Boundary Stabilization of a More General Kirchhoff Type Beam Equation

(1)

Boundary Stabilization of a More General

Kirchhoff-Type Beam Equation

*

Jianwen Zhang, Danxia Wang#

Department of Mathematics,Taiyuan University of Technology,Taiyuan, China Email: [email protected], #_{[email protected]} Received July 12, 2012; revised August 12, 2012; accepted September 12,2012

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 exis-tence 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 sys-tem.

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

1. Introduction

The problem is based on the equation

 



2



0 , d

l

tt xxxx x xx

u u   

_

u s t s u 0

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



2



0 d 0

l

tt xxxx x xx

u u M

_

u x u 

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 ex-ponential decay of the energy when a damping of the type a(x)ut 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(ut) was effective in Ω. To [6] considered the above kirchhoff-type beam equation under non-linear boundary conditions

 

0, _x

 

0, _xx

 

, 0

u t u t u l t 

 

2

 



 





 



0

, l , ,

xxx x x t

u l t M

_

u u l t  f u l t g u l t,

0

which the model is clamped at x = 0 and is supported x =

l. He proved the existence and decay rates of the solu-tions. A rather general kirchhoff-type beam equation



2



0 d 0 d )

tt xxxx xxxxt

l l

x x xt xx t

u u u

k u x u u x u u

 

  

 

 

_



_

 

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 con-cerned with the existence and uniqueness of solutions and the exponential decay property of energy on the nonlinear beam equation with external load

 







2



 

0 d ,

tt xxxx xxxxt t l

x xt x xx

u u u u

M u N u u x u q x

 

  

 

_

 t (1)

with nonlinear boundary conditions

 

0, xx

 

0, x

 

, 0

u t u t u l t  (2)

 

,

 

,



 

,





 



xxx xxxt t

u l t u l t  f u l t g u l t, (3) and initial conditions

 

_,0 0

 

u x u x and

 

_,0 1

 

t

u x u x (4)

2. Definition and Assumptions

*_{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.}

In this paper, our analysis is based on the Sobolev spaces

   

 



2 _0, ₀ ₀



x

(2)

   



4 _0, ₀ ₀



xx

W  u V H l u 

espectively equipped with the norm u V  uxx and xx xxxx

W . We assume that f, g:R → R are continuously differentiable functions such that

u  u  u

 

0

f s s

and f s s

 

2f sˆ

 

  0, s R (5) where

 

0

ˆ s _d

f s 

_

f z z and

 

0 0

g 

and



g r

 

g s

 





r s 



 r s 2,r s R, 

,

(6) for some ρ > 0.

Assume that the functions are non-negative functions and respectively satisfy

   





1

, 0

M  N  C 

M

 

0 0

and

 

0

ˆ l _d

M s 

_

M z z (7) N

 

0 0

and N s s

 

0 (8)

3. Existence and Uniqueness of Global

Solutions

Now we come to the following conclusions of the exis-tence and uniqueness of global solutions.

Theorem 1. Assume that the assumptions of (5)-(8)

and hold. Then for any

satisfying the compatibility condition

 

_, 2





_0,

  

_; 2 _0, q x t L  L l

0_, 1__W



u u

 



 





 

0 1 0 1

xxx xxx

u l u l  f u l g u l



(9)

There exists a function u satisfying (1)-(4) such that













 



2 _{0, ;} 0 _0, _; 2, _{0, ;} 2 _0,

u L_ _W _C _ V _W  _ L l

Proof. Let us solve the variational problem associated

with (1)-(4), which is given by: find u t

 

W such that

 





 







 



 

0 0 0 0

2

0 0

0

d d d

d d

+ , ,

, d

l l

d l

tt xx xx xxt xx t

l l

x x xt x x

t l

u x u x u x u x

M u N u u x u x

f u l t g u l t l

q x t x

l

     



  

 

_  _

 

  

 





(10)

for all  V. Let

 

_j _{be a complete orthogonal} sys-tem of W. For each mN, let us put



1 2



span , ,

m m

W    ， . We search for a function

 

1 m

m j

j

u t a t _j





where _{a t}j

 

m_{, and it satisfies the approximating equation} is a unknown function such that for any W

 

 





 







 



 

0 0 0 0

2

0 0

0

d d d

d d

, ,

, d

tt xx xxt t

x x xt x

t

l _m l _m l _m l _m

xx xx

l l

m m m m

x

m m

l

u x u x u x u x

M u N u u x u x

f u l t g u l t l

q x t x

d

     



  

 

_  _

 

 

_  _





(11)

with the initial conditions

 

₀ 0

m

u u and m

 

₀ 1 (12) t

u u

Thus (11) and (12) are equivalent to the Cauchy prob-lem of ODES in the variable t, which is known to have a local solution um₍_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 < tm) with m

 

t u t

  , we see that

 





 





 



 



 





 



 



 





 

2 2 2

0

0 0

2 2

0 0

ˆ

+ 2 d

2 d 2 , , d

ˆ

2 ,

ˆ

0 0 0

ˆ

2 , 0 2 , d

t

m m m

t xx xxt

t t _m _m

t t t t

m

m m

t xx

t l

m m

t

u u u s M z t

N z s z s s g u l s u l s s

f u l t

u u M z

f u l q x s u s



 



  

 



 

where

 

m 2 x

z t  u and

 

1 d 2 2 d

m

t x

z t u

t

 .

Considering that

 





 

2

, , ,

m m m

t t t

g u l s u l s  u l s 0,

 



t



t

 

0

N z s z s  and the initial conditions, we get

 

2 2 2

0 d 0 ,

t t

m m

t t

u 

_

u s s C 

_

q x s ds. Using Gronwall inequality, we have m 2 T

t

u Ce . Then there exists a constant M1 depending only on T such that

2 2

1

m m

t xx

u  u M . (13) for any t

 

0,T and for all m N .

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

(3)

 







 



 

   

 





 





 



 



2

0

0 0

|| 0 ||

0 0 0 (0)d

0 0 d 0 0 d

0 0 d ,0 0 d

0 0 0 0

0 0 0

m tt

l _m _m

t xx tt

l _m _m l _m _m

xxxxt tt xxxx tt

l _m _m l _m

t tt tt

m m ₀

tt t xx xxxx

m m

xxxxt t

u

M z N z u u x

u u x u u x

u u x q x u x

u M z N z u u

u u q

 

 

_  _

 

 

 

 _  _ 

  



ing the difference of (11) with t = t + ξand t = t, and re-placing ω by m







t t

u t um t



_{, we get}

m





 





 





 





 











 





















 













  









 



2 2

1

0

1 d ₊

2 d

, , ,

, ( , ,

d

m m m m

t t xx xx

m m m m

xxt xxt t t

m m m

t t

m m m

t t t

l _m _m

t t

u t u t u t u t

t

u t u t u t u t



,

m m t

f u l t f u l t u l t u l t

g u l t g u l t u l t u l t

I

q t q t u t u t x

 

   

 

 _ _ _ _ 

 

 

     

 

_   _  

 

_   _  





_

   

Thus there exists a positive constant M2 such that

 

0 2,

m tt

u M  m N . (14)

Estimate 3. Let us fix t, ξ > 0 such that ξ < T – t. Tak- where



















 





 



 









 



1 ₀ d

l _m _m _m

t x t x xt xt

m

I 

_

_M z t N z t _u t _M z t N z t _u t u t u t x.

where k3 > 0 is a constant. Putting Let us estimate I₁ . Since

 

0,

 

,

 

0, 0

m m m

x xx

u t u l t u t  ,

_{ }

_

_

_{ }

2

_

_

_{ }

2

, m m m m

m t uxx t uxx t u tt u tt

       

we have

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

, ,

m m m m m m

x x xx x

u l u u l u u l u

    xx (16)

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

 

4

 



d _, _, _,

d_tm t km t   t 0,T



e

(19)

















 









 







 









1

0

1 2 ₀

d d t

l _m _m _m _m

xx xx t t

l _m _m _m

xx t t

I M z t N z t

u t u t u t u t x

M M u t u t u t x

 



 

 _    _

    

     



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

 

,

 

0, k T4

m t m

    ₍₂₀₎ Dividing the above inequality by ξ2_{and letting}_ξ _→₀ gives

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

 





1 _0, M  C 

 

2 2



 

₀ 2

 

₀ 2



k T4

m m m m

xxt tt xxt tt

u t  u  u  u e _.

 





 







 

2 2

'

1 1 mx mx

m m

xx xx

M M u t u t

C u t u t

 



   

  

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

 

2 2

 

3, , 0

m m

xxt tt

u t  u M  m N  t ,T _. where η1 is between





2 m

x

u t and m

 

2 x

u t .

By the Mean value theorem, we also have





 





 

2 mxx mxx tm tm

M C u t  u t C u t  u t

      

With the estimates 1 - 3 we can use Lions-Aubin Lem- ma 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.

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





 

2





 

1 1 2

m m m m

xx xx t t

2

I k u t u t k u t u t

(17)

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.

A argument for f yields











 







 





 

2





 

3

, , ,

, ,

m m m

t t t

m m m m

xx xx t t

2 , m t

f u l t f u l t u l t u l t

k u t u t u l t u l t



  

   

     



(18)

4. The Exponential DECAY of the Energy of

System

(4)

 







 



2 2 2 1 1 2 2

1 _ˆ _ˆ _,

2

t xx

x

E t u t u t

M u t f u l t

 

 

where

 

0

ˆ s _d

M s 

_

M z z. 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

 

3 3

2 0 4

t t

E t  E e te .

To prove Theorem 3, we firstly introduce two lemmas. Let us define

 

2

0 d ₂ 0 d

l l

t

t uu x  u

 

_



_

x.

Then we have the following lemmas.

Lemma 1. Let Eε(t) = μE(t) + εψ(t). Then there exists a

constant k5 > 0 such that

 

5

 

, 0

E t_ E t k E t   .

Proof. By u   l ux , ux   l uxx and

xx

u l u there exists k5 > 0 such that

 

2 0 0 2 2 2 2 5 d d 2

2 2 2

l l

t

xx t xx

E t E t uu x u x

l _u _u l _u

k E t

              



2

where





2.

5 1

k   l

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

 

0

 

1

d

2

dtE t    E t  .

Proof. Taking the inner product of (1) with ut and

considering that N(s)s≥ 0, we have

 

2 2

d

dtE t   uxxt  ut .

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

 









 





 

1 2 2

0 0 0

2 ₂

0 0

d _d _d _d

d

, ,

l l

t xx xxt xx

l l

x x xt

t u x u x u u x

t

x

M u N u u x u

f u l t u l t

   

 





dx

Thus

 









 





 

2 2

2 2 2

0 0

2 2`

0

2 2 2

0 0 0

d

d d

d

d ( d )

, ,

l l

xxt xx xxt xx

l

xx t

l l l

x x x xt x

E t u u x u u dx

t

u x u

Set

 

2 2 2 2

0 d 0 d

l l

xxt xxt xx xxt xx

h u   u 

_

u x

_

u u x. Since

   

2 2 ₄ 2 2 2 ₃

 

2 2

xx xx xx

u u u

    0

      , we

have h u

 

xxt 0. Therefore

 









 





 





 

2 ₂ 2` 0

2 2 2

0 0 0

d _d d d d , , l xx t

l l l

x x x xt x

E t u x u

t

d

M u u x N u u x u

f u l t u l t

                  



x.

From the Mean value theorem, there exists a constant

λ1 such that





2

1

0 d 0 d

l l

x xt x

N u u x u x .

 



_



On writing



2



0 min , , 2

        , we have

 

0

 

1

d

2

dtE t    E t 

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



  k E t5

  

E t

 





  k E t5

  

. (21)

From Lemma 2, we have

 

0

 

d

2

dtE t    E t 1. (22)

Therefore

 

0

 

1

5

d 1

2

dtE t   k E t 

 

  _ _ 



  .

By Gronwall inequality and combing (21), we have



  

 



  

0 5 0 5 1 2 5 1 1 2 5 1 0 0 t k t k

k E t E t e

k E t e

                   _          _     _  _    _   _ .

Hence, for sufficiently small ε > 0

 

0 0

5 5

1 1

2 2

5 1

5 5

0 k t kt

k

E t E e e

k k              _  _     _   _           _.

On writing 5 0

2 3 5 5 2 , 0 k k k                

and 1 4 5 k       , d

M u u x N u u x u

f u l t u l t

                     



we have

 

3 3

2 0 4

t t

E t  E e te .

The proof of theorem 3 is now completed. x

.

REFERENCES

(5)

Vibration of Hinged Bars,” Journal of Applied Mechanics, Vol. 17, 1950, pp. 35-36.

[2] J. M. Ball, “Initial-Boundary Value Problems for an Ex-tensible Beam,” Journal of Mathematical Analysis and Applications, Vol. 42, No. 1, 1973, pp. 61-90.

doi:10.1016/0022-247X(73)90121-2

[3] L. A. Mederios, “On a New Class of Nonlinear Wave Equations,” Journal of Mathematical Analysis and Ap-plications, Vol. 69, No. 1, 1979, pp. 252-262.

[4] 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-#

[5] S. Kouemou Patcheu, “On a Global Solution and As-ymptotic Behavior for the Generalized Damped Extensi-ble Beam Equation,” Journal of Differential Equations, Vol. 135, No. 2, 1997, pp. 299-314.

doi:10.1006/jdeq.1996.3231

[6] 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

[7] J. M. Ball, “Stability Theory for an Extensible Beam,”