtained.
On the basis of previous research, rigid term strengthening becomes
and
, and this paper seeks a family of inertial manifolds. When defining the equivalence norm in space
, by making reasonable assumptions, it is obtained that the equation satisfies the spectral interval condition so that there is a family of inertial manifolds.
2. Preliminaries
For narrative convenience, we introduce the following symbols and assumptions:
Set
. Consider Hilbert space family
, whose inner product and norm are
and
, respectively. Apparently there are
The assumption is as follows:
Let
be a continuous function on interval
, and
:
, set
.
3. A Family of Inertial Manifolds
Definition 1 [12] lets
be the solution semigroup on Banach space
, and a subset
satisfies:
1)
is finite-dimensional Lipschitz popular;
2)
is positively unchanging,
;
3)
attracts the solution orbit exponentially, i.e. for any
, the existence constant
makes
.
Then
is called
is a family of inertial manifolds.
In order to describe the spectral interval condition, first consider that the nonlinear term
is integrally bounded and continuous, and has a positive Lipschitz constant
, and its operator A has several eigenvalues and eigenfunctions of the positive real part.
Definition 2 [12] Set operator
has several eigenvalues of positive real numbers, and
satisfies the Lipschitz condition:
The point spectrum of the operator A can be divided into two parts
and
, and
is finite,
and conditions
(6)
are satisfied.
Where the continuous projection
, there is orthogonal decomposition
, then the operator A satisfies the spectral interval condition.
Lemma 1
is a uniform bounded and integral Lipschitz continuous function.
Proof:
,
Similarly, there are
where l is the Lipschitz constant of
,
.
Lemma 2 [12] lets the sequence of eigenvalues
is a non-subtractive sequence, then
, for
,
and
are consecutive adjacent values.
In order to verify that the operator satisfies the spectral interval condition, so as to draw the conclusion that there is a family of inertial manifolds in questions (1)-(5), the following definitions and assumptions can be made first.
Based on the above relevant conditions, consider the first-order development equation equivalent to Equations (1)-(5), as follows:
(7)
Of which
,
In order to determine the eigenvalue of matrix operator
, first consider graph module
generated by inner product in
.
Where
, and
represent the conjugation of
respectively. In addition, operator
is monotonic, and for
, there is
Therefore,
is a nonnegative real number.
In order to further determine the eigenvalue of the matrix operator
, the following characteristic equation can be considered,
(8)
That is
Thus
meet the eigenvalue problem
Take the inner product of
and Equations (1) and (2) above respectively, with
That is
(9)
Equation (9) is a univariate quadratic equation about
. Replace
with
. For each positive integer j, Equation (8) has paired eigenvalues
where
is the characteristic root of
in
, then
.
If
, then
, the eigenvalues of operator
are all real numbers, and the corresponding eigenfunction form is
For convenience, mark for any
, there are
Theorem 1: Assumes that l is the Lipschitz constant of
. When
is sufficiently large, for
, the following inequality holds
(10)
Then all operators
satisfy the spectral interval condition (6).
Proof. Because
and the eigenvalues of
are positive real numbers,
and
are single increment sequences.
The following four steps are taken to prove theorem 1:
Step 1: Because
and
are non subtractive columns, according to lemma 2, there are
, for
,
and
are continuous adjacent values.
Therefore, there is N, so that
and
are continuous adjacent values, and the eigenvalue of
can be decomposed into
Step 2: Consider the corresponding decomposition of
into
The equivalent inner product
given below makes
orthogonal.
Further decompose
, of which
Because
and
are finite dimensional subspaces,
,
, and
and
are orthogonal, while
and
are not orthogonal,
and
are not orthogonal. So we need to redefine the equivalent norm on
, so that
and
are orthogonal. Order
.
Construct two functions
of which,
Among them
or
.
For
, then
For any k, there is
. According to hypothesis
, then
, that is,
is positive definite.
Similarly, for any
, there is
So there are
,
, then
is also positive definite.
Redefine the inner product of
:
(11)
where
and
are projections of
and
, respectively Here, Equation (11) is written as
Under the redefined inner product of
, to prove that
and
are orthogonal, we only need to prove that
and
are orthogonal, that is,
Because there are
, that is
(12)
Because of Equation (9), there are
So
.
Step 3: according to the orthogonal decomposition established above, let’s prove that
satisfies the spectral interval condition. First estimate the Lipschitz constant
of F, where
According to lemma 1,
are uniformly bounded and Lipschitz continuous, if
,
,
Then
Given
, we can get
So
(13)
From (13), if
(14)
Then the spectral interval condition (6) holds.
Step 4: according to the above paired eigenvalues, there are
(15)
Of which,
.
There are
, for
, let
, there are
(16)
And because of
, there are
(17)
According to the hypothesis (10) of Theorem 1 and Equations (13)-(17), there are
(18)
Theorem 1 is proved.
Theorem 2 [12] Through theorem1, operator
satisfies the spectral interval condition, and problems (1)-(5) have a family of inertial manifolds
, and
. The form is as follows,
where
is Lipschitz continuous and has Lipschitz constant
, and
represents the graph of
.