International Journal of Modern Nonlinear Theory and Application
Vol.05 No.04(2016), Article ID:72185,18 pages
The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping
Yunlong Gao, Yuting Sun, Guoguang Lin
Department of Mathematics, Yunnan University, Kunming, China
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: October 10, 2016; Accepted: November 20, 2016; Published: November 23, 2016
In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations:. At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.
Nonlinear Higher-Order Kirchhoff Type Equation, The Existence and Uniqueness, The Global Attractors, Hausdorff Dimensions, Fractal Dimensions
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:
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  studied spectral gap global solutions for degenerate Kirchhoff equations. Given a continuous function, they consider the Cauchy problem:
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  studied Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity:
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  studied the Global attractor for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity:
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  studied the global existence and blow-up of solutions for a higher-order nonlinear Kirchhoff-type hyperbolic equation:
where, is a bounded domain, with a smooth boundary and a unit outer normal v. Setting Assume that p satisfies the condition:
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  studied The Asymptotic Behavior of Solutions for a Nonlinear Higher Order Kirchhoff Type Equation:
where is an open bounded set of with smooth boundary and the unit normal vector. The function satisfies the following conditions:
where. Furthermore, there exists such that
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  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  .
For more related results we refer the reader to      . 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      , 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.
For convenience, we denote the norm and scalar product in by and;
, , , , ,
According to  , 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
(H3) There exist constant, such that
(H4) There exist constant, such that
For every, by (H1)-(H3) and apply Poincaré inequality, there exist constants, such that
where is independent of.
Lemma 1. Assume (H1)-(H3) hold, and. Then the solution of the problem (1.1) - (1.3) satisfies and
is the first eigenvalue of in, and, , , ,
. Thus, there exists and, such that
Proof. We take the scalar product in of equation (1.1) with. Then
After a computation in (2.10), we have
Collecting with (2.11) - (2.14), we obtain from (2.10) that
, by using Hölder in-
equality Young’s inequality and Poincaré inequality, we deal with the terms in (2.15) one by one as follow:
By (2.7), we can obtain
Because of, we can obtain
By (2.16) - (2.19), it follows from that
By Young’s inequality and, we have
By (2.22), we get
By (2.21) and substituting (2.23) into (2.20), we receive
Since and, we get
By (2.6) and (2.21), we have
Combining with (2.25) and (2.26), formula (2.24) into
We set. Then, (2.27) is simplified as
From conclusion (2.26), we know. So, by Gronwall’s inequality, we obtain
By generalized Young’s inequality, we have
Then, we get
By (2.26) and (2.30), we have
Combining with (2.29) and (2.31),we obtain
So, there exist and, such that
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
where, is the first eigenvalue of in,
. Thus, there exists and, such that
Proof. Taking L2-inner product by in (1.1), we have
After a computation in (2.37) one by one, as follow
By Young’s inequality, we get
Next to estimate in (2.41). By (H4): and Young’s inequality, we have
By and Embeding Theorem, then. So there exists
, such that. bounded by lemma 1. Then, (2.42) turns into
Collecting with (2.43), from (2.41) we have
By and Young’s inequality, we obtain
Integrating (2.38) - (2.40), (2.44) - (2.45), from (2.37) entails
By Poincaré inequality, such that. So, (2.46) turns into
First, we take proper, such that and by Lam- ma 1. Then, we assume that there exists, such that and
Then, formula is simplified
By Gronwall’s inequality, we get
On account of Lemma 1, we know is bounded. So the hypothesis is true. Namely, we prove that there are, makes
Substituting (2.50) into (2.47), we receive
where. By Gronwall’s inequality, we have
Let so we get
So, there exists and, such that
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
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
By multiplying (3.2) by, we get
Exploiting (3.4) - (3.6), we receive
In (3.7), according to Lemma 1 and Lemma 2, such that
where and are constants.
By (H4), we obtain
where is constant.
From the above, we have
For (3.10), because is bounded. Then, there exists, such that . So, we have
where By using Gron-
wall’s inequality for (3.11), we obtain
Hence , we can get That shows that
So we get the uniqueness of the solution.
3.2. Global Attractor
Theorem 3.2.  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
2) It exists a bounded absorbing set, namely, , it exists a constant, so that
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
is the bounded absorbing set of and satisfies
2), here and it is a bounded set,
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,
This shows that is uniformly bounded in.
(2) Furthermore, for any, when, we have
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):
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:
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  Assume H is a Hilbert space, is a compact set of H. is a continuous mapping, satisfy the follow conditions.
2) If is Fréchet differentiable, it exists is a bounded linear differential operator, that is
The proof of lemma 4.1 see ref.  is omitted here. According to Lemma 4.1, we can get the following theorem :
Theorem 4.1.   Let is the global attractor that we obtain in section 3.In that case, has finite Hausdorff dimensions and Fractal dimensions in
Proof. Firstly, we rewrite the equations (4.1), (4.2) into the first order abstract evolution equations in.
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:
where. The initial condition (4.5) can be written in the following form:
We take, then consider the corresponding n solutions: of the initial values: in the Equations (4.10) - (4.11). So there is
, 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
where is the inner product in.Then; .
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:
where meets:. Comprehensive above can be obtained:
, due to is a standard orthogonal basis in. So
Almost to all t, making
Let us assume that, is equivalent to Then
According to (4.19), (4.20), so
Therefore, the Lyapunov exponent of (or) is uniformly bounded.
From what has been discussed above, it exists, a and r are constants, then
According to the reference   , we immediately to the Hausdorff dimension and fractal dimension are respectively.
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.
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.
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11561076.
Cite this paper
Gao, Y.L., Sun, Y.T. and Lin, G.G. (2016) The Global Attractors and Their Hausdorff and Fractal Di- mensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping. International Jour- nal of Modern Nonlinear Theory and Appli- cation, 5, 185-202. http://dx.doi.org/10.4236/ijmnta.2016.54018
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