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The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping

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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

(H_{1}) Setting then

(2.1)

(H_{2}) If

(2.2)

where

(H_{3}) There exist constant, such that

(2.3)

(H_{4}) There exist constant, such that

(2.4)

(2.5)

where;

For every, by (H_{1})-(H_{3}) and apply Poincaré inequality, there exist constants, such that

(2.6)

(2.7)

where is independent of.

Lemma 1. Assume (H_{1})-(H_{3}) 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, (H_{1}) - (H_{4}) hold. If (H_{5}): , 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 L^{2}-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 (H_{4}): 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 (H_{1}) - (H_{4}) 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 (H_{4}), 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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*International Journal of Modern Nonlinear Theory and Application*,

**5**, 185-202. doi: 10.4236/ijmnta.2016.54018.

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