The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping ()
1. Introduction
In this paper, we are concerned with the existence of global attractor and Hausdorff and Fractal dimensions estimation for the following nonlinear Higher-order Kirchhoff-type equations:
(1.1)
(1.2)
(1.3)
where
is an integer constant, and
is a positive constant. Moreover,
is a bounded domain in
with the smooth boundary
and v is the unit outward normal on
.
is a nonlinear function specified later.
Recently, Marina Ghisi and Massimo Gobbino [1] studied spectral gap global solutions for degenerate Kirchhoff equations. Given a continuous function
, they consider the Cauchy problem:
(1.4)
(1.5)
where
is an open set and
and
denote the gradient and the Laplacian of u with respect to the space variables. They prove that for such initial data
there exist two pairs of initial data
for which the solution is global, and such that ![]()
Yang Zhijian, Ding Pengyan and Lei Li [2] studied Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity:
(1.6)
(1.7)
where
,
is a bounded domain
with the smooth boundary
,
and the nonlinearity
and external force term g will be specified. The main results are focused on the relationships among the growth exponent p of the nonlinearity
and well-posedness. They show that (i) even if p is up to the supercritical range,
that is,
, the well-posedness and the longtime behavior of the so-
lutions of the equation are of the characters of the parabolic equation; (ii) when
, the corresponding subclass G of the limit solutions exists
and possesses a weak global attractor.
Yang Zhijian, Ding Pengyan and Liu Zhiming [3] studied the Global attractor for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity:
(1.8)
(1.9)
where
is a bounded domain in
with the smooth boundary
,
,
and
are nonlinear functions, and
is an external force term. They prove that in strictly positive stiffness factors and supercritical nonlinearity case, there exists a global finite-dimensional attractor in the natural energy space endowed with strong topology.
Li Fucai [4] studied the global existence and blow-up of solutions for a higher-order nonlinear Kirchhoff-type hyperbolic equation:
(1.10)
(1.11)
(1.12)
where
,
is a bounded domain
, with a smooth boundary
and a unit outer normal v. Setting
Assume that p satisfies the condition:
(1.13)
Their main results are the two theorems:
Theorem 1. Suppose that
and condition (1.13) holds. Then for any initial data
the solution of (1.10) - (1.12) exists globally.
Theorem 2. Suppose that
and condition (1.12) holds. Then for any initial data
the solution of (1.10) - (1.12) blows up at finite time in
norm provided that
.
Li Yan [5] studied The Asymptotic Behavior of Solutions for a Nonlinear Higher Order Kirchhoff Type Equation:
(1.14)
(1.15)
(1.16)
where
is an open bounded set of
with smooth boundary
and the unit normal vector. The function
satisfies the following conditions:
(1.17)
(1.18)
where
. Furthermore, there exists
such that
(1.19)
At last, Li Yan studied the asymptotic behavior of solutions for problem (1.14) - (1.16).
For the most of the scholars represented by Yang Zhijian have studied all kinds of low order Kirchhoff equations and only a small number of scholars have studied the blow-up and asymptotic behavior of solutions for higher-order Kirchhoff equation. So, in this context, we study the high-order Kirchhoff equation is very meaningful. In order to study the high-order nonlinear Kirchhoff equation with the damping term, we borrow some of Li Yan’s [5] partial assumptions (2.1) - (2.3) for the nonlinear term g in the equation. In order to prove that the lemma 1, we have improved the results from assumptions (2.1) - (2.3) such that
. Then, under all assumptions, we prove
that the equation has a unique smooth solution ![]()
and obtain the solution semigroup
has global attractor
. Finally, we prove the equation has finite Hausdorff dimensions and Fractal dimensions by reference to the literature [7] .
For more related results we refer the reader to [6] [7] [8] [9] [10] . In order to make these equations more normal, in section 2 and in section 3, some assumptions, notations and the main results are stated. Under these assumptions, we prove the existence and uniqueness of solution, then we obtain the global attractors for the problems (1.1) - (1.3). According to [6] [7] [8] [9] [10] , in section 4, we consider that the global attractor of the above mentioned problems (1.1) - (1.3) has finite Hausdorff dimensions and fractal dimensions.
2. Preliminaries
For convenience, we denote the norm and scalar product in
by
and
;
,
,
,
,
, ![]()
According to [5] , we present some assumptions and notations needed in the proof of our results. For this reason, we assume nonlinear term
satisfies that
(H1) Setting
then
(2.1)
(H2) If
(2.2)
where ![]()
(H3) There exist constant
, such that
(2.3)
(H4) There exist constant
, such that
(2.4)
(2.5)
where
;
For every
, by (H1)-(H3) and apply Poincaré inequality, there exist constants
, such that
(2.6)
(2.7)
where
is independent of
.
Lemma 1. Assume (H1)-(H3) hold, and
. Then the solution
of the problem (1.1) - (1.3) satisfies
and
(2.8)
where
,
, ![]()
is the first eigenvalue of
in
, and
,
,
,
,
. Thus, there exists
and
, such that
(2.9)
Proof. We take the scalar product in
of equation (1.1) with
. Then
(2.10)
After a computation in (2.10), we have
(2.11)
(2.12)
(2.13)
(2.14)
Collecting with (2.11) - (2.14), we obtain from (2.10) that
(2.15)
Since
and
, by using Hölder in-
equality Young’s inequality and Poincaré inequality, we deal with the terms in (2.15) one by one as follow:
(2.16)
(2.17)
By (2.7), we can obtain
(2.18)
where ![]()
Because of
, we can obtain
(2.19)
By (2.16) - (2.19), it follows from that
(2.20)
By Young’s inequality and
, we have
(2.21)
(2.22)
By (2.22), we get
(2.23)
where ![]()
By (2.21) and substituting (2.23) into (2.20), we receive
(2.24)
Since
and
, we get
(2.25)
By (2.6) and (2.21), we have
(2.26)
where
.
Combining with (2.25) and (2.26), formula (2.24) into
(2.27)
We set
. Then, (2.27) is simplified as
(2.28)
where ![]()
From conclusion (2.26), we know
. So, by Gronwall’s inequality, we obtain
(2.29)
where ![]()
By generalized Young’s inequality, we have ![]()
Then, we get
(2.30)
By (2.26) and (2.30), we have
(2.31)
Combining with (2.29) and (2.31),we obtain
(2.32)
Then,
(2.33)
So, there exist
and
, such that
(2.34)
Lemma 2. In addition to the assumptions of Lemma 1, (H1) - (H4) hold. If (H5):
, and
. Then the solution
of the pro- blems (1.1) - (1.3) satisfies
, and
(2.35)
where
,
is the first eigenvalue of
in
,
and
,
,
. Thus, there exists
and
, such that
(2.36)
Proof. Taking L2-inner product by
in (1.1), we have
(2.37)
After a computation in (2.37) one by one, as follow
(2.38)
(2.39)
(2.40)
By Young’s inequality, we get
(2.41)
Next to estimate
in (2.41). By (H4):
and Young’s inequality, we have
(2.42)
By
and Embeding Theorem, then
. So there exists
, such that
.
bounded by lemma 1. Then, (2.42) turns into
(2.43)
Collecting with (2.43), from (2.41) we have
(2.44)
By
and Young’s inequality, we obtain
(2.45)
Integrating (2.38) - (2.40), (2.44) - (2.45), from (2.37) entails
(2.46)
By Poincaré inequality, such that
. So, (2.46) turns into
(2.47)
First, we take proper
, such that
and
by Lam- ma 1. Then, we assume that there exists
, such that
and
Then, formula is simplified
to
(2.48)
By Gronwall’s inequality, we get
(2.49)
On account of Lemma 1, we know
is bounded. So the hypothesis is true. Namely, we prove that there are
, makes
(2.50)
Substituting (2.50) into (2.47), we receive
(2.51)
Taking
, then
(2.52)
where
. By Gronwall’s inequality, we have
(2.53)
where ![]()
Let
so we get
(2.54)
Then
(2.55)
So, there exists
and
, such that
(2.56)
3. Global Attractor
3.1. The Existence and Uniqueness of Solution
Theorem 3.1. Assume (H1) - (H4) hold, and
,
,
. So Equation (1.1) exists a unique smooth solution
(3.1)
Proof. By the Galerkin method, Lemma 1 and Lemma 2, we can easily obtain the existence of Solutions. Next, we prove the uniqueness of Solutions in detail.
Assume
are two solutions of the problems (1.1) - (1.3), let
, then
and the two equations subtract and obtain
(3.2)
By multiplying (3.2) by
, we get
(3.3)
(3.4)
(3.5)
(3.6)
Exploiting (3.4) - (3.6), we receive
(3.7)
In (3.7), according to Lemma 1 and Lemma 2, such that
(3.8)
where
and
are constants.
By (H4), we obtain
(3.9)
where
is constant.
From the above, we have
(3.10)
For (3.10), because
is bounded. Then, there exists
, such that
. So, we have
(3.11)
where
By using Gron-
wall’s inequality for (3.11), we obtain
(3.12)
Hence , we can get
That shows that
(3.13)
That is
(3.14)
Therefore
(3.15)
So we get the uniqueness of the solution.
3.2. Global Attractor
Theorem 3.2. [10] Let E be a Banach space, and
are the semigroup operator on E.
, where I is a unit operator.Set
satisfy the follow conditions:
1)
is uniformly bounded, namely
, it exists a constant
, so that
(3.16)
2) It exists a bounded absorbing set
, namely,
, it exists a constant
, so that
(3.17)
where
and
are bounded sets.
3) When
,
is a completely continuous operator. Therefore, the semigroup operator S(t) exists a compact global attractor
.
Theorem 3.3. Under the assume of Lemma 1, Lemma 2 and Theorem 3.1, equations have global attractor
(3.18)
where
, ![]()
is the bounded absorbing set of
and satisfies
1)
;
2)
, here
and it is a bounded set,
(3.19)
Proof. Under the conditions of Theorem 3.1, it exists the solution semigroup S(t),
, here
.
(1) From Lemma 1 to Lemma 2, we can get that
is a bounded set that includes in the ball
,
(3.20)
This shows that
is uniformly bounded in
.
(2) Furthermore, for any
, when
, we have
(3.21)
So we get
is the bounded absorbing set.
(3) Since
is compact embedded, which means that the bounded set in
is the compact set in
, so the semigroup operator S(t) exists a compact global attractor
.
4. The Estimates of the Upper Bounds of Hausdorff and Fractal Dimensions for the Global Attractor
We rewrite the problems (1.1) - (1.3):
(4.1)
(4.2)
(4.3)
Let
, where
is a bounded domain in
with smooth boundary
, q is positive constant, and m is positive integer. The linearized equations of the above equations as follows:
(4.4)
(4.5)
Let
,
is the solution of problems (4.4) - (4.5). We can prove that the problems (4.4) - (4.5) have a unique solution
The equation (4.4) is the linearized equation by the Equation (4.17). Define the
mapping
, here
, let
,
, let
,
,
,
,
.
Lemma 4.1 [6] Assume H is a Hilbert space,
is a compact set of H.
is a continuous mapping, satisfy the follow conditions.
1)
;
2) If
is Fréchet differentiable, it exists is a bounded linear differential operator
, that is
![]()
The proof of lemma 4.1 see ref. [6] is omitted here. According to Lemma 4.1, we can get the following theorem :
Theorem 4.1. [6] [7] Let
is the global attractor that we obtain in section 3.In that case,
has finite Hausdorff dimensions and Fractal dimensions in
,that is
.
Let
, let
, is an isomorphic mapping. So let
is the global attractor of
, then
is also the global attractor of
, and they have the same dimensions. Then
satisfies as follows:
(4.6)
(4.7)
where ![]()
(4.8)
(4.9)
(4.10)
(4.11)
where
. The initial condition (4.5) can be written in the following form:
(4.12)
We take
, then consider the corresponding n solutions:
of the initial values:
in the Equations (4.10) - (4.11). So there is
. from
, we get
, here u is the solution of problems (4.1)-(4.3);
represents the outer product, Tr reprsents the trace,
is an orthogonal projection from the space
to the subspace spanned by
.
For a given time
, let
.
is the
standard orthogonal basis of the space
.
From the above, we have
(4.13)
where
is the inner product in
.Then
;
.
(4.14)
where
![]()
Now, suppose that
, according to theorem 3.3,
is a bounded absorbing set in
.
.
Then there is a
to make the mapping
. At the same time, there are the following results:
(4.15)
where
meets:
. Comprehensive above can be obtained:
(4.16)
, due to
is a standard orthogonal basis in
. So
(4.17)
Almost to all t, making
(4.18)
So
(4.19)
Let us assume that
, is equivalent to
Then
(4.20)
According to (4.19), (4.20), so
(4.21)
Therefore, the Lyapunov exponent of
(or
) is uniformly bounded.
(4.22)
From what has been discussed above, it exists
, a and r are constants, then
(4.23)
(4.24)
(4.25)
(4.26)
According to the reference [6] [7] , we immediately to the Hausdorff dimension and fractal dimension are respectively
.
5. Conclusion
In this paper, we prove that the higher-order nonlinear Kirchhoff equation with linear damping in
has a unique smooth solution
. Fur- ther, we obtain the solution semigroup
has global attractor
. Finally, we prove the equation has finite Hausdorff dimensions and Fractal dimensions in
.
Acknowledgements
The authors express their sincere thanks to the aonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. These contributions greatly improved the paper.
Fund
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11561076.