IJMNTAInternational Journal of Modern Nonlinear Theory and Application2167-9479Scientific Research Publishing10.4236/ijmnta.2012.13014IJMNTA-23094ArticlesEngineering Physics&Mathematics Boundary Stabilization of a More General Kirchhoff-Type Beam Equation ianwenZhang1*DanxiaWang1*Department of Mathematics, Taiyuan University of Technology, Taiyuan, China* E-mail:jianwen.z2008@163.com(IZ);danxia.wang@163.com(DW);28092012010397101July 12, 2012August 12, 2012 September 12, 2012© 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/

Simultaneously, considering the viscous effect of material, damping of medium, geometrical nonlinearity, physical nonlinearity, we set up a more general equation of beam subjected to axial force and external load. We prove the existence and uniqueness of global solutions under non-linear boundary conditions which the model is added one damping mechanism at l end. What is more, we also prove the exponential decay property of the energy of above mentioned system.

Kirchhoff-Type Beam; Non-Linear Boundary; Global Solutions; Exponential Decay
1. Introduction

The problem is based on the equation which was proposed by Woinowsky-krieger , as a model for vibrating beams with hinged ends. One of the first mathematical analysis for the equation was done by Ball , which was later extended to an abstract setting by defining a linear operator A by Medeiros . In , Tucsnak considered the above beam equation which clamped boundary and obtained the exponential decay of the energy when a damping of the type a(x)ut is effective near the boundary. In the same direction, Kouemon Patchen  obtained the exponential decay of the energy for above-equation when a nonlinear damping g(ut) was effective in Ω. To  considered the above kirchhoff-type beam equation under non-linear boundary conditions  which the model is clamped at x = 0 and is supported x = l. He proved the existence and decay rates of the solutions. A rather general kirchhoff-type beam equation was set up by Ball , who presented the existence and uniqueness of solution under linear boundary conditions. However the global solution and exponential decay for the more general beam equation is open under nonlinear boundary conditions. In the present work, we are concerned with the existence and uniqueness of solutions and the exponential decay property of energy on the nonlinear beam equation with external load

with nonlinear boundary conditions

and initial conditions and (4)

2. Definition and Assumptions

In this paper, our analysis is based on the Sobolev spaces , espectively equipped with the norm and . We assume that f, g:R → R are continuously differentiable functions such that and (5)

where and and (6)

for some ρ > 0.

Assume that the functions are non-negative functions and respectively satisfy and (7) and (8)

3. Existence and Uniqueness of Global Solutions

Now we come to the following conclusions of the existence and uniqueness of global solutions.

Theorem 1. Assume that the assumptions of (5)-(8) and hold. Then for any satisfying the compatibility condition

There exists a function u satisfying (1)-(4) such that Proof. Let us solve the variational problem associated with (1)-(4), which is given by: find such that

for all . Let be a complete orthogonal system of W. For each , let us put .

We search for a function where is a unknown function such that for any , and it satisfies the approximating equation

with the initial conditions and (12)

Thus (11) and (12) are equivalent to the Cauchy problem of ODES in the variable t, which is known to have a local solution um(t) in an interval [0, tm) (tm < T) for any given T > 0.

Estimate 1. By integration of (11) over [0, t] (t < tm) with , we see that where and .

Considering that , and the initial conditions, we get . Using Gronwall inequality, we have . Then there exists a constant M1 depending only on T such that

for any and for all .

In this paper, C is a constant independent of m, t and denotes different value in different mathematical expression.

Estimate 2. Integrating by parts (11) with and t = 0, and considering the compatibility condition (3) we get Thus there exists a positive constant M2 such that

Estimate 3. Let us fix t, ξ > 0 such that ξ < T – t. Taking the difference of (11) with t = t + ξ and t = t, and replacing ω by , we get where .

Let us estimate . Since we have

Noting that ΔM1 = M(z(t + ξ) – M(z(t)) and ΔM2 = N(zt(t + ξ)) – N(zt(t)), then integrating by parts we have Since , by the Mean value theorem, from estimates 1 and (16) we have where η1 is between and .

By the Mean value theorem, we also have Considering that M(z(t + ξ) ≤ C and N(z(t + ξ) ≤ C, we conclude that there exists constants k1 > 0 and k2 > 0 such that

A argument for f yields

where k3 > 0 is a constant. Putting and taking into account of (17)-(18) and the assumptions of g, we deduce from (15) that

where k4 = max{k1 + k3, k2}. Therefore

Dividing the above inequality by ξ2 and letting ξ → 0 gives .

From estimate 2 we find a constant M3 > 0 such that .

With the estimates 1 - 3 we can use Lions-Aubin Lemma to get the necessary compactness in order to pass (11) to the limit. Then it is a matter of routine to conclude the existence of the global solution in [0, T].

Theorem 2. The solution u(t) of theorem 1 is unique.

Proof. Let u, v be two solutions of (1)-(4) with the same initial data. Then writing p = u – v, putting ω = pt in (10) and using mean value theorem, chauchy-schwarz inequality and Gronwall inequality, we may get p = 0. Thus u = v.

4. The Exponential DECAY of the Energy of System

In order to establish our decay result, we define the energy of the system by where . We have Theorem 3. Let u(t) be the solution given by theorem 1 as q(x, t) = 0 and g = 0. And assume that N(s)s ≥ 0 and f(s)s ≥ 0. Then there exist constants λ2, λ4 > 0 and λ3 < 0 such that .

To prove Theorem 3, we firstly introduce two lemmas.

Let us define .

Then we have the following lemmas.

Lemma 1. Let Eε(t) = μE(t) + εψ(t). Then there exists a constant k5 > 0 such that .

Proof. By , and there exists k5 > 0 such that where .

Lemma 2. There exist constants λ0 > 0 and λ1 such that .

Proof. Taking the inner product of (1) with ut and considering that N(s)s ≥ 0, we have .

Taking the inner product of (1) with u, we have  Thus .

Set .

Since , we have . Therefore .

From the Mean value theorem, there exists a constant λ1 such that On writing , we have The proof of theorem 3. From lemma 1, we have

From Lemma 2, we have

Therefore .

By Gronwall inequality and combing (21), we have .

Hence, for sufficiently small ε > 0 .

On writing and we have .

The proof of theorem 3 is now completed.

REFERENCESNOTESReferencesS. Woinowsky-Krieger, “The Effect of Axial Force on the Vibration of Hinged Bars,” Journal of applied Mechanics, Vol. 17, 1950, pp. 35-36.J. M. Ball, “Initial-Boundary Value Problems for an Extensible Beam,” Journal of Mathematical Analysis and Applications, Vol. 42, No. 1, 1973, pp. 61-90. doi:10.1016/0022-247X(73)90121-2L. A. Mederios, “On a New Class of Nonlinear Wave Equations,” Journal of Mathematical Analysis and Applications, Vol. 69, No. 1, 1979, pp. 252-262.M. Tucsnak, “Semi-Internal Stabilization for a Nonlinear Euler-Bernoulli Equation,” Mathematical Methods in the Applied Sciences, Vol. 19, No. 11, 1996, pp. 897-907. doi:10.1002/(SICI)1099-1476(19960725)19:11<897::AID-MMA801>3.0.CO;2-#S. Kouemou Patcheu, “On a Global Solution and Asymptotic Behavior for the Generalized Damped Extensible Beam Equation,” Journal of Differential Equations, Vol. 135, No. 2, 1997, pp. 299-314. doi:10.1006/jdeq.1996.3231F. M. To, “Boundary Stabilization for a Non-Linear Beam on Elastic Bearings,” Mathematical Methods in the Applied Sciences, Vol. 24, No. 8, 2001, pp. 583-594. doi:10.1002/mma.230J. M. Ball, “Stability Theory for an Extensible Beam,” Journal of Differential Equations, Vol. 14, No. 3, 1973, pp. 61-90. doi:10.1016/0022-0396(73)90056-9