WELL-POSEDNESS AND STABILITY RESULT FOR TIMOSHENKO SYSTEM WITH THERMODIFFUSION EFFECTS AND TIME-VARYING DELAY TERM Текст научной статьи по специальности «Математика»

ISSN 2304-0122 Ufa Mathematical Journal. Vol. 16. No 2 (2024). P. 89-103.

doi:10.13108/2024-16-2-89

WELL-POSEDNESS AND STABILITY RESULT FOR TIMOSHENKO SYSTEM WITH THERMODIFFUSION EFFECTS AND TIME-VARYING DELAY TERM

A. RAHMOUNE, Ou. KHALDI, D. OUCHENANE, F. YAZID

Abstract. The main aim of the present paper is to investigate a new Timoshenko beam model with thermal and mass diffusion effects combined with a time-varying delay. Heat and mass exchange with the environment during a thermodiffusion in the Timoshenko beam, where the heat conduction is given by the classical Fourier law and acts on both the rotation angle and the transverse displacements. The heat conduction is given by the Cattaneo law. Under an appropriate assumption on the weights of the delay and the damping, we prove a well-posedness result, more precisely, we prove the existence of the weak solution. Then we proceed to the strong solution using the classical elliptic regularity and we get the result by applying the Lax-Milgram theorem, the Lumer-Phillips corollary and the Hille-Yosida theorem. We show the exponential stability result of the system in the case of nonequal speeds of wave propagation by using a multiplier technique combined with an appropriate Lyapunov functions. Our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical PDEs theory. To achieve our goals, we employ of the semigroup method and the energy method.

Keywords: Timoshenko beam, diffusion, time varying delay, existence and uniqueness, exponential stability.

Mathematics Subject Classification: 74K10, 37N15, 74F05, 65M60.

The shear deformation and rotational bending effects on the beam are described mathematically by the Timoshenko system. S. P. Timoshenko first introduced the following system

where ^ is the rotation angle of the beam's filament and (p is the transverse displacement of the beam. The coefficients p, Ip, E, I and K represent repsectively for the following quantities: the density (the mass per unit length), the polar moment of inertia of a cross section, Young's modulus of elasticity, the moment of inertia of a cross section and the shear modulus.

In [8] Rivera and Racke examined the following system

Using the energy approach, they established that this system is exponentially stable.

A. Rahmoune, Ou. Khaldi, D. Ouchenane, F. Yazid. Well-posedness and stability result for Timoshenko system with thermodiffusion effects and time-varying delay term. © Rahmoune A., Khaldi Ou., Ouchenane D., Yazid F. 2024. Submitted October 21, 2023.

1. Introduction

[10]:

piptt - k (<px + = 0, P24>tt - ot^xx + k (px + = 0, P3 dt - kdxx + ^tx = 0.

Time delays appear in many applications of the majority of phenomena naturally governed by partial differential equations problems relying not only on the present state but also on some past occurrences. The delay can cause an instability. To the best of our knowledge, Said Houari was the first who discussed the Timoshenko system with a time delay [9]. More precisely, in [9] the Timoshenko system

(pi^tt (x, t) - K (<px + (x, t) = 0,

\p2^tt (x, t) - b^xx (x, t) + K (tpx + (x, t) + ^t (x, t) + (x, t - T) = 0,

and under the assumption pi ^ p2 they proved the well-posedness and exponential decay. The work was extended upon by Kirane et al. [5], who also introduced the case of time-varying delay and established several estimates of general decay.

In [2], Apalara considered a one-dimensional Timoshenko system with linear friction damping and a constant delay operating on the displacement equation.

(Pi^tt (x, t) - K ((px + (x, t) + Pi^t (x, t) + (x, t - T) = 0, \p2^tt (x, t) - b^xx (x, t) + K (<£x + i>) (x, t) = 0.

Under appropriate assumptions on the weight of the delay and the wave speeds, the well-posedness and asymptotic stability results of the system were established. The stability results also showed that the dissipation through the frictional damping is strong enough to uniformly stabilize the system even in the presence of delay. This paper is similar to the one by Said Houari [9], but it includes a delay in the first equation, like the one considered in the manuscript.

We may believe that the dissipation cannot be fully explained by the thermal conduction in the Timoshenko beam and the area of diffusion in solids cannot be neglected. A natural question appears on what happens when the thermal effect and diffusion effect are included in Timoshenko beams. The diffusion is a random movement of a group of particles from high concentration areas to low concentration regions. The domains of strain, temperature, and mass diffusion cause the thermodiffusion in an elastic solid. Recently, Aouadi et al., in [1] studied the following problem

'piVtt (x,t) - K (<px + $)x (x,t) = 0,

p2^tt - OL^xx + k (ipx + - 716^ - 72Px = 0,

c9t + dPt - k9xx - 71^tx = 0, k ddt + rPt - hPxx - 72^ = 0.

Under different boundary conditions, they proved that the system is exponentially stable if and only if

- = -. (1.1)

Pi P2

Being inspired by previous works, the main goal of this paper is to demonstrate the well-posedness and establish a general energy decay, which results in the usual exponential decay. Our result is dependent on the kernel of the time varying delay term and the construction of an appropriate Lyapunov functional, which allows us to estimate the energy of the system. However, the study of the asymptotic behavior of the solution for various types of problems, such as the Timoshenko system [7], remains crucial. The study the exponential behavior of solutions to a variety of problems, such as the Timoshenko system [7], is still important.

In this article, we are intersted with the following problem

'pi<ptt (x, t) - K (yx + ip)x (x, t) + pi^t (x, t) + P2p(x, t - r(t)) = 0,

P2^tt - a^xx + k (yx + - ^idx - 72Px = 0, (1 2)

cdt + dPt - k9xx - = 0, .

^d0t + rPt - hPxx - ^tx = 0

(1.3)

(1.4)

subject to the initial conditions

ftp(x, 0) = p0(x), ^(x, 0) = ^0(x), 6(x, 0) = e°(x) for X e (0, L), [P(x, 0) = P°(x), <pt(x, 0) = <pl(x), ^t(x, 0) = ^l(x) for X e (0, L),

and a Dirichlet boundary conditions:

-p(0, t) = ^(0, t) = 0(0, t) = P(0, t) = 0, t> 0, _p(L, t) = ^(L, t) = d(L, t) = P(L, t) = 0, t> 0.

fc d\

We assume that the symmetric matrix A = ( ^ | is positive definite, that is,

5 = cr - d2 > 0. (1.5)

We note that the above relation implies

cB2 + 2d9P + rP2 > 0 (1.6)

for 9,P = 0. We also observe that condition (1.5) is needed to stabilize the system, when the diffusion effects are added to thermal effects.

In our opinion, the concept of the mass diffusion introduced into Timoshenko equations could have very significant physical effects other than body deformations. For example, recent studies focused on the effect of mass diffusion on the damping ratio in microbeam resonators, see, for instance, [9]. Moreover, the mass diffusion introduces a new critical thickness in addition to the conventional critical thickness of thermoelastic damping.

The explanations above indicate that the mass diffusion plays an important role in the clarification of the thermomechanical behaviour of Timoshenko model. To the best of the authors' knowledge, no theoretical or numerical simulation of the mass diffusion effects on the thermal vibration of the Timoshenko beam was done. And the goal of this work is to examine the effect of mass diffusion alongside the effect of temperature on the behaviour of the Timoshenko beam.

The paper is organized as follows. In Section 2 we demonstrate the well-posedness of our problem using the semi-group technique. We establish a general stability under a suitable conditions in Section 3.

2. Well-posedness

In this section we prove that the considered problem is well-posed. We introduce a new variable

z (x,p,t) = <pt (x,t - t(t)p), x e (0,1), p e (0,1), t> 0, and get following equation:

T(t)zt (x, P, t) + (1 - iJ(t)p)zp (x, P, t) = 0, (x, p, t) e (0,1) x (0,1) x (0, . Then problem (1.2) can be rewritten as

' pi^tt (x, t) - K (<px + (x, t) + Pi^t (x, t) + P2Z (x, 1, t) = 0, p2^tt - Oi^xx + k (<px + - - 72Px = 0,

< cdt + dPt - k9xx - ji^tx = 0, (2.1)

ddt + rPt - hPxx - ^tx = 0,

T(t)zt (x, p, t) + (1 - T,(t)p)zp (x, P, t) = 0, (x, p, t) e (0, L) x (0,1) x (0, ,

(2.3)

subject to the initial conditions

'p(x, 0) = p0(x), ^(x, 0) = ^°(x), 6(x, 0) = 9°(x) for x E (0, L), P(x, 0) = P0(x), ipt(x, 0) = p1(x), ^t(x, 0) = ^(x) for x E (0, L), (22) z (x, 0, t) = <pt (x,t) , x E (0, L), t> 0,

^ (x, p, 0) = fo (x, 1 - pr(0)), (x, p) E (0, L) x (0,1) , and the Dirichlet boundary conditions

V(0,i)= ^(0,t) = 0(0, t) = P (0,t) = 0 for t> 0, fi(L, t) = ^(L, t) = d(L, t) = P(L,t) = 0 for t> 0.

Here the function r(t) is supposed to satisfy the condition

0 < ro ^ t(t) ^ f for all t > 0. (2.4)

r'(t) ^ m< 1 for all t> 0, (2.5)

t E W2'™[0, (2.6)

We are going to study the well-posedness of the above problem. Namely, we provide sufficient conditions that ensuring the well-posedness. In order to do this, we follow procedures from recent paper [9], in which the Thimoshenko problem with a frictional damping was studied.

We rewrite system (2.1), (2.2), (2.3) as a first order system in order to apply the semigroup approach. Namely, we let U(t) = (<p(t),v(t),^(t),fi(t),d(t),P(t),z(t))T, and rewrite (2.1), (2.2), (2.3) as

'U' = A U,

U (0) = Uo := (p0(x),p1(x),^0(x),^1(x), 0o(x),Po(x),fo (x, 1 - pr(0)))T , where A is an operator defined as

A

M

V

d p

W

(

K

\

pi

P2

Pi (fixx + PiV PiZ ( ■ , 1)

p2 1 (a^xx - k (fix + + lidx + l2Px) -8~l ((d^2 - ni) fix - rndxx + dhPxx) -6-1 ((^71 - C72) fix + dnOxx - chPxx)

(T (t)p - 1)Zp

r (t)

V

/

on the domain

D (A) = {(p,v,^,fi,d,p,z)T E H : v = z (■ , 0) , in (0, where

H := (H2 (0, L) n H1 (0, L)) x H^ (0, L) x (H2 (0, L) n H^ (0, L)) x H^ (0, L) x (H2 (0, L) n H1 (0, L)) x (H2 (0, L) n H^ (0, L)) x L2 ((0, L); H1 (0, L)) . The energy space H is defined as

H :=H1 (0, L) x L2 (0, L) x H^ (0, L) x L2 (0, L) x L2 (0, L) x L2 (0, L) x L2 ((0,L); L2 (0,L)) .

For Uj = (<pj,Vj,-fij, fij, 9j, Pj, Zj)T E H and j = 1, 2, and a positive constant £ obeying

P2

m

< C < 2p1 -

P2

y/1 - m'

P2

< vr-

mp1,

(2.7)

v

the inner product in H is defined as

L

( Ui,U2)H = J P1V1V2 + + 0L-1,x—,x + k (<P1,X + -1 ) (<p2,x + -2)

0

+A

(pi) ■

L 1

Ol

)■{ O 2r2) dx + c;T(L)

00

dx + £r(i) z(x,p) z1 (x,p)dpdx.

The solvability result is formulated in the following theorem.

Theorem 2.1. Assume that p2 < V1 - mpi, then for any U0 e H there exists a unique solution U e C ([0, , H) of system (2.1), (2.2), (2.3). Moreover, if Uo e D (A), then

U eC ([0, , D (A)) n C1 ([0, , H).

Proof. We use the semigroup approach in the proof. In other words, we show how the operator A produces a C0-semigroup in H. We are going to demonstrate that the operator B(t) = A - 3(t)I is dissipative with

Jt' (i)2 +1

3® --MA+- ■ Indeed, for U = (tp,u,^,v,9,q, z) e D(A), we have

(2.8)

(AU,U)h =k J (<px + ^)xvdx + J (a^x + 7^ + 72P)xpdx oo

L L L

- p2 J z(x, 1)v dx -k J (ipx + tp)pdx + af px^x dx

o o o

L L _

+ k J (vx + p) (<Px + $)dx + J • (^J dx

oo

L i

oo

where

e = -<Ti ((dj2 - r7i) px - rndxx + dhPxx), $ = -<Ti ((dji - c72) px + dndxx - chPxx).

The last term in the right-hand side of (2.8) can be rewritten as l i l i

f J (At)P - 1) (*,,)<<** = / / (At)» - 1)11? (x, p)dpdx

o o o o

L

= 1 J {z2 (x, 1) (r'(t) - 1) - z2 (x, 0)} dx o

L i

At) 1 1 2

z (x, p)dpdx.

L

L

As a result, (2.8) becomes

{AU, U)h = - k 9x dx - h Pxdx - p1 v dx

- p2 z (x, 1) v dx -

o

L 1

^'(t) L 1 2

2

z (x,p) dpdx

o o

o

L L

+ ^J (r'(t) - 1)z2 (x, 1) dx - |J v2 (x) dx.

oo

By using Young inequality and (2.9) we obtain

{AU, U)h < - k 92xdx - h P^dx + -p1 +

(-

P2

2^/1-

m

L

^ J v2 (x) dx

L

+ (^^ - i Ji-^) / ^ (*, 1) dx + S(MAU, U)h.

(2.9)

In view of condition (2.7) we have

-P1 +

P2

2

£

- - < 0,

P2V1 - m (1 - m) < 0 2 ^ 2 < .

Hence, the operator A is dissipative.

Now we are going to prove that the operator XI - A is surjective for A > 0. We take an element F = (f1,f2,f3,f4,f5,fe,f7)T E H and we seek a solution U = (p,v,-fi,fi,9,p,z)T E D (A) to the equation

or equivalently

XU- AU = F

Xp - v = ¡1,

\ K , , , P1 P2 / r

Xv--(fixx + Wx) +--v +--Z ( ■ , 1) = f2,

P1 P1 P1

^ - fi = fз,

Xp2(fi - Offixx + k (fix + - l10x - l2Px = P2fi,

X89 + (d^2 - r^d) fix - rkdxx + dhPxx = X5P + (d^1 - C72) fix + dnOxx - chPxx = fife

(T'(f)p - 1) Z = f

Xz -jj^- zp = f7.

(2.10)

Assume that we have found p and ^ with the needed regularity. Then the first and third equations in (2.10) give

'v = Xp - f1,

fi = AV- /3. (211)

It is clear that v E Hi (0,1) , and fi E Hi (0,1). Moreover, we can find z as

z (x, 0) = v (x) for x E (0,1)

L

L

L

L

L

L

Following the lines of [6], by using the last equation in (2.10) we obtain

p

z (x, p) = v (x) e-xpT(t) + r(t)e-xpT(t) j f7 (x, s) eXsT(t) ds if r(t) = 0,

o

and

z(x, p) = v(x)er"(t) + er»(t) i Î7(x, s)J() e-r°(t) ds if r'(t) = 0,

J 1 - T'(t)s

where

r,(t) = X^)ln(1 - r'(t)p). T(t)

Using (2.11), we then get

and

z (x, p) = Xp (x) e-Apr(t) - fie-XpT(t) + r(t)e-xpT(t) J f7 (x, s) eXsT(t) ds if r'(t) = 0,

0

z(x, p) = Xper»(t) - fierp(t) + er»(t) [ îl(x, s)J() e-r°(t) ds if r\t) = 0.

1 - ( )

By the above identities we have

z(x, 1) = g(t)p(x) + zo(x),

Xe-Xt (t) if r'(t) = 0,

( )

Xer?(t) if r'(t) = 0.

and

zo (x)

- fie-XT(t) + r(t)e-Xt(t) f7 (x, s ) e XsT(t) ds if r'(t) = 0,

- fier»(t) + er»(t) i Î7(X, S]j[t] e-°(t) ds J 1 - T'(t)s

(2.12)

if ( ) = 0,

where x e (0, L). According to the above formula, z0 depends only on fi, i = 1,..., 7. The

following system can be satisfied by employing (2.10) and (2.11) with the functions p, rp, 9 and :

pi

,ß2

X2 + —X + g(t)— ) p - — (pxx + -x) = f2 + ( X + — ) fi--zo(x), Pi Pi J Pi V Pi J Pi

P2

X2p2^ - OL1pxx + k (px +4>) - Ji9X - ^Px = p2 (Xf3 + /4), XS9 + (dj2 - r7i) - rn9xx + dhPxx = 5f5 + (d^2 - Hi) f3,x, ,X8P + (dji - C72) + dn9xx - chPxx = 8f6 + (dji - cj2) f:i,x. Solving system (2.13) is equivalent to finding

( p,^,9,p) e H2 (0, L) nHi (0, L) x H2 (0,1) n Hi (0,L) x Hi (0,L) x Hi (0,L)

(2.13)

i

i

such that

L

J ((X2P1 + P1X + g(t)[i2) pw + K (fix + fi) Wx) dx

L

= (p 1 ¡2 + ( Ap 1 + P1) f1 - p2Zo(x))wdx,

<

o

L L

, 2

1P2^ - a^xx + k(px + fi) - Ox - ^Px) xdx = P2 (14 + A/3) x dx, (2.14)

oo

L L

( XSB + (dj2 - m) fix - rrJxx + dhPxx) W1 dx = (8 f5 + (dj2 - m) f3,x) W1 dx,

x - x x

oo

L L

J ( XS P + (dj1 - 0^2) fix + duBxx - chPxx) X1 = J ($h + (dl1 - c72) f3,x) X1 dx,

oo

for all (W,X,W1,X1) E H1 (0,L) x H^ (0,L) x H1 (0,L) x H^ (0,L). Hence, problem (2.14) is equivalent to

C((p,^,e,p), (w,x,w1,x1)) = HW,X,W1,X1), (2.15)

where a bilinear from

C : (H1 (0, L) x H1 (0, L) x H1 (0, L) x H1 (0, L))2 ^ R

and a linear form

I : H1 (0, L) x H1 (0, L) x H1 (0, L) x H^ (0, L) ^ R

are defined as

L

<((p,fi,0,p) , (W,X,W1,X1)) = J ((>?P1 +P1^ + g(t)P2) PW + K (fix + "fi)wx) dx

o

L

2

p2rfi - ca^xx + k(px + fi) - hdx - j2Px) X dx

o

L

J ( XSB + (dj2 - H!) ^ - rrJxx + dhPxx) W1 dx o

L

j ( XS P + (dj1 - 0^2) fix + dKdxx - chPxx) X1, o

and

L L

I (w, X, W1, X1) = J (p 1/2 + (Xp 1 + P1) f1 - P2Zo(x)) w dx + J p2 (/4 + A/3) x dx

oo

L L

+ J ($h + (dl2 - rj1) f3,x) W1dx + J (8f6 + (dj1 - C72) f3,x) X1 dx, oo

where zo (x) satisfies the equation in (2.12). The continuous and coercive character of ( is easily verified, and I is continuous, so applying the Lax-Milgram theorem, we deduce that for all

(w,x,wi,xi) E H1 (0,L) x H1 (0,L) x H1 (0, L) x H1 (0,L)

problem (2.15) possesses a unique solution

( p, fi, 9,p) E H1 (0, L) x H1 (0, L) x H1 (0, L) x H1 (0, L).

Applying the classical elliptic regularity, by (2.14) we find that

( p, fi, 9,p) E H2 (0, L) x H2 (0, L) x H1 (0, L) x H1 (0, L).

The operator XI - A is hence surjective for each A > 0. Now the statement of the theorem follows from the Hille-Yosida theorem. The proof is complete. □

3. Exponential stability for /2 < VT—m/ii.

In this section we show the exponential stability of system (2.1), (2.2), (2.3) under the assumption — d> and the condition of nonequal wave speeds of propagation

- = -. (3.1)

Pi P2

Our approach is based on an appropriate Lyapunov functional using the energy technique, which results in the needed exponential decay.

We first observe that ( <p,fi,6,p, z) satisfies the same system (2.1), (2.2) and (2.3) and £ still satisfies

12 ^^ 2/i — . (3.2)

y/1 — m y/1 — m

The functional energy of the problem(2.1), (2.2) reads as

L

E(t) =1 I [pP + P2ф2 + -Ф1 + « (Px + Ф)2 + св2 + 2ddP + rP2] dx

2

0

L 1

(3.3)

+ ^ ( 'J J z2(x,p, t) dp dx. 00

We multiply the first equation in (2.1) by fit, the second equation by fit, the third equation in (2.1) by 9, and the fourth equation in (2.1) by q. Then we integrate by parts and we get

L

1 d 1 f

2 dt 2 [Pl<Pt + p2X^ + + K (Px + Ф)2 + 002 + 2dOP + rP2] dx

0 (3.4)

L L L L K '

= — к J в2х dx — hj P2 dx — h J p2 (x, t) dx — p2 J pt (x, t) z (x, 1, t) dx. 0 0 0 0

We multiply the last equation in (2.1) by £ z and z and integrate the result over (0, L) x (0,1) with respect to p and x respectively. This gives

l i l i

!A J J T{t)z2 t) ipix = J J (At)f - l)ztf t) dpdx

0 0 0 0

L i

+ ^'(t) J J 2,2 (X,P, t) dpdx 00

l i

= 2 J j W(T'(^)^ - 1)Z<2 (X,P, dpdx (3.5)

00 L

= 2 J ^ (X, 0,f) - ^ (X, 1, ^ dx 0

L

+ J z2 (x, 1, t) dx.

0

By (3.3), (3.4) and (3.5) we find

L L L

—= - k J d^dx - hj P2 dx - - J ¥2 (x, t) dx

0 0 0

L L

+ ( - - + ) [ z2 (x, 1, t) dx - p2 f ft (x, t) z (x, 1, t) dx.

22

00

Using the Young inequality, we rewrite (3.6) as

L L L

^ - K J eldx - h f px dx - (»i - 2 - / (x, f) dx

0

L

+ (§(/«) -1) + ^i-™) f ^ (x. 1,o<fc.

2

0

In view of (3.2), (2.4), (2.5) and (2.6) we conclude there exists C > 0 such that

L L ( L L

^ -k [ 62x dx - hi P2 dx - C < [ $ (x, t) dx + [ z2 (x, 1, t) dx

(3.6)

dt

According to the last inequality, the function E does not increase in t. Now we are in position to formulate our main result.

Theorem 3.1. Assume (1.1), (3.1), (2.5), (2.6) and p2 < V1 - mpi. Then, for any solution to problem (1.2), (1.3), (1.4) there are two positive constants C and 7 independent oft such that

E (t) ^ Ce-lt for all t ^ 0.

To establish the exponential decay of the solution, it is sufficient to construct a functional L (t), which is equivalent to the energy E (t) and satisfies

< -AL (t) for all 0 dt w

with some constant A > 0.

In order to find such functional, we first introduce another functional defined as

L L

1 (t) = J (pifitfi + p2fitfi) dx + y J p2 dx. (3.7)

oo

We then have the following estimate.

Lemma 3.1. Assume that conditions (1.6) and (3.1) hold and ( p,pt,fi,fit,9,P) is the solution to problem (2.1), (2.2), (2.3). If £i > 0, we then have the estimate

L L L

(t) <pi J p2dx + P2 f fi^dx + (^-k + ^^ J p2x dx o o o

L L L L

- ^ f fildx + Q J Oldx + C2 J P^dx - k J fi2 dx (3.8)

o o o o

L L

- 2 k J fipx dx + J z<2 (x, 1, t) dx.

o o

Proof. We calculate the derivative of 1 (t):

L L L

—1 (t) = pi J (p2 + pptt) dx + P2 J (fit2 + fifitt) dx + pi J ppt dx. o o o

It follows from (2.1) and (2.1) that

L L L L

^y piptpdx =pi J p2 dx — k J pi dx — k J fipx dx 0 000

L L

— Pi J pptdx — p2 J pz (x, 1, t) dx, 00

and

L L L

TT pifitfidx = p2 ttf dx + (afixx — k(px + fi) + 71^ + ^2Px)fidx

(3.9)

~Jt I P2^tfidx = p2 I u,x , J (u^xx

0 0 0

L L L L L

=P2 dx — k fipxdx — k fi2 dx — a fildx — (jid + j2P) fix dx.

Summing (3.9) and (3.8), we get

L L L L

A

~dt

-Hi) =„y ^fc + ^J -aj mdX-„J ^fc^0

0 0 0 0 L L L L

(3.10)

- J (h® + 12P) Vx dx - k J V dx - k J <pxdx - 2k J Vfx dx. 0 0 0 0 Using the Young and Poincare inequalities, we arrive at

L L L L

-J (lid + l2P) Vx dx <CiJ e2xdx + C2Jp2 dx + 2 y Vl dx, (3.11)

0 0 0 0 L L L

- p2 J ipz (x, 1, t) dx ^ —i J + J z2 (x, 1, t) dx. (3.12)

0 0 0 Substituting (3.11), (3.12) into (3.10), we arrive at (3.14). The proof is complete. □

Now, we define one more functional:

L

J (t) = P2 J Vt^dx, (3.13)

0

where

-l!Ux = cB + dP, u(0) = u(L) = 0.

Lemma 3.2. Let the assumptions of Lemma (3.1) hold true. Then the functional J defined by (3.13) satisfies

L L L L

dJ p2 f ,2 a ( 2 k [ 2 k [ 2

~^~(t) < - -J n™ + 16 VXdx + ^ <PX dx + 4 V dx

0 0 0 0 L L L

k

2 „ „ „

0 0 0

(3.14)

+ — I Vfx dx + C3 92 dx + C4 P2 dx.

Proof. We take the derivative of (4.7) and we get

L L

(t) = J p2Vt^tdx + J p24>ttwdx := Ji(t) + J2 (t). (3.15)

00 Employing (3.15) and Young inequality, we find:

L L

Ji(t) := J PiVwt dx = - —J Vt9-i (k6xx + <yiVxt) dx

0 0

L L L L

p2 j Vtdx - ^ j VtOx dx ^ J Vtdx + C(i) j B2X dx. 0 0 0 0

It follows from (3.15) that

L L

MO :=J ^ «to = / (afe - k fe + + ^ + u

oo

L L L

= - a J uxfix dx - k J (px + fi) u dx + J u (71 Bx + j2Px) dx. o o o

By using Young and Poincare's inequalities, we arrive at

L L L L

-a J uxfixdx < ^J fix dx + C(2) j 9'2xdx + C(3) J P%dx. (3.16)

0 o o o

L L L

1 u (71 Ox + i2Px) dx < C(4) f ex dx + C(5) f P2 dx, (3.17)

and

0

L L L L

—k J ( Px + fi)udx ^^J (px + fi)2dx + C(6) J dXdx + C(7) J Px2dx

0 0 0 0

L L L

k f k f k f

= 8 / P2x dx + 8 / ^2 dx +4 / fiPxdx 000 L L

+ C(6) d2xdx + C(7) Px2 dx (3.18)

X

0 0 L L L

k f k f k f

^ 8 & dx + 4 / fi2 dx + ^ fipxdx

000 L L

+ C(6) j 92xdx + C(7) J P2 dx. 00

Substituting (3.16), (3.17) and (3.18) into (3.15), we obtain (3.14) with

C3 = c(1) + c(2) + c(4) + c(6), C4 = c(3) + c(5) + c(7). The proof is complete. □

Our next step is to define a Lyapunov functional L(t) (t) and prove that it is equivalent to an energy functional E.

Lemma 3.3. Under the assumptions of Lemma 3.1, there exists a constant ß0 > 0 such that

( N -ßo)E(t) ^ L(t) ^ (N + ßo)E(t), for all t^ 0, (3.19)

where L(t) is a Lyapunov functional defined by

L (t) = NE (t) + X (t) + 4^ (t), (3.20)

and N > ß0 is a sufficiently large constant.

Proof. Young, Poincare, and Cauchy-Schwarz inequalities show that

L L L L

mi ^ dx + f/V dx + dx + ^Jp2 dx,

0 0 0 0 L L L

\J(t)\ ^ igdx + Ci d2dx + C2 P2dx.

P2

0 0 0 Hence, there exists a constant ¡0 > 0 such that

|L(t) - NE(t)i = \I(t) + 4^(i)| ^ 30E(t), and this implies estimate (3.19). The proof is complete. □

Theorem 3.2. Let the assumptions of Lemma 3.1 hold. Then there exist positive constants v0,v\ such that the energy functional satisfies

E(t) ^ vE(0)e-V0t for all t > 0. (3.21)

Proof. It follows from (3.7), (3.14) and (3.20) that for each t > 0 we have the inequality

L L

jtL(t) ^ - ( kN -Cx - 4C3) J 92x dx - (hN -C2 - 4C4) J P2 dx

00 L L L

- p2 I ^ dx - ( ^ - ß2°£l\ [ yldx - ^ f ^dx

,2 2 J J rx 4

0

1 1

fk P2C' eA

U )

- ( CN -pi )J ¿dx -(on - J z2 (x, 1, t) dx. 0 0

We choose small enough such that

k > P2C'£I 2 > 2 .

Then we choose N large enough such that

- 4 C3 C2 - 4C4 pi P2

N > sup

[C - 4C3 C2 - 4C4 pi p2 \ \ k ' h ' C' 2C£i J

Then there exists a positive constant c such that

and by using (3.19) it yields

d

-LW < -EM

dL(t) ^ -L(t)

with some positive constant (. Now estimate (3.21) follows by using (3.19) and this completes the proof. □

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Abdelaziz Rahmoune, Department of Mathematics, Faculty of Science,

Laboratory of Pure and Applied Mathematics Amar Telidji University, Laghouat 03000, Algeria E-mail: a.rahmoune@lagh-univ.dz

Oussama Khaldi, Department of Mathematics, Faculty of Science,

Laboratory of Pure and Applied Mathematics Amar Telidji University, Laghouat 03000, Algeria E-mail: oussama.khaldi@lagh-univ.dz

Djamel Ouchenane, Department of Mathematics, Faculty of Science,

Laboratory of Pure and Applied Mathematics Amar Telidji University, Laghouat 03000, Algeria E-mail: d.ouchenane@lagh-univ.dz

Fares Yazid,

Department of Mathematics, Faculty of Science,

Laboratory of Pure and Applied Mathematics Amar Telidji University, Laghouat 03000, Algeria E-mail: f.yazid@lagh-univ.dz