Uniform stability for thermoelastic systems with boundary time-varying delay

J. Math. Anal. Appl. 383 (2011) 490–498

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Uniform stability for thermoelastic systems with boundary time-varying delay Muhammad I. Mustafa King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, P.O. Box 860, Dhahran 31261, Saudi Arabia

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Article history: Received 27 January 2011 Available online 30 May 2011 Submitted by M. Nakao

In this paper we consider a thermoelastic system with boundary time-varying delay. Using the energy method, we show, under suitable assumptions, that the damping effect through heat conduction is still strong enough to uniformly stabilize the system even in the presence of boundary time-varying delay. Our result improves earlier results existing in the literature. © 2011 Elsevier Inc. All rights reserved.

Keywords: Thermoelasticity Time-varying delay Stability Exponential decay

1. Introduction In this paper we are concerned with the following problem

⎧ autt (x, t ) − du xx (x, t ) + βθx (x, t ) = 0, ⎪ ⎪ ⎪ ⎪ bθt (x, t ) − kθxx (x, t ) + β u xt (x, t ) = 0, ⎪ ⎪ ⎪ ⎨ u (0, t ) = u ( L , t ) = θ(0, t ) = 0, x   ( L , t ) + k1 θ( L , t ) + k2 θ L , t − τ (t ) = 0, θ ⎪ x ⎪ ⎪ ⎪ ⎪ u (x, 0) = u 0 (x), ut (x, 0) = u 1 (x), θ(x, 0) = θ0 (x), ⎪ ⎪ ⎩ θ( L , −t ) = f 0 (t ),

in (0, L ) × (0, ∞), in (0, L ) × (0, ∞), t  0,

(1.1)

t  0, x ∈ (0, L ),





t ∈ 0, τ (0)

a thermoelastic system with boundary time-varying delay associated with initial data u 0 , u 1 , θ0 and history function f 0 in suitable function spaces. Here, u is the displacement, θ is the temperature difference from the reference value, a, b, d, k, β are positive constants, k1 and k2 are nonnegative constants, and the time-varying delay τ is a positive bounded differentiable function of t, with τ  (t ) < 1. This implies that (t − τ (t ) > −τ (0)) which justifies the domain assigned in (1.1) for the history function f 0 . We study the asymptotic behavior for the solution of (1.1) and look for sufficient conditions that guarantee the uniform stability of this system. Time delays arise in many applications because, in most instances, physical, chemical, biological, thermal, and economic phenomena naturally depend not only on the present state but also on some past occurrences. In recent years, the control of PDEs with time delay effects has become an active area of research, see for example [1,26], and the references therein. In many cases it was shown that delay is a source of instability and even an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable in the absence of delay unless additional conditions or control terms have been used. The stability issue of systems with delay is, therefore, of theoretical and practical importance.

E-mail address: [email protected]. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.05.066

©

2011 Elsevier Inc. All rights reserved.

M.I. Mustafa / J. Math. Anal. Appl. 383 (2011) 490–498

491

In the absence of delay, it is well known (see [3,9,18]) that the one-dimensional linear thermoelastic system associated with various types of boundary conditions decays to zero exponentially. For the multi-dimensional case, we have the pioneering work of Dafermos [5], in which he proved an asymptotic stability result; but no rate of decay has been given. The uniform rate of decay for the solution in two- or three-dimensional space was obtained by Jiang, Rivera and Racke [12] in special situation like radial symmetry. Lebeau and Zuazua [16] proved that the decay rate is never uniform when the domain is convex. Thus, in order to solve this problem, additional damping mechanisms are necessary. In this aspect, Pereira and Menzala [25] introduced a linear internal damping effective in the whole domain, and established the uniform decay rate. A similar result was obtained by Liu [17] for a linear boundary velocity feedback acting on the elastic component of the system, and by Liu and Zuazua [19] for a nonlinear boundary feedback. Oliveira and Charao [24] improved the result in [25] by including a weak localized dissipative term effective only in a neighborhood of part of the boundary and proved exponential decay rate when the damping term is linear and polynomial decay rate for nonlinear damping term. For more literature on the subject, we refer the reader to books by Jiang and Racke [11] and Zheng [29]. Introducing a delay term in the boundary feedback of the thermoelastic system makes the problem different from that considered in the literature. The presence of a delay term may turn such well-behaved system into a wild one. In [6,7], the authors showed that a small delay in a boundary control of certain hyperbolic systems could be a source of instability, and to stabilize these systems involving input delay terms, additional control terms will be necessary. In this aspect, Datko, Lagnese, and Polis [7] examined the following problem

utt − u xx + 2aut + a2 u = 0, u (0, t ) = 0, with a, k,

in (0, 1) × (0, ∞),

u x (1, t ) = −kut (1, t − τ ),

t>0

τ positive real numbers. Through a careful spectral analysis, they have shown that, for any a > 0, if k satisfies

0
1 − e −2a 1 + e −2a

then the spectrum of this system lies in Re w  −β , where β is a positive constant depending on the delay τ , and the uniform stability of the system is obtained. Regarding the following system of wave equation with a linear boundary damping term and a boundary time-varying delay

⎧ in Ω × (0, ∞), ⎨ utt − u = 0, u (x, t ) = 0, x ∈ Γ0 , t > 0,   ⎩ ∂u ∂ v (x, t ) = −a0 ut (x, t ) − aut x, t − τ (t ) , x ∈ Γ1 , t > 0

(1.2)

it is well known, in the absence of delay (a = 0, a0 > 0), that this system is exponentially stable, see [13–15,30]. In the presence of delay (a > 0), Nicaise, Pignotti, and Valein [22] examined system (1.2), with positive bounded function τ (t ), and proved under the assumptions

τ  (t )  q < 1,

a<



1 − qa0

(1.3)

that system (1.2) is exponentially stable. This generalizes the results obtained in [20] for the wave equation with boundary or internal constant time delay. Some instability results were also given in [20] if the second inequality in (1.3) is not satisfied. See also [2] for treatment to this problem in more general abstract form. We also recall the result by Yung, Xu, and Li [28], where the authors proved the same result as in [20] for the one space dimension by adopting the spectral analysis approach. If the boundary condition on Γ1 in (1.2) is replaced by

∂u (x, t ) = −a0 ut (x, t ) − ∂v

τ2 a(s)ut (x, t − s) ds,

x ∈ Γ1 , t > 0

τ1

using distributed delay, an exponential stability result has been obtained in [21] under the condition

τ2 a(s) ds < a0 . τ1

The heat equation with internal constant, distributed, or time-varying delay was also studied in [4,8,10,27] via Lyapunov method. For boundary delay, the authors in [23] considered

⎧ in (0, π ) × (0, ∞), ⎨ ut − u xx = 0, u (0, t ) = 0, t > 0,   ⎩ u x (π , t ) = −a0 u (π , t ) − au π , t − τ (t ) , t > 0

and established, under the same assumptions (1.3), an exponential decay result.

(1.4)

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M.I. Mustafa / J. Math. Anal. Appl. 383 (2011) 490–498

Our aim in this work is to investigate (1.1) and establish exponential decay result under suitable assumptions (see (2.4)) on the delay term that are weaker than those in (1.3). In fact, we show, even if k1 = 0, that we still have uniform stability √ 1−q

provided k2 < L , where q is such that τ  (t )  q < 1. Our result is evidently applicable to system (1.4), and so it improves the result obtained in [23]. The presence of time-varying delay term brings some additional difficulties in the proof. We first need to specify our assumptions and suitably define the modified energy that possesses a dissipative nature under these assumptions. Then, to establish our exponential decay result, our main idea is to construct a Lyapunov functional L equivalent to the modified energy E satisfying

a1 E (t )  L(t )  a2 E (t ) for two positive constants a1 , a2 and

L (t )  −α L(t ) for some α > 0. A careful choice of the multipliers and the sequence of estimates in the energy method will give the desired result. The paper is organized as follows. In Section 2, we present our assumptions and treat the well-posedness issue. Then, in Section 3, we state and prove our main result. 2. The well-posedness of the problem Let us introduce the following new variable





z(ρ , t ) = θ L , t − τ (t )ρ ,

(ρ , t ) ∈ (0, 1) × (0, ∞).

Then, problem (1.1) is equivalent to

⎧ au (x, t ) − du (x, t ) + βθ (x, t ) = 0, tt xx x ⎪ ⎪ ⎪ ⎪ bθt (x, t ) − kθxx (x, t ) + β u xt (x, t ) = 0, ⎪ ⎪   ⎪  ⎪ ⎪ ⎨ τ (t ) zt (ρ , t ) + 1 − τ (t )ρ zρ (ρ , t ) = 0, u x (0, t ) = u ( L , t ) = θ(0, t ) = 0, ⎪ ⎪ ⎪ θx ( L , t ) + k1 θ( L , t ) + k2 z(1, t ) = 0, z(0, t ) = θ( L , t ), ⎪ ⎪ ⎪ ⎪ u (x, 0) = u 0 (x), ut (x, 0) = u 1 (x), θ(x, 0) = θ0 (x), ⎪ ⎪   ⎩ z( p , 0) = f 0 τ (0)ρ ,

in (0, L ) × (0, ∞), in (0, L ) × (0, ∞), in (0, 1) × (0, ∞), t  0,

(2.1)

t  0, x ∈ (0, L ),

ρ ∈ (0, 1).

We consider the following Hilbert spaces





V 1 = v ∈ H 1 (0, L ): v ( L ) = 0 ,







V 2 = v ∈ H 1 (0, L ): v (0) = 0 ,



V 3 = v ∈ H 2 (0, L ): v x (0) = v ( L ) = 0 . Then, with U = (u , v , θ, z), where v = ut , system (2.1) can be written in the following form



U t + A (t )U = 0, t > 0, U (0) = U 0 = (u 0 , u 1 , θ0 , f 0 ),

(2.2)

where, with the Hilbert space H = V 1 × L 2 (0, L ) × L 2 (0, L ) × L 2 (0, 1), the time-dependent operator A (t ) : D ( A (t )) → H is given by

β β d k 1 − τ  (t )ρ A (t )U = − v , − u xx + θx , − θxx + v x , zρ a a b b τ (t )



with domain











D A (t ) = (u , v , θ, z) ∈ V 3 × V 1 × H 2 (0, L ) ∩ V 2 × H 1 (0, 1): θ( L ) = z(0), θx ( L ) + k1 θ( L ) + k2 z(1) = 0



which is independent of the time t, i.e. D ( A (t )) = D ( A (0)) for all t > 0. We assume, for some constants ( K , M , R , q) and all t > 0,



0 < K  τ (t )  M ,

τ  (t )  q < 1 and

k2 <



1 − q k1 +

1 L

   τ (t )  R ,

 .

(2.3)

(2.4)

M.I. Mustafa / J. Math. Anal. Appl. 383 (2011) 490–498

493

Let ξ be a fixed positive constant satisfying



max k1 , √



k2

1

< ξ < k1 + .

1−q

(2.5)

L

Then, for U = (u , v , θ, z), V = (u , v , θ, z) ∈ H , we equip H with the time-dependent inner product

L

1

(U , V )t =

(du x u x + av v + bθθ) dx + kξ τ (t ) 0

z(ρ ) z(ρ ) dρ . 0

The existence and uniqueness result reads as follows. Theorem 2.1. For any U 0 ∈ H , problem (2.2) has a unique solution U ∈ C ([0, +∞); H ). Moreover, if U 0 ∈ D ( A (0)), then







U ∈ C [0, +∞); D A (0)

  ∩ C 1 [0, +∞); H .

Proof. By making the transformation R

U = e 2K t U , we need to look at the problem



U t + A (t )U = 0,

(2.6)

U (0) = U 0 where

A (t ) = A (t ) +

R 2K

I

with the same domain of A (t ). Using the semigroup method, the well-posedness of system (2.6) can be established by following similar steps as those used in [23] provided we prove, for a fixed t, that the operator A (t ) is monotone under the weaker assumption (2.4). For this purpose, we find, for any U ∈ D ( A (t )), that



A (t )U , U

 t

L

1 θx2 dx + kk1 θ 2 ( L ) + kk2 θ( L ) z(1) + kξ

=k 0



1 − τ  (t )ρ z(ρ ) zρ (ρ ) dρ +

R 2K

(U , U )t

0

L θx2 dx + kk1 θ 2 ( L ) + kk2 θ( L ) z(1) +

=k



kξ  2



1 − τ  (t ) z2 (1)

0

−

kξ 2

2

θ (L) +

kξ τ  (t )

1 z2 (ρ ) dρ +

2

R 2K

(U , U )t .

0

Notice that

τ  (t ) kξ τ (t ) 2τ (t )

1 z2 (ρ ) dρ +

R 2K

 (U , U )t 

0

R 2K

−

 1 |τ  (t )| kξ τ (t ) z2 (ρ ) dρ  0. 2τ (t ) 0

Also, using Hölder and Young’s inequalities gives

L θx2 dx



1

 L

θx dx

L

0

2

1

= θ 2 (L)

(2.7)

L

0

and

1 2 2 ξ k2 θ( L ) z(1)  − θ 2 ( L ) − k z (1). 2 2ξ 2 Combining all the above, we deduce that



A (t )U , U

 t

    k22 1 k(1 − q) 2 ξ −  k k1 + − ξ θ 2 ( L ) + z2 (1) L 2ξ (1 − q)

which, in view of (2.5), implies that ( A (t )U , U )t  0. Therefore, A is monotone.

2

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M.I. Mustafa / J. Math. Anal. Appl. 383 (2011) 490–498

3. Exponential decay In this section we state and prove our main result. For this purpose we establish several lemmas. We use c, throughout this section, to denote a generic positive constant. Lemma 3.1. Let (u , θ, z) be the solution of (2.1). Then the energy functional, defined by

E (t ) =

1

L





aut2 + du 2x + bθ 2 dx +

2

1

kξ

z2 (ρ , t ) dρ ,

τ (t )

2

0

(3.1)

0

satisfies, for some positive constant m,



 L



θx2 dx + z2 (1, t )

E (t )  −m

(3.2)

.

0

Proof. Using Eqs. (2.1) and integrating by parts yield

L



E (t ) =





ut (du xx − βθx ) + du x u xt + θ(kθxx − β u xt ) dx

0

kξ + τ  (t ) 2

1

1 2

z (ρ , t ) dρ − kξ 0





1 − τ  (t )ρ z(ρ , t ) zρ (ρ , t ) dρ

0

L

kξ 

θx2 dx − kk1 θ 2 ( L , t ) − kk2 θ( L , t ) z(1, t ) −

= −k

2



1 − τ  (t ) z2 (1, t ) +

kξ 2

θ 2 ( L , t ).

0

By (2.5) and (2.7), we find that

L

μ := L [ξ − k1 ] satisfies 0 < μ < 1 and L

θx2 dx = −k(1 − μ)

−k 0

L θx2 dx − kμ

0

θx2 dx 0

L

L θx2 dx − kL [ξ

= −k(1 − μ)

θx2 dx

− k1 ]

0

0

L θx2 dx − k[ξ − k1 ]θ 2 ( L , t ).

 −k(1 − μ) 0

This leads to

E  (t )  −k(1 − μ)

L θx2 dx −

kξ 2

θ 2 ( L , t ) − kk2 θ( L , t ) z(1, t ) −

kξ 2

(1 − q) z2 (1, t )

0

L θx2 dx −

= −k(1 − μ)

kξ

 θ( L , t ) +

2

k2

ξ

2 −

z(1, t )

k(1 − q) 2ξ

0

L θx2 dx −

 −k(1 − μ)

k(1 − q) 2ξ

0

Hence, using (2.5), our conclusion holds.

2

 ξ2 −

k22

(1 − q)

 z2 (1, t ).

 ξ2 −

k22

(1 − q)

 z2 (1, t )

M.I. Mustafa / J. Math. Anal. Appl. 383 (2011) 490–498

495

Now we are going to construct a Lyapunov functional L equivalent to E, with which we can show the desired result. Lemma 3.2. Let (u , θ, z) be the solution of (2.1). Then, the functional F 1 , defined by

L F 1 (t ) := a

uut dx, 0

satisfies, for some positive constant m0 , the estimate

L



L u 2x dx + a

F 1 (t )  −m0 0

L ut2 dx + c

0

θx2 dx.

(3.3)

0

Proof. Direct computations, using (2.1), yield

F 1 (t ) = a

L

L ut2 dx +

0

L

0

L ut2 dx − d

u (du xx − βθx ) dx = a 0

L u 2x dx − β

0

u θx dx. 0

Using Young’s and Poincaré’s inequalities, we find that



L

L ut2 dx − d

F 1 (t )  a 0

By choosing

L u 2x dx +

ε

0

L u 2x dx +

θx2 dx.

Cε

0

0

ε small enough, (3.3) is established. 2

Lemma 3.3. Let (u , θ, z) be the solution of (2.1). Then, the functional F 2 , defined by

 L  x F 2 (t ) := −ab θ ut (s, t ) ds dx, 0

0

satisfies, for some positive constant m1 , the estimate

L



L ut2 dx + δ

F 2 (t )  −m1 0

L u 2x dx +

θx2 dx + cz2 (1, t )

Cδ

0

(3.4)

0

for any 0 < δ < 1. Proof. By exploiting Eqs. (2.1) and integrating by parts, we have

 x

L



F 2 (t ) = −a

(kθxx − β u xt ) 0

 L  x ut (s, t ) ds dx − b θ (du xx − βθx ) ds dx 

0

L

0

ut2 dx + akk1 θ( L , t )

= −aβ

0

L

0

L ut dx + akk2 z(1, t )

0

L ut dx + ak

0

L θx ut dx − bd

0

L θ 2 dx.

θ u x dx + bβ 0

0

Then, the use of Young’s inequality gives



L

F 2 (t )  −aβ 0

Next, we choose

L ut2 dx +

ε 0



L ut2 dx + δ

u 2x dx + 0



L 2

2

θx2 dx

C ε θ ( L , t ) + z (1, t ) + 0

ε small enough and use (2.7) and Poincaré’s inequality to get (3.4). 2

L

L 2

θ 2 dx.

θ dx + bβ

+ Cδ 0

0

496

M.I. Mustafa / J. Math. Anal. Appl. 383 (2011) 490–498

Lemma 3.4. Let (u , θ, z) be the solution of (2.1). Then, the functional F 3 , defined by

1 F 3 (t ) := τ (t )

e −τ (t )ρ z2 (ρ , t ) dρ ,

0

satisfies, for some positive constant γ0 , the estimate

1



L 2

F 3 (t )  −γ0 τ (t )

θx2 dx.

z (ρ , t ) dρ + c 0

(3.5)

0

Proof. We differentiate with respect to t to find that

F 3 (t ) = τ  (t )

1

e −τ (t )ρ z2 (ρ , t ) dρ − τ (t )τ  (t )

0

1

ρ e−τ (t )ρ z2 (ρ , t ) dρ + 2τ (t )

1

0

e −τ (t )ρ z(ρ , t ) zt (ρ , t ) dρ .

0

Using the third equation of (2.1) and integrating by parts give

1 2τ (t )

e

−τ (t )ρ

1 z(ρ , t ) zt (ρ , t ) dρ = −2

0





1 − τ  (t )ρ e −τ (t )ρ z(ρ , t ) zρ (ρ , t ) dρ

0

1 =−





1 − τ  (t )ρ e −τ (t )ρ

∂ 2 z (ρ , t ) dρ ∂ρ

0



 = θ ( L , t ) − 1 − τ (t ) e −τ (t ) z2 (1, t ) − τ  (t ) 2



1

e −τ (t )ρ z2 (ρ , t ) dρ

0



1

+ τ (t )τ (t )

ρe

−τ (t )ρ 2

1

z (ρ , t ) dρ − τ (t )

0

e −τ (t )ρ z2 (ρ , t ) dρ .

0

Therefore, we obtain

 F 3 (t ) = θ ( L , t ) − 1 − τ (t ) e −τ (t ) z2 (1, t ) − τ (t ) 



2



1

e −τ (t )ρ z2 (ρ , t ) dρ .

0

Then, we make use of (2.3) and (2.7) to infer 

L

F 3 (t )  L

θx2 dx − e − M

1

0

which gives (3.5).

z2 (ρ , t ) dρ

τ (t ) 0

2

We are now ready to state and prove our main result. Theorem 3.5. Assume that (2.3) and (2.4) hold and (u , θ, z) is the solution of (2.1). Then, there exist positive constants c 0 , c 1 such that the energy functional satisfies

E (t )  c 0 e −c1 t . Proof. For N 1 , N 2 > 1, let

L(t ) := N 1 E (t ) + F 1 (t ) + N 2 F 2 (t ) + F 3 (t ). By combining (3.2)–(3.5), we obtain

(3.6)

M.I. Mustafa / J. Math. Anal. Appl. 383 (2011) 490–498

L (t )  −(m1 N 2 − a)

L

L ut2 dx − (m0 − δ N 2 )

0

497

L u 2x dx − (mN 1 − 2c − C δ N 2 )

0

θx2 dx 0

1 z2 (ρ , t ) dρ − (mN 1 − cN 2 ) z2 (1, t ).

− γ0 τ (t ) 0

At this point, we choose N 2 large enough so that

γ1 := (m1 N 2 − a) > 0, then δ small enough so that

γ2 := (m0 − δ N 2 ) > 0. Next, we choose N 1 large enough so that

γ3 := (mN 1 − 2c − C δ N 2 ) > 0 and

(mN 1 − cN 2 ) > 0. So, we arrive at

L (t )  −γ1

L

L ut2 dx − γ2

0

L u 2x dx − γ3

0

1 θx2 dx − γ0 τ (t )

0

z2 (ρ , t ) dρ 0

which, using Poincaré’s inequality, yields

L (t )  −c  E (t )

(3.7)

for some constant c  > 0. On the other hand, we find that

        L(t ) − N 1 E (t )   F 1 (t ) + N 2  F 2 (t ) +  F 3 (t )  L L   x 1     a |uut | dx + abN 2 θ ut (s, t ) ds  dx + τ (t ) e −τ (t )ρ z2 (ρ , t ) dρ   0

L c

0





0

0

1

u 2x + ut2 + θ 2 dx + τ (t )

0

z2 (ρ , t ) dρ 0

 c E (t ). Therefore, we can choose N 1 even larger (if needed) so that

L(t ) ∼ E (t ).

(3.8)

Hence, (3.7) and (3.8) lead to

L (t )  −c  L(t ). A simple integration on (0, t ), then another use of (3.8) give (3.6).

2

Remarks. (1) For the special case of a constant delay, τ (t ) ≡ α and nentially stable under the unique condition

k2 < k1 +

1 L

α is a positive constant, we conclude that system (1.1) is expo-

.

(2) It is evident that the conditions (2.3) and (2.4) are sufficient to guarantee the uniform stability of system (1.4) studied in [23]. Our assumption (2.4) is weaker than the assumption

k2 <



1 − qk1

imposed in [23]. This extends the stability region of the system.

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M.I. Mustafa / J. Math. Anal. Appl. 383 (2011) 490–498

Acknowledgments The author thanks King Fahd University of Petroleum and Minerals (KFUPM) for its continuous support. This work has been funded by KFUPM under Project # IN 101002.

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