IJMNTAInternational Journal of Modern Nonlinear Theory and Application2167-9479Scientific Research Publishing10.4236/ijmnta.2016.54018IJMNTA-72185ArticlesEngineering Physics&Mathematics The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping YunlongGao1*YutingSun1GuoguangLin1Department of Mathematics, Yunnan University, Kunming, China* E-mail:gyl0813101x@163.com(YG);101120160504185202October 10, 2016Accepted: November 20, November 23, 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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
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:

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.

2. Preliminaries

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

(H2) If

where

(H3) There exist constant, such that

(H4) There exist constant, such that

where;

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

where, ,

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

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:

By (2.7), we can obtain

where

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

where

By (2.21) and substituting (2.23) into (2.20), we receive

Since and, we get

By (2.6) and (2.21), we have

where.

Combining with (2.25) and (2.26), formula (2.24) into

We set. Then, (2.27) is simplified as

where

From conclusion (2.26), we know. So, by Gronwall’s inequality, we obtain

where

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

Then,

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,

and, ,

. 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

to

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

Taking, then

where. By Gronwall’s inequality, we have

where

Let so we get

Then

So, there exists and, such that

3. Global Attractor3.1. 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

That is

Therefore

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

where,

is the bounded absorbing set of and satisfies

1);

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.

1);

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

,that is.

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

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

. 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

where is the inner product in.Then; .

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:

where meets:. Comprehensive above can be obtained:

, due to is a standard orthogonal basis in. So

Almost to all t, making

So

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.

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.

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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