**International Journal of Modern Nonlinear Theory and Application** Vol.1 No.3(2012), Article ID:23094,5 pages DOI:10.4236/ijmnta.2012.13014

Boundary Stabilization of a More General Kirchhoff-Type Beam Equation^{*}

Department of Mathematics, Taiyuan University of Technology, Taiyuan, China

Email: jianwen.z2008@163.com, ^{#}danxia.wang@163.com

Received July 12, 2012; revised August 12, 2012; accepted September 12, 2012

**Keywords:** Kirchhoff-Type Beam; Non-Linear Boundary; Global Solutions; Exponential Decay

ABSTRACT

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.

1. Introduction

The problem is based on the equation

which was proposed by Woinowsky-krieger [1], as a model for vibrating beams with hinged ends. One of the first mathematical analysis for the equation

was done by Ball [2], which was later extended to an abstract setting by defining a linear operator A by Medeiros [3]. In [4], 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)u_{t} is effective near the boundary. In the same direction, Kouemon Patchen [5] obtained the exponential decay of the energy for above-equation when a nonlinear damping g(u_{t}) was effective in Ω. To [6] 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 [7], 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

(1)

with nonlinear boundary conditions

(2)

(3)

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

(9)

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

(10)

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

(11)

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 u^{m}(t) in an interval [0, t_{m}) (t_{m} < T) for any given T > 0.

**Estimate 1.** By integration of (11) over [0, t] (t < t_{m}) with, we see that

where and.

Considering that

,

and the initial conditions, we get

.

Using Gronwall inequality, we have. Then there exists a constant M_{1} depending only on T such that

. (13)

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 M_{2} such that

. (14)

**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

(16)

Noting that ΔM_{1} = M(z(t + ξ) – M(z(t)) and ΔM_{2} = N(z_{t}(t + ξ)) – N(z_{t}(t)), then integrating by parts we have

Since, by the Mean value theorem, from estimates 1 and (16) we have

where η_{1} 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 k_{1} > 0 and k_{2} > 0 such that

(17)

A argument for f yields

(18)

where k_{3} > 0 is a constant. Putting

and taking into account of (17)-(18) and the assumptions of g, we deduce from (15) that

(19)

where k_{4} = max{k_{1} + k_{3}, k_{2}}. Therefore

(20)

Dividing the above inequality by ξ^{2} and letting ξ → 0 gives

.

From estimate 2 we find a constant M_{3} > 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 = p_{t} 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 λ_{2}, λ_{4} > 0 and λ_{3} < 0 such that.

To prove Theorem 3, we firstly introduce two lemmas.

Let us define .

Then we have the following lemmas.

**Lemma 1.** Let E_{ε}(t) = μE(t) + εψ(t). Then there exists a constant k_{5} > 0 such that

.

Proof. By, and there exists k_{5} > 0 such that

where.

**Lemma 2.** There exist constants λ_{0} > 0 and λ_{1} such that

.

**Proof.** Taking the inner product of (1) with u_{t} 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 λ_{1} such that

On writing, we have

**The proof of theorem 3.** From lemma 1, we have

. (21)

From Lemma 2, we have

. (22)

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.

REFERENCES

- S. 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)90121-2
- L. 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.1996.3231
- F. 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.230
- J. 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

NOTES

^{*}This work is supported by Natural Science Foundation of Shanxi Provice, and the Natural Science Foundation for Young Scientists of Shanxi Province (2010011008, 2011021002-2).

^{#}Corresponding author.