On Uniform Decay of Solutions for Extensible Beam Equation with Strong Damping

FENG Bao-wei,ZHANG Ming,LIANG Tie-wang,LI Hai-yan

(1.College of Economic Mathematics,Southwestern University of Finance and Economics,Chengdu 611130,China;2.College of Information Science and Technology,Donghua University,Shanghai201620, China;3.Department of Inforwation and Arts Design,Henan Forestry Vocational College,Luoyang 471002,China)

On Uniform Decay of Solutions for Extensible Beam Equation with Strong Damping

FENG Bao-wei1,ZHANG Ming2,LIANG Tie-wang3,LI Hai-yan2

(1.College of Economic Mathematics,Southwestern University of Finance and Economics,Chengdu 611130,China;2.College of Information Science and Technology,Donghua University,Shanghai201620, China;3.Department of Inforwation and Arts Design,Henan Forestry Vocational College,Luoyang 471002,China)

This paper investigates the existence and uniform decay of global solutions to the initial and boundary value problem with clamped boundary conditions for a nonlinear beam equation with a strong damping.

extensible beam equation;global existence;uniqueness;uniform decay

§1. Introduction

In this paper we study a nonlinear extensible beam equation with a strong damping

whereαandγare positive constants,Ω⊆ℝn(n≥1)is a bounded domain with smooth boundary∂Ω,νis the unit outward normalon∂Ω.Here M(·)is a function.

In recent years,many mathematicians have paid their attention to the global wellposedness and energy decay for nonlinear wave equations,hyperbolic systems and viscoelastic equations. Firstly,we recall some results concerning extensible beam equations.In 1950,Woinowsky-Krieger[1]proposed the one-dimensional nonlinear equation of vibration of beams,which is given by

where L is the length of the beam andα,β,γare positive physical constants.The nonlinear part of(1.4)represents for the extensible eff ect for the beam whose ends are held a fixed distant apart in its transverse vibrations.In 1970s,mathematical analysis for global existence and asymptotic behavior of these extensible beams have been studied by Ball[23],Dickey[4], Medeiros[5]and the references therein.

The general form of(1.4)reads

where M(·)is a function satisfying some conditions.When functions f,g,h=0 in(1.5),many researches have been devoted to the study ofextensible beam equations,see,e.g.,Brito[6],Ma[7], Oliveira and Lima[8],Rivera[910],etc.Patcheu[11]studied the existence and decay property of global solutions to the Cauchy problem of(1.5)at the abstract level with f=0.Tusnack[12]investigated(1.5)with clamped boundary conditions and f=0 and obtained the exponential decay of the energy when a damping of type a(x)utis effective near the boundary.Cavalcanti et al[13]investigated the global existence and asymptotic behavior of the initial and boundary value problem of(1.5)with clamped boundary conditions.They established the globalexistence of solutions by the fixed point theorem and continuity arguments.They proved that the IBVP possesses a unique globalweak solution u,with(u,ut)∈C(ℝ+;H20×L2),provided that M∈C1,

with 1＜ ρ,r≤ n/n−2 if n≥ 3;ρ,r＞ 1 if n=1,2.Under the basic research in[13], Ma and Narciso[14]subsequently studied the existence of global attractor to the IBVP of(1.5), with clamped boundary conditions.Recently,Yang[15]studied the global existence,stability and long time dynamics of solutions to the IBVP of(1.5).The author proved the IBVP is globally well-posed provided that either the growth exponent p of the source term f(u)is nonsupercritical or p is supercritical but is dominated by the growth exponent q of the nonlinear damping g(ut).Moreover,the author also investigated the existence offinite-dimensionalglobal attractors and exponential attractors.For the related work,one can refer to[16-19]and the references therein.

To our best knowledge,the global existence for extensible beam equations with a strong damping were not previously considered.So,the objective of this work is to establish the global existence and uniqueness of initial boundary value problem(1.1)～(1.3)with clamped boundary conditions.Moreover,we discuss the uniform decay of energy.

The outline of this paper is as follows.In Section 2,we give some preparations for our consideration and our main result.The statement and the proof of our main result will be given in Section 3.

§2.Preliminaries and Main Result

In this section,we give some preparations for our consideration and our main result.

We define

We assume M(·):ℝ+→ℝ+is a C2function satisfying

(if M(z)is monotone nondecreasing).

The energy of problem(1.1)～(1.3)is given by

For convenience,we denote the norm and scalar product in L2(Ω)by‖·‖and(·,·),respectively.C1denotes a general positive constant,which may be different in different estimates.

Now we are in a position to state our main result.

Theorem 1 Let(2.3)hold.Ifthe initialcondition(u0,u1)∈H,then problem(1.1)～(1.3) admits a unique strong solution to(u(t),ut(t))∈H,such that for any t＞0,

Moreover there exist positive constants C2and C3such that the energy E(t)of problem (1.1)～(1.3)satisfies

§3.Proof of Main Result

In this section,we are going to complete the proof of Theorem 1.Firstly,we show the global existence and uniqueness of solutions for problem(1.1)～(1.3).

Using the similar method as in[13],we can establish the global existence of solution (u(t),ut(t))∈H with a minimalmodification,we omit the detailed proofand(2.5)can also beobtained.Next we prove the uniqueness of solutions.

Let(u(t),ut(t))and(v(t),vt(t))be two global solutions of problem(1.1)～(1.3)with respect to initial data(u0,u1)and(v0,v1)respectively.Letω(t)=u(t)−v(t).Thenω(t)verifies

Multiplying(3.1)by utand integrating the result overΩwith respect to x,we have

The direct computation gives

It follows from(2.5)that

Noting that the interpolation inequality

we can get

whereξ1=θ‖▽u‖2+(1−θ)‖▽v‖2andθ∈(0,1).Then we see that

Differentiating(3.1)with respect to t,multiplying the result byωttand integrating by parts overΩ,we arrive at

It is easy to see that

By H¨older’s inequality,the interpolation inequality and(2.5),we conclude

whereξ2=(1−θ1)‖▽u‖2+θ1‖▽v‖2andθ1∈(0,1).

Similarly,

withξ3=(1−θ2)‖▽u‖2+θ2‖▽v‖2andθ2∈(0,1).

Combining(3.6)～(3.11),we obtain

Now we define

Then we can easily know that G(t)is equivalent to

It follows from(3.12)～(3.13)that

which,along with(3.13),gives

From(3.1),we can derive

which,combined with(3.14),yields

This shows that the solution of problem(1.1)～(1.3)depend continuously on the initial data.

In what follows,we shall show the uniform decay of energy defined in(2.4).

Multiplying(1.1)by utand integrating the resulting equation overΩ,we get

Let us define the functional

For everyε＞0,we define the perturbed modified energy by

Firstly,we claim that forε＞0 suffi ciently small

Indeed,by Young’s inequality and Poincar´e’s inequality,we get

Then we have

which,takingε＞0 smallenough,gives(3.18).

On the other hand,by(2.4),we have

Using(1.1),integration by parts and Young’s inequality,we see that for anyδ＞0,

which,noting that‖Δu(t)‖2≤C1E(t)and takingδ＞0 small enough,implies

It follows from(3.15),(3.17)～(3.18)and(3.21)that

Takingε＞0 so smallthatγ−C1ε≥0 and using(3.18),we shallsee below,

and

Integrating(3.23)and using inequality(3.18),we obtain

This concludes the proof of Theorem 1.

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tion:35L25,35L57

1002–0462(2014)01–0151–08

Chin.Quart.J.of Math. 2014,29(1):151—158

date:2013-06-15

Supported by the NNSF of China(11271066,11326158);Supported by the grant of Shanghai Education Commission(13ZZ048);Supported by the Doctoral Innovational Fund of Donghua University(BC201138)

Biography:FENG Bao-wei(1985-),male,native of Shangqiu,Henan,a Ph.D.candidate of Donghua University,engages in nonlinear evolution equations and infinite-dimensional dynamical systems.

CLC number:O175.29 Document code:A