URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLOCAL APPROACH TO PROBLEMS ON LONGITUDINAL VIBRATION IN A SHORT BAR LUDMILA S

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

NONLOCAL APPROACH TO PROBLEMS ON LONGITUDINAL VIBRATION IN A SHORT BAR

LUDMILA S. PULKINA, ALEXANDER B. BEYLIN

Abstract. In this article, we consider a problem with dynamic nonlocal con- ditions for a forth-order PDE with dominating mixed derivative. This problem is closely related to vibration problems, in particular, to longitudinal vibra- tion in a short bar. The existence and uniqueness of a generalized solution are proved.

1. Introduction

We study a nonlocal problem for a forth-order PDE with dominating mixed derivative

Lu≡σ(x)∂²u

∂t² − ∂

∂x

a(x)∂u

∂x − ∂

∂x

b(x) ∂³u

∂t²∂x

=f(x, t). (1.1) This equation is closely related to the problem of longitudinal vibration of a short thick bar. Vibration problems are of great importance in engineering and have been studied by many researches. The majority of works deals with second order hyperbolic equation. Initial-boundary problems for wave equation has been studied comprehensively and became classical [19].

However this model is not strictly correct for vibration of a thick short bar as is shown by Rayleigh [18]. But many machine components may be interpreted just as a thick short bar. For a more precise analysis of the longitudinal vibrations in a thick short bar we need to take into account the transverse deformations.

Mathematical model of longitudinal vibration considering the effect of transverse movements in a thick short bar is calledRayleigh barand is based on the equation (1.1). Some results of studying of initial-boundary problems for (1.1) can be find in [3, 6].

In this article we do a next step to make this model more precise. To this end we propose to define more exactly boundary conditions from the following reasoning.

The assumption on dimension of the bar suggests that there exists certain con- nection between values of a required solution in different boundary points. Such effect was found by Steclov [20] for heat equation. A relation connecting values of a solution to a PDE in various boundary points is a nonlocal condition.

Thus we suggest a nonlocal approach to study longitudinal vibration of a short thick bar. Note that nonlocal approach is in agreement with survey and results of

2010Mathematics Subject Classification. 35L10, 35L20, 35L99.

Key words and phrases. Nonlocal problem; longitudinal vibration; forth order equation;

dynamical boundary condition; generalized solution.

c

2019 Texas State University.

Submitted October 24, 2018. Published February 18, 2019.

1

experiments analyzed in [2] and turn out to be often more precise in mathematical modeling. Motivated by this, we consider the problem with nonlocal dynamical boundary conditions [1, 5, 7, 8, 9, 10, 11, 12, 14, 16, 21].

Note that there is close connection between nonlocal boundary conditions of the form to be dealt with below and nonlocal integral conditions [4].

2. Statement of the problem

Consider the longitudinal vibration of a thick short bar. Suppose that the bar represents the solid of revolution around the axisOx. Denote byu(x, t) the longi- tudinal displacements subject to determination. Let the exciting distributed force be f(x, t). Suppose that the left and right ends of the bar, x= 0 and x=l, are attached to the immovable ground with the help of the point massesM1, M2 and springs. In addition we take into account the resistance of medium. The latter implies the presence ofutin the boundary conditions. Lagrangian of Rayleigh bar is constructed in [17, p. 158-184]. Hamilton variational principle and elementary manipulation lead to the equation

σ(x)u_tt−(a(x)u_x)_x−(b(x)u_ttx)_x=f(x, t), (2.1) where

σ(x) =ρ(x)A(x), a(x) =A(x)E(x), b(x) =ρ(x)ν²(x)Ip(x),

A(x) is the cross-section area,ρ(x) is the mass density of the bar,E(x) is Young’s modulus,Ip(x) is the polar moment of inertia, ν is the Poisson coefficient.

The main object of this article is the following problem: find inQT = (0, l)× (0, T) a solution to (2.1) satisfying the initial conditions

u(x,0) = 0, u_t(x,0) = 0 (2.2)

and the nonlocal boundary conditions

a(0)ux(0, t) +b(0)uxtt(0, t) =

α11u(0, t) +α12u(l, t) +β11ut(0, t) +β12ut(l, t) +M1utt(0, t), a(l)u_x(l, t) +b(l)u_xtt(l, t) =

α₂₁u(0, t) +α₂₂u(l, t) +β₂₁u_t(0, t) +β₂₂u_t(l, t)−M₂u_tt(l, t).

(2.3)

Some particular cases of the problem (2.1)–(2.3), namely whenα12=α21=βij = 0 and for special form of coefficients of (2.1), are considered in [6]. In [3] the generalized solvability of (2.1)–(2.3) when α12 = α21 = βij = 0 is proved. The main goal of our paper is to determine conditions under which there exists a unique solution to the problem (2.1)–(2.3), that is to the problem with nonlocal dynamical conditions.

To prove solvability of nonlocal problem (2.1)–(2.3) we suggest an approach which enables us to use many well-known techniques. We define a notion of a weak solution for (2.1)–(2.3) and show that under some assumptions on data there exists a unique weak solution.

It is convenient here to list main assumptions on the data.

(H1) a, b, σ∈C¹[0, l], a(x)≥a0>0, b(x)≥b0>0, σ(x)≥σ0>0;

(H2) f, ft∈C( ¯QT);

(H3) Mi >0, i= 1,2.

Remark 2.1. Positiveness of coefficientsa, b, σandMiis a consequence of physical significance of them.

Remark 2.2. We consider homogeneous initial conditions only to simplify calcu- lations and without loss of generality.

Denote

Γ0={(x, t) :x= 0, t∈[0, T]}, Γl={(x, t) :x=l, t∈[0, T]}, W(QT) ={u:u∈W₂¹(QT), uxt∈L2(QT), ut∈L2(Γ0∪Γl)},

V(Q_T) ={v:v∈W(Q_T), v(x, T) = 0}.

Now we define a solution of the problem using a standard method [13, p. 92]:

integrating by parts an identity RT 0

Rl

0(Lu−f)v dx dt = 0 where u(x, t) satisfies (2.1)–(2.3),v∈C²(Q_T)∩C¹( ¯Q_T) we obtain the equality

Z T

0

Z l

0

(−σ(x)utv_t+a(x)u_xv_x−b(x)u_xtv_xt)dx dt +

Z T

0

v(0, t)[α11u(0, t) +α12u(l, t) +β11ut(0, t) +β12ut(l, t)]dt

− Z T

0

v(l, t)[α₂₁u(0, t) +α₂₂u(l, t) +β₂₁u_t(0, t) +β₂₂u_t(l, t)]dt

−M1

Z T

0

ut(0, t)vt(0, t)dt−M2

Z T

0

ut(l, t)vt(l, t)dt

= Z T

0

Z l

0

f v dx dt.

(2.4)

Note that all integrals in (2.4) exist also foru∈W(Q_T),v∈V(Q_T). Hence, (2.4) is suitable for a definition of a generalized solution to the problem (2.1)–(2.3).

Definition 2.3. A functionu∈W(QT) is said to be a weak solution to the problem (2.1)–(2.3) ifu(x,0) = 0 and for everyv∈V(QT) the identity (2.4) holds.

3. Main results

Theorem 3.1. Under assumptions (H1)–(H3)there exists a unique weak solution to problem (2.1)–(2.3)if

α₁₁ξ₁²+ 2α₁₂ξ₁ξ₂−α₂₂ξ₂²≥0, ξ= (ξ₁, ξ₂)∈R².

Proof. Uniqueness. Let u1, u2 be two weak solutions of (2.1)–(2.3). Then u = u₁−u₂ satisfiesu(x,0) = 0 and the identity

Z T

0

Z l

0

(−σ(x)utvt+a(x)uxvx−b(x)uxtvxt)dx dt +

Z T

0

v(0, t)[α11u(0, t) +α12u(l, t) +β11ut(0, t) +β12ut(l, t)]dt

− Z T

0

v(l, t)[α₂₁u(0, t) +α₂₂u(l, t) +β₂₁u_t(0, t) +β₂₂u_t(l, t)]dt

−M1

Z T

0

ut(0, t)vt(0, t)dt−M2

Z T

0

ut(l, t)vt(l, t)dt= 0

(3.1)

holds. Let

v(x, t) = (Rt

τu(x, η)dη, 0≤t≤τ,

0, τ ≤t≤T (3.2)

whereτ ∈[0, T] is arbitrary. After integrating (3.1) by parts we obtain Z l

0

[σu²(x, τ) +av²_x(x,0) +bu²_x(x, τ)]dx

+α11v²(0,0)−α22v(0,0)v(l,0) +M1u²_t(0, τ) +M2u²_t(l, τ)

=−2β₁₁ Z τ

0

u²(0, t)dt−2(β₁₂−β₂₁) Z τ

0

u(0, t)u(l, t)dt

+ 2β22

Z τ

0

u²(l, t)dt−2(α12+α21) Z τ

0

u(0, t)u(l, t)dt.

(3.3)

Under the assumptions of this Theorem,

α₁₁v²(0,0)−α₂₂v²(l,0) + 2α₁₂v(0,0)v(l,0)≥0, M₁u²_t(0, τ) +M₂u²_t(l, τ)>0.

We consider the right side of (3.3) and estimate each term. To this end we use Cauchy inequality and obtain

2

Z τ

0

u(0, t)u(l, t)dt ≤

Z τ

0

[u²(0, t) +u²(l, t)]dt, (3.4) 2

Z τ

0

u(0, t)v(l, t)dt ≤

Z τ

0

[u²(0, t) +v²(l, t)]dt. (3.5) Thus from (3.3),

Z l

0

[σu²(x, τ) +av²_x(x,0) +bu²_x(x, τ)]dx

≤(2|β11|+|β12|+|β21|+|α12|+|α21|) Z τ

0

u²(0, t)dt + (2|β22|+|β12|+|β21|)

Z τ

0

u²(l, t)dt+ (|α12|+|α21|) Z τ

0

v²(l, t)dt.

To proceed with the estimate, we derive some inequalities. As for anyu∈W(Q_T) representations

u(0, t) = Z 0

x

uξ(ξ, t)dξ+u(x, t), u(l, t) = Z l

x

uξ(ξ, t)dξ+u(x, t) hold we easily get the inequalities

u²(0, t)≤2l Z l

0

u²_x(x, t)dx+ 2u²(x, t), u²(l, t)≤2l Z l

0

u²_x(x, t)dx+ 2u²(x, t).

Integrating both with respect toxover (0, l) we obtain u²(zi, t)≤2l

Z l

0

u²_x(x, t)dx+2 l

Z l

0

u²(x, t)dx, i= 0,1, z0= 0, z1=l. (3.6) Using the same procedure we obtain

v²(l, t)≤2l Z l

0

v²_x(x, t)dx+2 l

Z l

0

v²(x, t)dx.

From ((3.2)) it follows that v²(x, t)≤τ

Z τ

0

u²(x, t)dt, v_x²(x, t)≤τ Z τ

0

u²_x(x, t)dt, then

v²(l, t)≤2lτ Z τ

0

Z l

0

u²_x(x, t)dx dt+2τ l

Z τ

0

Z l

0

u²(x, t)dx dt. (3.7) DenoteA=|α12|+|α21|,B=P2

i,j=1|βij|,

m0= min{a0, b0, σ0}, M = 2 max{Bl+AlT,B+AT l }.

Taking into account (3.6) and (3.7) , from (3.3) we obtain m0

Z l

0

[u²(x, τ) +v²_x(x,0) +u²_x(x, τ)]dx≤M Z τ

0

Z l

0

(u²+u²_x)dx dt and therefore

m₀ Z l

0

[u²(x, τ) +u²_x(x, τ)]dx≤M Z τ

0

Z l

0

(u²+u²_x)dx dt.

Thus from Gronwall’s inequality, we haveu(x, τ) = 0 for allτ∈[0, T]. Hence there exists at most one weak solution to the problem (2.1)–(2.3).

Existence. We prove the existence part in several steps. First, we construct ap- proximations of the generalized solution by the Faedo-Galerkin method. Second, we obtain a priori estimates to guarantee weak convergence of approximations. Finally, we show that the limit of approximations is the required solution.

Letw_k(x)∈C²( ¯Ω) be a basis inW₂¹(Ω). We define approximations as follows, u^m(x, t) =

m

X

k=1

ck(t)wk(x) (3.8)

and shall seekck(t) from relations Z l

0

σu^m_ttwj+au^m_xw⁰_j+bu^m_xttw_j⁰

dx+M1u^m_tt(0, t)wj(0)−M2u^m_tt(l, t)]wj(l) + [α₁₁u^m(0, t) +α₁₂u^m(l, t) +β₁₁u^m_t (0, t) +β₁₂u^m_t (l, t)]w_j(0)

−[α21u^m(0, t) +α22u^m(l, t) +β21u^m_t (0, t) +β22u^m_t (l, t)]wj(l)

= Z l

0

f wjdx.

(3.9)

For every m the relations (3.9) represent a system of second-order ODE’s with respect tock(t) and after substituting (3.8) we can rewrite it in the form

m

X

k=1

[A_kjc⁰⁰_k(t) +B_kjc⁰_k(t) +D_kjc_k(t)] =f_j(t), (3.10) where

A_kj = Z l

0

(σw_kw_j+bw_k⁰w_j⁰)dx+M₁w_k(0)w_j(0) +M₂w_k(l)w_j(l);

B_kj=β₁₁w_k(0)w_j(0) +β₁₂w_k(l)w_j(0)−β₂₁w_k(0)w_j(l)−β₂₂w_k(l)w_j(l);

Dkj= Z l

0

aw_k⁰w⁰_jdx+α11wk(0)wj(0) +α12wk(l)wj(0)

−α₂₁w_k(0)w_j(l)−α₂₂w_k(l)w_j(l);

f_j(t) = Z l

0

f(x, t)w_j(x)dx.

Adding the initial data,

ck(0) = 0, c⁰_k(0) = 0 (3.11)

we obtain Cauchy problem for (3.10). Now we show that Cauchy problem (3.10)–

(3.11) is solvable.

Consider a matrixA= (A_kj)^m_k,j=1 and verify that it is positive definite. To this end we introduce a quadratic form

q=

m

X

k,j=1

A_kjξ_kξ_j,

where ξ_k, ξ_j are coefficients of sums ξ = Pm

i=1ξ_iw_i(x). Rearrange this quadratic form using representations of the coefficientsA_ij:

q=

m

X

k,j=1

Z l

0

(σw_kw_jξ_kξ_j+bw_k⁰w_j⁰ξ_kξ_j)dx+M₁w_k(0)w_j(0)ξ_kξ_j+M₂w_k(l)w_j(l)ξ_kξ_j. After changing the order of summing and integrating we obtain

q= Z l

0

σ|ξ|²+b|ξx|²

dx+M1|ξ(0)|²+M2|ξ(l)|².

We know that σ, b,M1, M2 are positive. Now note that quadratic formqvanishes only if ξ = 0. Hence ξk = 0 ∀k = 1, . . . , m by virtue of linearity independence ofwk(x). Consequently the matrixA is positive definite and the system (3.10) is solvable with respect toc⁰⁰_k(t). The conditions of Theorem imply that the coefficients of (3.10) are bounded and f ∈ L2(QT). These facts guarantee the solvability of Cauchy problem (3.10)–(3.11). Moreover,c⁰⁰_k ∈L₂(0, T). Thus, the approximation {u^m(x, t)}is constructed.

We need now to derive an a priori estimate. To this end we multiply (3.9) by c⁰_j(t), sum with respect toj= 1, . . . , m, integrate over (0, τ) and obtain

Z τ

0

Z l

0

(σ(x)u^m_ttu^m_t +a(x)u^m_xu^m_xt+b(x)u^m_xttu^m_xt)dx dt +

Z τ

0

h

α11u^m(0, t)u^m_t (0, t) +α12u^m(l, t)u^m_t (0, t) +β11(u^m_t (o, t))² +β12u^m_t (0, t)u^m_t (l, t)i

dt− Z τ

0

[α21u^m(0, t)u^m_t (l, t) +α22u^m(l, t)u^m_t (l, t) +β21u^m_t (0, t)u^m_t (l, t) +β22(u^m_t (l, t))²]dt+M1

Z τ

0

u^m_tt(0, t)u^m_t (0, t)dt

−M₂ Z τ

0

u^m_tt(l, t)u^m_t (l, t)dt+ (α₁₂+α₂₁) Z τ

0

u^m_t (l, t)u^m(0, t)dt

= Z τ

0

Z l

0

f u^m_t dx dt.

(3.12)

After integrating by parts in (3.12) we obtain Z l

0

[σ(x)(u^m_t (x, τ))²+a(x)(u^m_x(x, τ))²+b(x)(u^m_xt(x, τ))²]dx +α11(u^m(0, τ))²+ 2α12u^m(0, τ)u^m(l, τ)−α22(u^m(l, τ))² +M1(u^m_t (0, τ))²+M2(u^m_t (l, τ))²

= 2β22

Z τ

0

(u^m_t (l, t))²dt−(α12+α21) Z τ

0

u^m_t (l, t)u^m(0, t)dt + 2(β21−β12)

Z τ

0

u^m_t (0, t)u^m_t (l, t)dt−2β11

Z τ

0

(u^m_t (0, t))²dt + 2

Z τ

0

Z l

0

f u^m_t dx dt.

Under assumption (H1) the left-hand side of this equality is positive. Using Cauchy, Cauchy-Bunyakovskii inequalities and (3.6), (3.7) we derive from the last equality the inequality

m0

Z l

0

[(u^m_t (x, τ))²+ (u^m_x(x, τ))²+ (u^m_xt(x, τ))²]dx+α11(u^m(0, τ))² + 2α₁₂u^m(0, τ)u^m(l, τ)−α₂₂(u^m(l, τ))²+M₁(u^m_t (0, τ))²

+M₂(u^m_t (l, τ))²

≤M Z τ

0

Z l

0

[(u^m_t )²+ (u^m_x)²+ (u^m_xt)²]dx dt+ Z τ

0

Z l

0

f²dx dt.

(3.13)

Applying Gronwall’s inequality to (3.13) we obtain Z l

0

[(u^m_t (x, τ))²+ (u^m_x(x, τ))²+ (u^m_xt(x, τ))²]dx≤m⁻¹₀ e^Cτkfk²_L

2(Q_τ)

whereC=M/m0. It is easy to see that from this inequality after integrating over (0, T) we obtain

Z T

0

Z l

0

[(u^m_t (x, τ))²+ (u^m_x(x, τ))²+ (u^m_xt(x, τ))²]dx dt≤M⁻¹(e^CT−1)kfk²_L

2(Q_T). From (3.13) it also follows that

M1

Z T

0

(u^m_t (0, t))²dt+M2

Z T

0

(u^m_t (l, t))²dt≤T e^CTkfk²_L2(QT).

As f ∈ L₂(Q_T) then kfkL₂(Q_T) is finite: kfkL₂(Q_T) ≤ k. Thus the obtained inequalities lead to the required estimate

ku^mk_W(QT)≤P (3.14)

whereP =k²max{M⁻¹(e^CT −1), T e^CT}and does not depend onm.

As W(QT) is Hilbert space then the estimate (3.14) enables state that we can extract from approximationsu^m(x, t) a subsequence weakly convergent inW(QT).

For technical reasons we do not change notation for it.

At a final step we show that the limit of extracted subsequence is the required weak solution to the problem (2.1)–(2.3).

Multiplying (3.9) bydj∈C²[0, T], summing fromj= 1 toj=mand integrating with respect tot from 0 toT we obtain

Z T

0

Z l

0

[σu^m_ttη+au^m_xη_x+bu^m_xttη_x]dx dt+ Z T

0

η(0, t)h

α₁₁u^m(0, t) +α12u^m(l, t) +β11u^m_t (0, t) +β12u^m_t (l, t) +M1u^m_tt(0, t)i

dt

+ Z T

0

η(l, t)h

α₂₁u^m(0, t) +α₂₂u^m(l, t) +β₂₁u^m_t (0, t) +β₂₂u^m_t (l, t)

−M₂u^m_tt(l, t)i dt

= Z T

0

Z l

0

f η dx dt

(3.15)

where η(x, t) = Pm

j=1d_j(t)w_j(x). Because of obtained estimates we are able to pass to the limit in (3.15) to obtain (3.1) for v(x, t) =η(x, t). Taking into account that the set of all functions of the form Pm

j=1d_j(t)w_j(x) is dense in V(Q_T) we conclude that (3.1) holds for everyv∈V(QT). This means thatu(x, t), weak limit of the subsequenceu^m(x, t), is the required solution to the (2.1)–(2.3). The proof

is complete.

Acknowledgments. We want to thank the anonymous referees for their comments that improved this article, and Professor Julio G. Dix for managing our submission to the EJDE.

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[20] Steclov, V. A.;The problem of cooling in inhomogeneous solid, Communications of Kharkov mathematical Society. 1896. Vol. 5, issue 3-4, p. 136-181.

[21] Zhang, Zhifei;Stabilization of the wave equation with variable coefficients and a dynamical boundary control, Electron. J. Differential. Equ., vol. 2016 (2016), no.27, pp. 1–10.

Ludmila S. Pulkina

Samara National Research University, Samara, Russia E-mail address:[email protected]

Alexander B. Beylin

Samara State Technical University, Samara, Russia E-mail address:[email protected]