The Long-Term Dynamic Behavior of Solutions to a Class of Generalized Higher-Order Kirchhoff-Type Coupled Wave Equations ()
1. Introduction
In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations:
(1)
(2)
the boundary conditions:
(3)
(4)
the initial conditions:
(5)
where
is a bounded domain in
with smooth boundary
,
is a known function,
are nonlinear source term and the external force interference terms,
is real number.
Recently, the global attractor and its dimension estimation for Kirchhoff type equations have been favored by many scholars. Many scholars have done a lot of research on this kind of problems and obtained good results [1] [2] [3].
Lin Guoguang, Gao Yunlong [1] 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.
Guoguang Lin, Ming Zhang [2] studied the initial boundary value problem for a class of Kirchhoff-type coupled equations:
they obtained the existence of the global attractor and a precise estimate of upper bound of Hausdorff dimension.
Lin Guoguang, Yang Lujiao [3] studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms:
by assuming the nonlinear source terms
and Kirchhoff stress term
, the author verified the appropriateness of the solution and proved the existence of the global attractor, obtained the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor.
For more significant research results about the global attractor and its dimension estimation of Kirchhoff equation, please refer to the literature [4] - [18].
2. Existence and Uniqueness of Solutions
The following symbols and assumptions are introduced for the convenience of statement:
In order to obtain our results, we consider system (1)-(5) under some assumptions on
and
. Precisely, we state the general assumptions:
(A1)
is not decreasing function and for positive constants
,
(1)
,
(2)
is a non-negative Lipschitz function, L is associated with the Lipschitz constant
,
.
(A2) For any
,
, there exist
,
,
, such that
Lemma 1 Assuming (A1)-(A2) are true, letting
,
, then there is a solution
for problem (1)-(5) which has the following properties:
(i)
;
(ii)
(6)
where
.
(iii) There are normal numbers
and
, such that
(7)
Proof: Let
inner product with Equation (1),
(8)
according to the hypothesis (A1), using the Young inequality, Holder inequation, Poincare inequality, etc., there are
(9)
(10)
(11)
(12)
where
or
.
Similarly, letting
inner product with Equation (2), the treatment of each item is similar to (9)-(11), and the above results are sorted out,
(13)
using the Young inequality, the Poincare inequality, and the assumptions (A2), The individual items in Equation (13) are treated as follows:
(14)
by the Poincare inequality has
(15)
where
is the first eigenvalue with homogeneous Dirichlet boundary conditions of
, in the same way
(16)
(17)
where
,
,
.
Let
,
, there are normal numbers
, such that
(18)
let
, there are
(19)
by Gronwall inequality,
(20)
so there are normal numbers
and
, such that
(21)
Lemma 1 is proved.
Lemma 2 Assuming (A1)-(A2) are true,
,
,
, then there is a solution
for problem (1)-(5), which has the following properties:
(i)
;
(ii)
(22)
where
.
(iii) There are normal numbers
and
, such that
(23)
Proof: Let
inner product with Equation (1),
(24)
according to the hypothesis (A1), using the Young inequality, Holder inequation, Poincare inequality, etc., there are
(25)
(26)
(27)
where
or
.
Similarly, letting
inner product with Equation (2), the treatment of each item is similar to (25)-(27), using Young inequality, Poincare inequality, and assumption (A2), the individual terms are treated as follows:
(28)
(29)
Then sort out the result of appeal and other inner product items, get
(30)
using the Young inequality, Poincare inequality, and assumption (A2), the individual terms in Equation (30) are treated as follows:
(31)
(32)
where
,
,
.
Let
,
, there are normal numbers
, such that
(33)
let
, get
(34)
by Gronwall inequality,
(35)
so there are normal numbers
and
, such that
(36)
Lemma 1 is proved.
Theorem 1 Assuming (A1)-(A2) is true,
,
,
, then the initial boundary value problem (1)-(5) has a unique solution
(37)
Proof: According to literature [9] and Galerkin method, combining with lemma 1 and lemma 2, we can easily obtain the existence of solutions.
Next, prove the uniqueness of the solution:
Assuming
are the two solutions of the problem (1)-(5), letting
, obtain that
(38)
where
.
Let
inner product with the first two equations in (38) and obtain,
(39)
using the Young inequality, Poincare inequality, as well as the assumptions (A2), for processing on the type of individual items as follows:
(40)
similarly, we obtain
(41)
(42)
(43)
Through the (40)-(43), finally will become
(44)
Let
, there are normal numbers
, where
, such that
(45)
using the Gronwall inequality, we have
(46)
(47)
so
(48)
Theorem 1 is proved.
3. The Family of Global Attractors and Dimension Estimation
Theorem 2 [9] Assume
is a Banach space,
is the operator semigroup,
,
,
, where I is the identity operator. If
satisfies
1) Semigroup
is uniformly bounded in E, i.e.
, exists a constant
such that when
, there is
;
2) There exists a bounded absorbing set B in E, that is, for any bounded set
, there exists a constant
, such that
, (49)
3)
is completely continuous operator.
Then operator semigroup
has compact global attractor A.
Theorem 3 Let
is a solution semigroup generated by the initial boundary value problems (1)-(5) under the hypothesis of lemma 1 and lemma 2, then the initial boundary value problems (1)-(5) have the family of global attractors. There are compact sets satisfying:
and
where
,
1)
,
2)
attracts all bounded sets of
, that is, any bounded set
, having
, where
.
then compact set
are called the family of global attractors of semigroup
.
Proof: From lemma 1, lemma 2, for any bounded set
and
, the equation has solution semigroups
, and
(50)
where
,
, shows that
is uniformly bounded in
;
Further,
is a
bounded absorption set of semigroup
;
is compactly embedded in
, i.e., the bounded set in
is a compact set in
, so the operator semigroup
is completely continuous operator. Then there exists a global attractor family of equations
Theorem 3 is proved.
After the family of global attractors are obtained, in order to estimate the Hausdroff dimension and Fractal dimension of the family of global attractors, the initial boundary value problem (1)-(5) is linearized and obtain that
(51)
where
,
is the solution of the initial boundary value problem (51).
Given
,
, for any
, there exists a unique solution
to the linear initial boundary value problem (51).
Lemma 3 For any
,
, the mapping
is Frechet differentiable. The derivative on
is a linear operator on
,
where
is the solution of the problem (51).
Proof: suppose
,
with
,
, we denote
First, we can prove a Lipschitz property of
on the bounded sets on
, that is
(52)
We now consider the difference
is the solution to problem (53),
(53)
where
(54)
(55)
where
.
Some items are treated as follows
(56)
Let
inner product with the first equation in (53),
and inner product with the second equation in (53), obtain
which implies that
(57)
Analogously,
(58)
Further,
(59)
where,
Similarly
(60)
Based on the above Equations (57)-(60), it is sorted out that
(61)
where
,
by Gronwall inequality
(62)
so as
in
, there are
The differentiability of S(t) is proved.
The next step will be used in demonstrating the process of dimension estimation. it seems obvious that the Equations (1)-(5) also can be written as
(63)
where
,
,
,
(64)
(65)
Consider the first variation equation of (63)
(66)
where
,
,
, and
is the solution to problem (63), and
(67)
(68)
(69)
Theorem 4 Under the condition of Theorem 3, the global attractors of initial boundary value problems (1)-(5) have finite dimensional Hausdroff dimension
and fractal dimension, and then
,
.
Proof: For any fixed
, assume that
are
elements in
, and
are
solutions of the linearized Equation (66) with an initial value
, where
is a natural number. It can be obtained by calculation
(70)
where
represents the outer product, tr represents the trace of the operator,
represents the orthogonal projection from
to
.
At a given time
, let
are the standard orthogonal basis of space
, then define the inner product on
Through the above conditions, can get
by the Holder inequality, Young and Poincare inequality
(71)
(72)
(73)
Based on the above Equations (71)-(73), it is sorted out that
(74)
let
we can obtain
(75)
for almost all times t, there is
,
, so
(76)
because of
(77)
(78)
Therefore, the Lyapunov exponent
on set
is uniformly bounded, and
(79)
so
(80)
(81)
further
(82)
Thus, can obtain
,
.