Global Attractor for a Class of Nonlinear Generalized Kirchhoff-Boussinesq Model

ABSTRACT

In this
paper, we study the long time behavior of solution to the initial boundary
value problem for a class of Kirchhoff-Boussinesq model flow . We first prove the
wellness of the solutions. Then we establish the existence of global attractor.

Received 15 December 2015; accepted 14 March 2016; published 17 March 2016

1. Introduction

In this paper, we are concerned with the existence of global attractor for the following nonlinear plate equation referred to as Kirchhoff-Boussinesq model:

(1.1)

(1.2)

(1.3)

where is a bounded domain in, and are positive constants, and the assumptions on will be specified later.

Recently, Chueshov and Lasiecka [1] studied the long time behavior of solutions to the Kirchhoff-Boussinesq plate equation

(1.4)

with clamped boundary condition

(1.5)

with where v is the unit outward normal on. Here is the damping parameter, the mapping and the smooth functions and represent (nonlinear) feedback forces acting upon the plate, in particular,

When and, also considering the (1.4) with a strong damping, then (1.4) becomes a class of Krichhoff models arising in elastoplastic flow,

(1.6)

which Yang Zhijian and Jin Baoxia [2] studied. In this model, Yang Zhijian and Jin Baoxia gained that under rather mild conditions, the dynamical system associated with above-mentioned IBVP possesses in different phase spaces a global attractor associated with problem (1.6), (1.2) and (1.3) provided that g and h satisfy the nonexplosion condition,

(1.7)

(1.8)

with, , , and and there exist constant such that

(1.9)

Zhijian Yang, Na Feng and Ro Fu Ma [3] also studied the global attractor for the generalized double dispersion equation arising in elastic waveguide model

(1.10)

In this model, g satisfies the nonexplosion condition,

(1.11)

where is the first eigenvalue of the, and as; as.

T. F. Ma and M. L. Pelicer [4] studied the existence of a finite-dimensional global attractor to the following system with a weak damping.

(1.12)

with simply supported boundary condition

(1.13)

and initial condition

(1.14)

where, and,

For more related results we refer the reader to [5] -[8] . Many scholars assume,

to make these equations more normal; we try to make a different hypothesis (specified Section 2), by combining the idea of Liang Guo, Zhaoqin Yuan, Guoguang Lin [9] , and in these assumptions, we get the uniqueness of solutions, then we study the global attractors of the equation.

2. Preliminaries

For brevity, we use the follow abbreviation:

with, and, where are the -based Sobolev spaces and are the completion of in for. The notation for the H-inner product will also be used for the notation of duality pairing between dual spaces.

In this section, we present some materials needed in the proof of our results, state a global existence result, and prove our main result. For this reason, we assume that

(H_{1}),

(2.1)

(2.2)

where, , and when,

(2.3)

where as; as; and as.

(H_{2}) and, is the first eigenvalue of the.

Now, we can do priori estimates for Equation (1.1).

Lemma 1. Assume (H_{1}), (H_{2}) hold, and,. Then the solution of the problem (1.1)-(1.3) satisfies, and

(2.4)

where, , and , thus there exists and

, such that

(2.5)

Remark 1. (2.1) and (2.1) imply that there exist positive constants and, such that

(2.6)

Proof of Lemma 1.

Proof. Let, then v satisfies

(2.7)

Taking H-inner product by v in (2.7), we have

(2.8)

Since and, by using Holder inequality, Young’s inequality and

Poincare inequality, we deal with the terms in (2.8) one by one as follow,

(2.9)

(2.10)

and

(2.11)

(2.12)

(2.13)

By (2.9)-(2.13), it follows from that

(2.14)

By (2.6), we can obtain

(2.15)

Substituting (2.15) into (2.14), we receive

(2.16)

By using Holder inequality, Young’s inequality, and (H_{2}), we obtain

(2.17)

(2.18)

Then, we have

(2.19)

Because of, we get

(2.20)

Substituting (2.20) into (2.19) gets

(2.21)

Taking, then

(2.22)

where, by using Gronwall inequality,we obtain

(2.23)

From (H_{1}):, and as; as; as, we have, according to Embedding Theorem, then, let, then we have

(2.24)

Then

(2.25)

So, there exists and, such that

(2.26)

■

Lemma 2. In addition to the assumptions of Lemma 1, if (H_{3}):, , then the solution of the problem (1.1)-(1.3) satisfies, and

(2.27)

where, , and, thus there exists

and, such that

(2.28)

Proof. Taking H-inner product by in (2.7), we have

(2.29)

Using Holder inequality, Young’s inequality and Poincare inequality, we deal with the terms in (2.29) one by one as follow,

(2.30)

(2.31)

and

(2.32)

(2.33)

Substituting (2.30)-(2.33) into (2.29), we can obtain that

(2.34)

By using Holder inequality, Young’s inequality, and (H_{1}), (H_{3}), we obtain

(2.35)

(2.36)

By using Gagliardo-Nirenberg inequality, and according the Lemma 1, we can get

Then, we have

(2.37)

By using the same inequality, we can obtain

(2.38)

By using Gagliardo-Nirenberg inequality, and according the Lemma 1, we can get

, Then, by using Young’s inequality, we have

(2.39)

where, then

(2.40)

Substituting (2.35), (2.37), (2.40) into (2.34), we receive

(2.41)

Because of, we get

(2.42)

Taking, then

(2.43)

where, by Gronwall inequality, we have

(2.44)

Let, so we get

(2.45)

Then

(2.46)

So, there exists and, such that

(2.47)

■

3. Global Attractor

3.1. The Existence and Uniqueness of Solution

Theorem 3.1. Assume that,

where, and is the first eigenvalue of the, and when,

where as; as; as.

, , and.

Then the problem (1.1)-(1.3) exists a unique smooth solution

Remark 2. We denote the solution in Theorem 3.1 by. Then composes a continuous semigroup in.

Proof of Theorem 3.1.

Proof. By the Galerkin method and Lemma 1, we can easily obtain the existence of Solutions. Next, we prove the uniqueness of Solutions in detail. Assume are two solutions of (1.1)-(1.3), let, then and the two equations subtract and obtain

(3.1)

Taking H-inner product by in (3.1), we get

(3.2)

By (H_{1}), (H_{2})

(3.3)

(3.4)

where.

By using Gagliardo-Nirenberg inequality, and according the Lemma 1,we can get

Then, we have

(3.5)

Substituting (3.3), (3.5) into (3.2)

(3.6)

Taking

Then

(3.7)

By using Gronwall inequality, we obtain

(3.8)

So, we can get because of.

That shows that

That is

Therefore

We get the uniqueness of the solution. So the proof of the Theorem 3.1. has been completed. ■

3.2. Global Attractor

Theorem 3.2. [10] Let X be a Banach space, and are the semigroup operator on X., here I is a unit operator. Set satisfy the follow conditions.

1) is bounded, namely, it exists a constant, so that

2) It exists a bounded absorbing set, namely, , it exists a constant, so that

here and B are bounded sets.

3) When, is a completely continuous operator.

Therefore, the semigroup operators S(t) exist a compact global attractor A.

Theorem 3.3 Under the assume of Theorem 3.1, equations have global attractor

where, B 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, here,.

(1) From Lemma 1-Lemma 2, we can ge 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) exist a compact global attractor A. Theorem 3.3 is proved. ■

Acknowledgements

The authors express their sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. These contributions greatly improved the paper.

Funding

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11161057.

Cite this paper

Lv, P. , Lou, R. and Lin, G. (2016) Global Attractor for a Class of Nonlinear Generalized Kirchhoff-Boussinesq Model.*International Journal of Modern Nonlinear Theory and Application*, **5**, 82-92. doi: 10.4236/ijmnta.2016.51009.

Lv, P. , Lou, R. and Lin, G. (2016) Global Attractor for a Class of Nonlinear Generalized Kirchhoff-Boussinesq Model.

References

[1] Chueshov, I. and Lasiecka, I. (2006) Existence, Uniqueness of Weak Solutions and Global Attractors for a Class of Nonlinear 2D Kirchhoff-Boussinesq Models. AIM Journals, 15, 777-809.

[2] Yang, Z. and Jin, B. (2009) Global Attractor for a Class of Kirchhoff Models. Journal of Mathematical Physics, 50, Article ID: 032701.

[3] Yang, Z., Feng, N. and Ma, T.F. (2015) Global Attractor for the Generalized Double Dispersion Equation. Nonlinear Analysis, 115, 103-116.

http://dx.doi.org/10.1016/j.na.2014.12.006

[4] Ma, T.F. and Pelicer, M.L. (2013) Attractors for Weakly Damped Beam Equations with p-Laplacian. Discrete and Continuous Dynamical Systems. Supplement, 525-534.

[5] Yang, Z. and Liu, Z. (2015) Exponential Attractor for the Kirchhoff Equations with Strong Nonlinear Damping and Supercritial Nonlinearity. Applied Mathematics Letters, 46, 127-132.

http://dx.doi.org/10.1016/j.aml.2015.02.019

[6] Kloeden, P.E. and Simsen, J. (2015) Attractors of Asymptotically Autonomous Quasi-Linear Parabolic Equation with Spatially Variable Exponents. Journal of Mathematical Analysis and Applications, 425, 911-918.

http://dx.doi.org/10.1016/j.jmaa.2014.12.069

[7] Silva, M.A.J. and Ma, T.F. (2013) Long-Time Dynamics for a Class of Kirchhoff Models with Memory. Journal of Mathematical Physics, 54, Article ID: 021505.

[8] Lin, G.G., Xia, F.F. and Xu, G.G. (2013) The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays. International Journal of Modern Nonlinear Theory and Application, 2, 209-218.

http://dx.doi.org/10.4236/ijmnta.2013.24029

[9] Guo, L., Yuan, Z.Q. and Lin, G.G. (2014) The Global Attractors for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Linear Damping and Source Terms. International Journal of Modern Nonlinear Theory and Application, 4, 142-152.

http://dx.doi.org/10.4236/ijmnta.2015.42010

[10] Lin, G.G. (2011) Nonlinear Evolution Equation. Yunnan University Press.

[1] Chueshov, I. and Lasiecka, I. (2006) Existence, Uniqueness of Weak Solutions and Global Attractors for a Class of Nonlinear 2D Kirchhoff-Boussinesq Models. AIM Journals, 15, 777-809.

[2] Yang, Z. and Jin, B. (2009) Global Attractor for a Class of Kirchhoff Models. Journal of Mathematical Physics, 50, Article ID: 032701.

[3] Yang, Z., Feng, N. and Ma, T.F. (2015) Global Attractor for the Generalized Double Dispersion Equation. Nonlinear Analysis, 115, 103-116.

http://dx.doi.org/10.1016/j.na.2014.12.006

[4] Ma, T.F. and Pelicer, M.L. (2013) Attractors for Weakly Damped Beam Equations with p-Laplacian. Discrete and Continuous Dynamical Systems. Supplement, 525-534.

[5] Yang, Z. and Liu, Z. (2015) Exponential Attractor for the Kirchhoff Equations with Strong Nonlinear Damping and Supercritial Nonlinearity. Applied Mathematics Letters, 46, 127-132.

http://dx.doi.org/10.1016/j.aml.2015.02.019

[6] Kloeden, P.E. and Simsen, J. (2015) Attractors of Asymptotically Autonomous Quasi-Linear Parabolic Equation with Spatially Variable Exponents. Journal of Mathematical Analysis and Applications, 425, 911-918.

http://dx.doi.org/10.1016/j.jmaa.2014.12.069

[7] Silva, M.A.J. and Ma, T.F. (2013) Long-Time Dynamics for a Class of Kirchhoff Models with Memory. Journal of Mathematical Physics, 54, Article ID: 021505.

[8] Lin, G.G., Xia, F.F. and Xu, G.G. (2013) The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays. International Journal of Modern Nonlinear Theory and Application, 2, 209-218.

http://dx.doi.org/10.4236/ijmnta.2013.24029

[9] Guo, L., Yuan, Z.Q. and Lin, G.G. (2014) The Global Attractors for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Linear Damping and Source Terms. International Journal of Modern Nonlinear Theory and Application, 4, 142-152.

http://dx.doi.org/10.4236/ijmnta.2015.42010

[10] Lin, G.G. (2011) Nonlinear Evolution Equation. Yunnan University Press.