Long Time Behavior of a Class of Generalized Beam-Kirchhoff Equations ()
1. Introduction
We study the initial boundary value problem of the following higher order Beam-Kirchhoff equation:
(1.1)
(1.2)
(1.3)
where
is a positive integer,
is a bounded region in
with a smooth homogeneous Dirichlet boundary,
represents the boundary of
, and
is the external force term.
is a strongly damped term,
is a constant greater than 0,
is a given non-negative function, and
.
Kirchhoff type equation is a hot topic in the research of mathematical physics equations in recent years, and many scholars have done research on this kind of equation. Igor Chueshov [1] studied the long-term behavior of the solution of the following nonlinear strongly damped Kirchhoff wave equation:
In the energy space
, the author can find that the growth of exponential p of strong nonlinear
is supercritical, that is, when
,
, there is
, where when
, the growth of exponential p for
is supercritical. In
, under the locally strong topology, a global attractor is established that is finite-dimensional. Especially in the non-supercritical case: 1) The partial strong topology will become a strong topology; 2) In
, the exponential attractor can be obtained due to the feature of strong stability estimation. In addition, Chueshov [1] also considers the well-fitting solution of the Kirchhoff equation with structural damping term
at the abstract level and the long-term dynamic system, where
.
Guoguang Lin, Yunlong Gao [2] studied the long time behavior of the solutions of a class of higher-order Kirchhoff type equation
.
For Kirchhoff dissipation term, Newton binomial theorem is used to process analysis. In dimension estimation, a novel method is used to deal with the variational problem. The lemma and theorem are obtained by adding
to both sides of the variational equation at the same time, and the existence of global attractor and exponential attractor and Hausdorff dimension estimation are proved.
Guoguang Lin and Zhuoqian Li [3] studied the family of global attractor and its dimension estimation of a class of high-order nonlinear Kirchhoff equation
.
They use Sobolev embedding theorem
to deal with source term
, analyzing Kirchhoff stress term by case treatment, and prove that compacting family of global attractor exists in solution semigroup by means of the family of global attractor correlation theory. Of course, there are many studies on the family of global attractor higher order Kirchhoff equations.
Zhijian Yang and Zhiming Liu [4] explored the long-term behavior of the solution of the Kirchhoff equation
with nonlinear strong damping and critical term. Under the premise of critical nonlinearity, the adaptability of the solution, the existence of the global attractor and exponential attractor of the solution exist in the energy space
. Their innovation is that the obtained results improve Yang’s research results.
Guigui Xu, Libo Wang and Guoguang Lin [5] explored the inertial manifold of the strongly damped wave equation:
The Fadeo-Galerkin method and the uniformly compact method are used to prove the existence and uniqueness of the strong solution to this problem. The existence of the inertial manifold is proved under the assumption of a certain spectral interval and a sufficiently small delay.
Naimen [6] studied the Kirchhoff type elliptic boundary value problem with Sobolev critical growth:
When
and
, The Sobolev embedding theorem
is used to deal with the nonlinear source term
, and the corresponding energy functional is reduced to the critical value level, so the equation satisfies the compactness condition.
Masamro [7] studied the initial boundary value problem of a class of Kirchhoff type equation
containing dissipative terms and damping terms, and proved the existence of the global solution of the equation under the initial boundary value condition by using Galerkin’s method, where
is a bounded region with a smooth boundary
, and
when
and
are present.
Matsugama and Ikehata [8] used the potential well method to discuss the existence of the global solution of the Kirchhoff equation
with damping term and the attenuation estimation of the global solution, where
Nakao and Zhijian Yang [9] studied the long time behavior of the solution of the Kirchhoff type equation
with strong damping term. When
satisfies the local Lipschitz condition, they prove the existence of the global attractor by proving the existence of the bounded absorption set and the asymptotic compactness of the corresponding continuous semigroup. Where
and
have order
with respect to u.
2. Background Knowledge and Assumptions
In this paper, we need the following mathematical notation:
,
,
,
.
,
, Closure of infinitely differentiable Spaces in
.
and
, where
is constant.
is the first eigenvalue of
on
.
The inner product and norm expressed by
and
respectively, that is,
,
.
Note the global attractor from
to
as
, while
is the bounded absorption set in
, where
.
Lemma 2.1. [10] (Holder inequality) set
We have
.
Lemma 2.2. [10] (Poincare inequality) If
is a bounded open subset, then
where
is the first eigenvalue of
on
.
Lemma 2.3. [10] (Young Inequality) For any real number
, then
,
Let function
satisfy the condition:
(A1)
,
,
Where
is the positive constant;
(A2)
,
,
Where
is the positive constant;
(A3)
3. The Existence and Uniqueness of the Global Solution
Lemma 3.1. Assuming that condition (A1), (A2), (A3) is true,
,
, then the initial boundary value problem (1.1)-(1.3) has a global smooth solution
and is satisfied
where
,
Ream
So there is a non-negative real number
and
that makes
Prove we set
and take the inner product of both sides of Equation (1.1) and v, we get
(3.1)
This is obtained by using the Holder inequality, the Young inequality, the Poincare inequality and the terms of condition (A1), (A2), (A3) in successive processing (3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
Synthetically available
(3.8)
Ream
In a result
(3.9)
It’s derived from the Gronwall inequality
(3.10)
So there exist
and
to make
.
Lemma 3.2. Assuming that condition (A1), (A2), (A3) is true and
,
is true, then the global solution
of the initial boundary value problem (1.1)-(1.3) is satisfied
where
,
Then there are positive constants
and
that make
Prove set
, and take the inner product of both sides of Equation (1.1) with
, i.e.
(3.11)
By using Holder inequality, Young inequality, Poincare inequality and conditions (A1), (A2), (A3), the terms in Equation (3.11) are processed successively
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
In summary can be obtained
From the Gronwall inequality:
(3.18)
So there exist
and
to make
Theorem 3.3. (existence and uniqueness of solutions) Under the conditions of lemma 3.1 and Lemma 3.2, and assuming
,
,
, then the initial boundary value problem (1.1)-(1.3) has a unique global solution
Proof: Galerkin’s finite element method is used to prove the existence of the global solution.
The first step is to construct an approximate solution.
Let
, where
is the eigenvalue of
with a homogeneous Dirichlet boundary on
, and
is the eigenfunction determined by the corresponding eigenvalue
, and the orthonormal basis of H is formed by the eigenvalue theory known
.
Construct the approximate solution
of problem where
is determined by the following nonlinear system of ordinary differential equations
(3.19)
When the initial condition
is satisfied and
is satisfied,
, in
is known from the basic theory of ordinary differentiation that the approximate solution
exists on
.
The second step is prior estimation.
Recording
, multiply both sides of the equation
, by
, and sum over j to get
1)
, have
2)
, have
then
in
Middle bounded.
The third step is the limiting process.
In space
, choosing a subcolumn
from sequence
causes
to converge weakly * in
.
From the space compact embedding theorem we know that
is compactly embedded in
, and
is strongly convergent almost everywhere in
.
Let
and take the limit, which can be obtained from the above equation
(3.20)
Weakly convergent in
. Due to
Thus
converges in
, and
is the conjugate space of the infinitely differentiable space of
.
is weakly convergent in
.
is weakly convergent in
.
is weakly convergent in
.
is weakly convergent in
.
In particular,
is weakly convergent in
, and
is weakly convergent in
. For all j and
,
(3.21)
can be derived.
Therefore, the existence of the weak solution of the problem (1.1)-(1.3) is obtained, the existence is proved, and the uniqueness of the solution below is obtained.
Let
be two solutions to the problem (1.1)-(1.3), and let
have
(3.22)
(3.23)
(3.24)
By
, then the original formula
(3.25)
(3.26)
So there exist
to make that
(3.27)
From Gronwall’s inequality:
(3.28)
In a result, we have
and the Uniqueness obtained.
4. The Existence and Dimension Estimation of the Family of Global Attractor
Having studied the existence and uniqueness of global solutions for problems (1.1)-(1.3), the following proves the existence of the family of global attractor and estimates the hausdorff dimension and fractal dimension.
Theorem 4.1. [10] [11] Let E be a Banach space and semigroup
satisfy the following conditions:
1) The semigroup
is uniformly bounded in E, that is
, when
, there is a constant
, such that
is true;
2) There exists a bounded absorption set
in E;
3)
is a completely continuous operator, and the semigroup
has a compact complete attractor
.
By changing the Banach space E from the above theorem to the Hilbert space
, the existence theorem of the family of global attractor is obtained.
Theorem 4.2. If the global smooth solution of the problem (1.1)-(1.3) satisfies the conditions of Lemma 3.1, Lemma 3.2 and theorem 3.3 in the existence and uniqueness of the solution, then the problem (1.1)-(1.3) has a family of global attractor
where
is the bounded absorption set in
and satisfies:
1)
.
2)
(
is a bounded set), where
,
where
is the solution semigroup generated by problem (1.1)-(1.3).
Proof: It is necessary to verify the condition (1), (2), (3) of theorem 4.1, under the condition of theorem, the equation has a solution semigroup
. The bounded set known as
and contained in ball
,
(4.1)
Then
is uniformly bounded within
. For any
,
It follows that
is the bounded absorption set of a semigroup
.
Since
is compactly embedded, that is the bounded set in
is the compactly set in
, the solution semigroup
is a completely continuous operator, so there exists a family of global attractor
of the solution semigroup
.
Theorem 4.2 is proved.
To linearize Equation (1.1)-(1.3), consider the following initial boundary value problem:
(4.2)
(4.3)
(4.4)
where
,
,
,
is the solution of problem (1.1)-(1.3) obtained by
, it can be shown that for any
, the linearized initial boundary value problem (4.2)-(4.3) has a unique solution
.
Lemma 4.3.
, The map
is Frechet differentiable on
, and the Frechet differential in
is a linear operator
.
Proof: Let
,
and
give the Lipchitz continuity of
on
, i.e.
If
, then
Subtract from the three formulas to get
(4.5)
where
(4.6)
(4.7)
, Use the mean value theorem to deal with
(4.8)
where
Make
and
, Then the above formula is
(4.9)
Similarly deal with
(4.10)
where
Similarly deal with
Recording
,
.
So we can obtain
(4.11)
Let
and each of them take the inner product: where
.
(4.12)
(4.13)
(4.14)
Due to
So there are
(4.15)
(4.16)
(4.17)
(4.18)
So we obtain
(4.19)
(4.20)
when
,
.
Theorem 4.4. Under the condition of Theorem 4.2, the family of global attractor
of the problem (1.1)-(1.3) has the Hausdorff dimension and the Fractal dimension and
,
.
Proof: Let
be an isomorphic mapping, then
, where
,
,
The Frechet differentiability of
is known from lemma 4.3 to estimate the Hausdorff dimension and Fractal dimension of problem (1.1)-(1.3). Consider the variational equation
of Equation (4.2) under initial conditions, where
(4.21)
where ream
,
,
For A fixed
, let
be n elements of
, and let
be n solutions of the linear Equation (4.2) with corresponding initial values of
. Get
where
represents the outer product,
represents the trace of the operator, and
is the orthogonal projection from space
to
.
Given
, let
be the orthonormal basis of
.
Define the inner product on
as
, (4.22)
(4.23)
Let
,
. (4.24)
Since
,
is an orthonormal basis,
(4.25)
For any t, there’s a
, (4.26)
So there is
. (4.27)
Let
Then
, Therefore, the Lyapunov exponent
of
is uniformly bounded, and
make
,
(4.28)
,
,
(4.29)
It can be concluded that N-dimensional volume elements decay exponentially in
and
,
, then the Hausdorff dimension and Fractal dimension of the family of global attractor are finite, and theorem 4.4 can be proved.
5. Conclusion
In this paper, we studied a class of generalized Bean-Kirchhoff equations, proved the existence and uniqueness of the global solution of this class of equations by Galerkin method, and further proved the existence of the family of global attractor and its Hausdorff dimension and Fractal dimension estimation, which has certain scientific significance. And there’s more to study.