Effects of Saint-Venant type and uniqueness and continuous dependence results for incremental elastodynamics

International Journal of Non-Linear Mechanics 34 (1999) 159 — 167

Effects of Saint-Venant type and uniqueness and continuous dependence results for incremental elastodynamics Stan Chirit,a\ !,* Ludovico Nappa" ! Mathematical Seminarium, University of Ias,i, 6600 Ias,i, Romania "Istituto di Costruzioni, Facolta+ di Architettura, Via Monteoliveto 3, 80134 Napoli, Italy

Abstract This paper establishes a result of Saint—Venant type for the solutions of the initial boundary-value problems of incremental elastodynamics. Such a result is used to study the uniqueness and continuous data dependence of solutions. Both the cases of bounded and unbounded bodies are discussed. One characteristic feature of the uniqueness and continuous dependence results for unbounded bodies is that they are established without any a priori artificial conditions concerning the growth of solutions at infinity. ( 1998 Elsevier Science Ltd. All rights reserved Keywords: Saint-Venant principle; Uniqueness; Continuous dependence; Incremental elastodynamics

1. Introduction The equations governing incremental perturbations in elastodynamics for bounded bodies have been extensively studied by Knops and Wilkes [1] in connection with the stability problem. The Ho¨lder continuous dependence of solutions with respect to incremental body forces and initial data is established by Chirit,a [2] for the initial boundary-value problems of incremental elastodynamics for unbounded bodies. The method of proof is based on a suitable coupling of the logarithmic convexity [1] and the weight function method [3]. Spatial exponential decay of solutions with respect to the axial coordinate has been established

* Corresponding author. Tel: 00 40 32 213041.

by Galdi et al. [4] in the linearised static theory of small elastic deformation superposed upon large ones for a semi-infinite prismatic cylinder under appropriate conditions of growth at infinity. Furthermore, Knops et al. [5] have studied the asymptotic behaviour of solutions for a general class of non-compact regions of arbitrary shape and they have established an alternative growth and decay behaviour (i.e. at least algebraic growth or decay) without a priori asymptotic conditions. The subject of Saint-Venant’s principle has been comprehensively surveyed by Horgan and Knowles [6] and by Horgan [7, 8]. In this paper we establish a principle of SaintVenant-type characterising the spatial behaviour of the solution to the initial boundary-value problems of incremental elastodynamics. We study the incremental deformation of a body for which the

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S. Chirit, a\ , L. Nappa / International Journal of Non-Linear Mechanics 34 (1999) 159–167

primary state is obtained from the reference state by a non-linear motion. This means that we consider the case when the elasticity tensor depends on the spatial as well as the time variable. We discuss both cases of bounded and unbounded bodies. We assume that the incremental body forces and the initial and boundary data of the considered initial boundary-value problem have a bounded support DK on the given time interval [0, ¹], ¹'0. Then T we prove that, for each fixed t3[0, ¹], the spatial behaviour of the corresponding solution as measured by an appropriate weighted integral measure depending on t and on the distance r to the support region DK , is described by the following: (i) the T integral measure vanishes for r*C(t) and (ii) for r)C(t) the integral measure decays spatially with the distance r, the decay rate being controlled by the factor (1!C~1(r)/t), where C(t)'0 is an increasing function determined in terms of the maximum eigenvalue of the elasticity tensor and the mass density. The above result leads to a proof of uniqueness and continuous data dependence theorems, which, for unbounded bodies, are obtained without any artificial a priori asymptotic conditions. Thus, in specialising for the exterior problem and the whole space problem, our results improve or otherwise complement some previous uniqueness and continuous data dependence theorems in Ref. [2].

2. Statement of problem, hypotheses and other preliminaries Throughout this paper we consider an elastic solid occupying the reference configuration in the region B of Euclidean three-dimensional space E3. 0 We assume B to be a properly regular region 0 whose boundary surface is the piecewise smooth surface LB . As usual the motion of the body is 0 referred to the reference configuration and a fixed set of rectangular cartesian axes, relative to which B is at rest. 0 We consider apart from the reference configuration B , two other states: the primary state B and 0 the secondary state B . In what follows, we call * incremental [1] those quantities associated with the difference of motions between the secondary

and primary states. Thus, if the point X in the K reference configuration B moves to x in the pri0 i mary state and to y in the secondary, then i u "y !x are the incremental displacement comi i i ponents. As it is well known [1], the system of governing equations for linearised elasticity referred to the reference configuration B is 0 o u¨ "t #o F in B ][0,R), 0 i Ki,K 0 i 0

(2.1)

t "A u in B ][0,R), Ki iKjL j,L 0

(2.2)

where o (X) (*o '0, o "const.) is the mass 0 1 1 density associated with reference configuration B , 0 t are the incremental first Piola—Kirchhoff stress Ki tensor components measured per unit area in the reference configuration B , F are the components 0 i of the incremental body force per unit mass in the reference configuration B , and, finally, A are 0 iKjL the elasticities which satisfy the symmetry relation A "A . iKjL jLiK

(2.3)

Throughout this paper we shall use the following notation: Latin indices have the range 1, 2, 3; summation over repeated subscripts is implied; subscripts preceded by a comma denote partial differentiation with respect to the corresponding cartesian coordinate X ; and a superposed dot K denotes the time derivative. The system we study in this paper is that of a finitely deformed elastic body whose primary state is a time—dependent configuration. It follows that the elasticities A are functions of time as iKjL well as position [1], viz., A "A (X, t), iKjL iKjL

(2.4)

and are determined explicitly by the finite deformation of the body and the strain energy function. In what follows, we shall assume that the elastic coefficients A (X, t) are continuously differentiable iKjL functions with respect to X and t and bounded on BM ][0,R) together with its first time derivative. 0 Moreover, we assume that A are the compoiKjL nents of a positive definite tensor. Then, it follows that we have k m m )A m m )k m m , ∀m , . iK iK iKjL iK jL M iK iK iK

(2.5)

S. Chirit, a\ , L. Nappa / International Journal of Non-Linear Mechanics 34 (1999) 159–167

where k (X, t)'0 and k (X, t)'0 are the min. M imum and maximum eigenvalues of A (X, t), reiKjL spectively. Further, we note that the above hypotheses upon A imply that there exists iKjL j(t)'0 so that

or

AQ m m )jA m m , iKjL iK jL iKjL iK jL In fact, we can set

c (X, q)O0 for some q3[0, ¹]; i

G K

∀m . iK

(2.6)

KH

1 j(t)"9 max sup AQ (X, t) . (2.7) k (X, t) iKjL M i,j,K,L BÒ . For later convenience we introduce the notation t j*(t)" j(s) ds. (2.8) 0 It should be remarked that when the primary state is a time-independent configuration then we have j(t)"0 and hence j*(t)"0. In this case inequality (2.6) is identically satisfied. On the other hand, by means of relations (2.2) and (2.5), we deduce

P

t t "A u t )(A t t )1@2(A u u )1@2 Ki Ki iKjL j,L Ki iKjL Ki Lj iKjL i,K j,L )(k t t )1@2(A u u )1@2 (2.9) M Ki Ki iKjL i,K j,L and, therefore, we get t t )k A u u . (2.10) Ki Ki M iKjL i,K j,L Along with Eqs. (2.1) and (2.2) we shall assume that the following standard initial and boundary conditions hold: u (X, 0)"a (X), uR (X, 0)"b (X), X3BM , (2.11) i i i i 0 u (X, t)"c (X, t) on & ][0,R), i i 1 t (X, t)N (X)"d (X, t) on & ][0,R), (2.12) Ki K i 2 where &1 X& "LB and & W& "0. Further1 2 0 1 2 more, N are the components of the outward unit K normal vector on & and a (X), b (X), c (X, t) and 2 i i i d (X, t) are prescribed sufficiently smooth functions. i Let us consider a given time interval [0, ¹], ¹'0. We introduce the set DK of all points in T BM so that 0 (i) if X3B , then 0 a (X)O0 or b (X)O0 (2.13) i i

F (X, q)O0 for some q3[0, ¹]; i

161

(2.14)

(ii) if X3& , then 1 (2.15)

and (iii) if X3& , then 2 d (X, t)O0 for some q3[0, ¹]. i

(2.16)

Roughly speaking, DK represents the support of the T initial and boundary data and the incremental body forces. In what follows, we shall assume that DK is a bounded subset of BM . T 0 We choose a non-empty properly regular subset DK * of BM in such a way that DK LDK *LBM . We T 0 T 0 T further specify that, if the support of the data, DK , is T non-empty, then we can choose DK * to be the T smallest regular region in BM which includes DK ; in 0 T particular, DK *"DK if DK also happens to be a T T T regular region. If DK is empty, then DK * may be T T chosen in an arbitrary manner. In what follows, we assume that DK * is chosen in T such a way that it satisfies the above requirements. On this basis, we introduce the set D , defined by r D "MX3BM : DK * W&(X, r)O0N, r 0 T

(2.17)

where &(X, r) is the open ball with radius r and center at X. Further, we note that D is a properly r regular region in BM . Moreover, we denote by 0 R the part of B contained in B CD . Let S be r 0 0 r r the surface of separation between D and R , whose r r unit normal vector is oriented towards the exterior of D . r 3. A principle of Saint-Venant type This section examines a principle of Saint-Venant type for the incremental displacement u which i is a solution of the initial boundary-value problem considered in the above section, under the hypothesis that the data have a bounded support. Our approach, which relies upon the derivation and integration of a first-order partial differential

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162

inequality, is developed in terms of the function t

P0PS e~j (s) tKi uR i NK da ds,

I(r, t)"!

where LB(rL , r) is the boundary surface of the properly regular region B(rL , r) defined by

*

B(rL , r)"R L CR . r r

r

r*0, 0)t)¹,

(3.1)

which is well defined for unbounded bodies on the basis of our hypotheses in the above section. For bounded bodies we set r3[0, ¸] in Eq. (3.1), where ¸"maxMminM[(X !½ )2#(X !½ )2 1 1 2 2 #(X !½ )2]1@2: (½ , ½ , ½ )3DK *N: T 3 3 1 2 3 (X , X , X )3BM N. (3.2) 1 2 3 0 In this last case we can see that relations (2.15), (2.16) and (3.1) imply that I(¸, t)"0, 0)t)¹.

(3.3)

It follows immediately from Eq. (3.1) that L I(r, t)"! Lt

PS

An application of the divergence theorem in Eq. (3.7) combined with the use of relations (2.1) and (2.2) yields

#A

uR u !o F uR ) d» ds. iKjL i,K j,L 0 i i

1 I(r, t)!I(rL , t)"! 2

where

S

l (t) M , o 1

l (t)"supMk (X, t): X3BM N. (3.6) M M 0

Proof. On the basis of the relations (2.12), (2.15)—(2.17), from Eq. (3.1), we deduce that t

P0P©B(r,r)e~j (s)tKi uR i NK da ds,

I(r, t)!I(rL , t)"!

*

L

0)rL )r, 0)t)¹,

(3.7)

*

L

u u ) d» iKjL i,K j,L

PP

(3.4)

u u ) iKjL i,K j,L

#A !AQ

Lemma 3.1. For bounded or unbounded bodies, the function I(r, t) defined by relation (3.1) satisfies the partial differential inequality

c(t)"

PB(r,r)e~j (t)(o0uR iuR i

1 t e~j*(s)[j(o uR uR ! 0 i i 2 0 B(rL ,r)

r

K

(3.9)

An integration by parts in Eq. (3.9) and relation (2.3) and the definitions of DK and D (that is, T r relations (2.11) and (2.13)—(2.17)) imply that

#A

L L I(r, t) #c(t) I(r, t))0, r*0, 0)t)¹, Lt Lr (3.5)

*

L

The partial differential inequality satisfied by I(r, t) is established in the following lemma.

K

t

P0PB(r,r)e~j (s)(o0uR iu¨ i

I(r, t)!I(rL , t)"!

e~j*(t)t uR N da, Ki i K

r*0, 0)t)¹.

(3.8)

u u ] d» ds, iKjL i,K j,L

0)r')r, 0)t)¹.

(3.10)

Thus, by a direct differentiation with respect to r in Eq. (3.10), we get 1 L I(r, t)"! 2 Lr

PS e~j (t)(o0uR iuR i#AiKjLui,Kuj,L) da *

r

1 t ! e~j*(s)[j(o uR uR #A u u ) 0 i i iKjL i,K j,L 2 0 Sr

PP

!AQ

u u ] da ds, iKjL i,K j,L

r*0, 0)t)¹.

(3.11)

On the other hand, by an application of the Schwarz and arithmetic—geometric mean inequalities in conjunction with relation (2.10), from

S. Chirit, a\ , L. Nappa / International Journal of Non-Linear Mechanics 34 (1999) 159–167

while Eq. (3.16) gives

Eq. (3.4), we get

K

K P

L 1 I(r, t) ) Lt 2 1 ) 2

PS

r

∀e'0.

A

B

1 e~j*(t) euR uR # t t da i i e Ki Ki

Sr

A

e~j*(t)

B

e k o uR uR # MA u u da, 0 i i o e iKjL i,K j,L 1 (3.12)

By choosing e"(o k )1@2 and by using relations 1 M (3.6) and (3.11), we deduce

K

163

K

G

L L I(r, t) )!c(t) I(r, t) Lt Lr

A P

B

t I(r , t )*I r # Òc(q) dq, 0 "0. 0 0 0 0

(3.19)

If we make r to tend to infinity in Eqs. (3.18) and 0 (3.19), we deduce that relation (3.14) holds true and the proof is complete. h Corollary 3.2. For bounded or unbounded bodies, the function I(r, t) is positive for all r*0 and t3[0, ¹].

1 t e~j*(s)[j(o uR uR #A u u ) # 0 i i iKjL i,K j,L 2 0 Sr

Proof. If we set r"¸ for bounded bodies or we make rPR for unbounded bodies, from relations (3.3), (3.10) and (3.14), we obtain

!AQ

I(r, t)"J(r, t)

PP

H

u u ] da ds , iKjL i,K j,L

(3.13)

so that, in view of relations (2.6) and (2.7), we conclude that the relation (3.5) holds true and the proof is complete. h

CA P

BD

(3.15)

CA P

BD

(3.16)

t d I r # c(q) dq, t 0 dt tÒ and t d I r ! c(q) dq, t 0 dt tÒ We further choose

)0, 0)t)¹

*0, 0)t)¹.

t r * Òc(q) dq. (3.17) 0 0 Since t *0, from Eqs. (3.1) and (3.15), we deduce 0 that

P

A P

B

t I(r , t ))I r ! Òc(q) dq, 0 "0, 0 0 0 0

(3.18)

(3.20)

where

PR e~j (t)(o0uR iuR i#AiKjLui,Kuj,L) d» 1 t e~j (s)[j(o uR uR #A u u ) # 0 i i iKjL i,K j,L 2P P 0 R

1 J(r, t)" 2

*

r

*

Corollary 3.1. For unbounded bodies, the function I(r, t) defined by relation (3.1) satisfies the relation lim I(r, t)"0, 0)t)¹. (3.14) r?= Proof. Let us consider r *0 and t 3[0, ¹]. Then 0 0 relation (3.5) implies

for r*0, 0)t)¹,

r

!AQ r*0,

u u ] d» ds, iKjL i,K j,L

0)t)¹.

(3.21)

To obtain Eq. (3.20) we have used the fact that, by means of Eq. (3.8), we have B(r, ¸)"R for r bounded bodies and B(r,R)"R for unbounded r bodies. In view of relations (2.6) and (2.7) we deduce that J(r, t) is a positive function and, therefore, by means of relation (3.20), I(r, t) is a positive function for r*0 and t3[0, ¹]. h Theorem 3.1. Principle of Saint-»enant type. For bounded or unbounded bodies and for each fixed t3[0, ¹], we have I(r, t)"0 for r*C(t)

(3.22)

and

A

B

C~1(r) I*(r, t)) 1! I*(0, t) for r)C(t), (3.23) r

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164

in the last integral in Eq. (3.31) so that we get

where t I*(r, t)" I(r, q) dq, 0

(3.24)

t C(t)" c(s) ds. 0

(3.25)

P

P

Proof. In view of the Lemma 3.1, from Eq. (3.5) we get L L I(r, t)#c(t) I(r, t))0, r*0, 0)t)¹, Lt Lr (3.26) and hence, with Eq. (3.25), we have d [I(C(t), t)])0, 0)t)¹. dt

(3.27)

Then, for any t3[0, ¹], from Eq. (3.27) we get I(C(t), t))I(0, 0)"0.

(3.28)

On the other hand, by means of relations (2.5)—(2.7) and (3.11) it results that for each fixed t3[0, ¹], I(r, t) is a non—increasing function with respect to r. Thus, we have I(r, t))I(C(t), t) for r*C(t).

(3.29)

From relations (3.28) and (3.29) and by using the Corollary 3.2, we conclude that relation (3.22) holds true. Let us consider now t 3[0, ¹], and r *0 such 1 1 that r )C(t ). If we define t* by 1 1 1 t*"C~1(r ), 1 1

(3.30)

it follows that the inequality r )C(t ) implies that 1 1 t*)t . Then relations (3.22) and (3.24) give 1 1 t t I*(r , t )" ÇI(r , q) dq" ÇI(r , q) dq. 1 1 1 1 0 t*1

P

P

(3.31)

Further, we make the change of variables

A

t* q(s)" 1! 1 t 1

B

s#t*, 1

(3.32)

A

BP

tÇ t* I*(r , t )" 1! 1 I(r , q(s)) ds. (3.33) 1 1 1 t 0 1 Now, we take r "r and t "q in relation (3.15) 0 1 0 and we note that s)q, so that we obtain

A P

B

s I(r , q))I r # c(m) dm, s . 1 1 q Since

(3.34)

s (3.35) r # c(m) dm*0 1 q and I(r, t) is a non-increasing function with respect to r, it follows that

P

A P

B

s I r # c(m) dm, s )I(0, s), 0)s)t . (3.36) 1 1 q Thus, relations (3.24), (3.33), (3.34) and (3.36) imply that relation (3.23) holds true and the proof is complete. h Remark 3.1. In view of relation (3.20), it can be seen that the above theorem can be formulated in terms of the volume measure J(r, t) as defined by Eq. (3.21). Remark 3.2. ¹he above analysis becomes more transparent when the primary state is a time-independent configuration. In that case we have to set c(t)"c"const., C(t)"ct and j*(t)"0. Remark 3.3. ¹he surface integral measure (3.1) was motivated by similar quantities used by Flavin et al. [9] and Chirit, a\ and Quintanilla [10] for classical elastodynamics.

4. A uniqueness result Theorem 3.1 leads to a uniqueness result for the solutions of the initial boundary-value problems in incremental elastodynamics. Theorem 4.1. For bounded or unbounded bodies, the initial boundary-value problem of incremental

S. Chirit, a\ , L. Nappa / International Journal of Non-Linear Mechanics 34 (1999) 159–167

elastodynamics, as defined by relations (2.1), (2.2), (2.11) and (2.12), has at most one solution. Proof. It is easy to see that the difference of two solutions corresponding to the same incremental body forces and satisfying the same initial and boundary conditions is a solution of system (2.1), (2.2), (2.11) and (2.12) with vanishing data. Therefore, the support of these data is empty for each ¹3[0,R), and in this case the non-empty set DK * can be chosen in an arbitrary manner. Thus we T choose the non-empty set DK * to be a subset of T LB and in this way we have I*(0, t)"0 for all 0 t3[0, ¹]. Further, relations (3.20)—(3.23) and the vanishing initial conditions imply that u (X, t)"0 for all X3BM and t3[0, ¹]. (4.1) i 0 The uniqueness result follows now from relation (4.1) and the fact that ¹'0 can be chosen in an arbitrary manner. h Remark 4.1. ¹he above uniqueness result remains valid for the case when the region B is the whole 0 three-dimensional space. In order to prove this we choose the set DK * in an arbitrary manner in the T three-dimensional space, and then we use the procedure described in the above section to obtain for the difference of two solutions a relation of type (3.22). On this basis and by using relations (2.1), (2.2) and (2.11) with vanishing data, we deduce that 1 2

5. Continuous data dependence results We assume in this section that & "0 and so we 1 shall consider the initial traction boundary-value problem as defined by relations (2.1), (2.2), (2.11) and t (X, t)N (X)"d (X, t) on LB x[0,R). (5.1) Ki K i 0 We further assume that the given data F (X, t), i d (X, t), a (X) and b (X) are sufficiently regular and i i i that they have a bounded support DK on the time T interval [0, ¹], ¹'0. Moreover, we assume that they satisfy the relations

PBÒo0 ai d»"0, PBÒ o0 bi d»"0

(5.2)

and

PBÒo0Fi d»#P©BÒdi da"0,

0)t)¹.

(5.3)

We note that, according to our assumptions in the above, all integrals occurring in Eqs. (5.2) and (5.3) are well-defined even in the case when B is un0 bounded. By an integration of Eq. (2.1) over B and by 0 taking into account result (3.22) and relations (2.11), (5.1), (5.2) and (5.3), we deduce that

PBÒ o0ui d»"0,

0)t)¹.

(5.4)

On the basis of relation (5.4) we can make use of Poincare´’s inequality and the trace theorem [11] in order to deduce that

P

e~j*(t)(o uR uR #A u u ) d» 0 i i iKjL i,K j,L BÒ 1 t e~j*(s)[j(o uR uR #A u u ) # 0 i i iKjL i,K j,L 2 0 BÒ !AQ u u ] d» ds"0 iKjL i,K j,L

165

PP

(4.2)

and hence u (X, t)"0 for all X3BM and t3[0,R). i 0 Remark 4.2. It can be seen from the above analysis that for unbounded bodies the uniqueness result is established without any kind of a priori artificial conditions concerning the growth of solutions at infinity.

P©BÒuiui da)"PBÒui,Kui,K d»,

(5.5)

P©BÒuiui da)lPBÒAiKjLui,Kuj,L d»,

(5.6)

where " is a positive parameter depending on D . C(T) On combining relations (2.5) and (5.5) we obtain

where

G NC

l"max " *0,T+

inf k (X, t) . BM Ò

DH

.

(5.7)

S. Chirit, a\ , L. Nappa / International Journal of Non-Linear Mechanics 34 (1999) 159–167

166

PBÒe~j (t)(o0uR iuR i#AiKjLui,Kuj,L) d» t e~j (s)[j(o uR uR #A u u ) #1 2P P 0 i i iKjL i,K j,L 0 BÒ

#

*

¼(t)"1 2

*

!AQ u u ] d» ds, 0)t)¹. iKjL i,K j,L

(5.8)

P0¼(s) ds)2t C¼(0)!P B aidi(0) daD t #2t2 P0PB o0e~j (s)FiFi d» ds t 1@2 #2l e~j (s)d d da ds i i GAP0P B B t 1@2 2 #t PC 0P B e~j (s)(jdi!dQ i) (jdi!dQ i) da dsD H , t

0

*

0

*

L

0

*

L

0

0)t)¹.

(5.9)

Proof. On the basis of Eqs. (2.1), (2.2) and (5.1) and by using Theorem 3.1, we deduce the identity t

P0PBÒo0 e~j (s)Fi uR i d» ds

¼(t)"¼(0)# t

*

P0P BÒe~j (s)uR idi da ds.

#

*

D

t ¼(s) ds"t ¼(0)! u (0)d (0) da i i 0 ©BÒ t s o e~j*(q)F uR d» dq ds # 0 i i 0 0 BÒ

PPP

By means of the Cauchy—Schwarz inequality and the arithmetic—geometric mean inequality and by using estimate (5.6), we get

P0¼(s) ds)tC¼(0)!P©BÒaidi(0) daD t2 t o e~j (s)F F d» ds # i i 2b P P 0 0 BÒ 1 t e~j (s)[bo uR uR #alA u u ] d» ds # 0 i i iKjL i,K j,L 2P P 0 BÒ t 1@2 1 # e~j (s)d d da ds i i 2a GAP P B 0 ©BÒ t 1@2 2 #t CP0P©BÒe~j (s)(jdi!dQ i) (jdi!dQ i ) da dsD H , *

*

*

*

0)t)¹,

(5.12)

where a, b are positive parameters. Choosing b"al"1 , from Eq. (5.12) we deduce estimate (5.9) 2 and the proof is complete. h Remark 5.1. An estimate analogous to Eq. (5.9) can be established in a similar manner for the case when B is the whole space. 0 Remark 5.2. It is worth noting that the continuous dependence estimate (5.9) is established for infinite bodies without any restriction on the growth of solutions at infinity. ¹his was possible because of the results described by relations (3.20)—(3.22) and this is based, in fact, upon the hypothesis that the external data have a bounded support.

t

P

(5.11)

Remark 5.3. Relations (3.21) and (5.8) imply that

Further, we get

C

*

(5.10)

L

P

*

t

Theorem 5.1. ¸et u be a solution of the initial i boundary-value problem (2.1), (2.2), (2.11) and (5.1) associated with the (bounded or unbounded) regular region B and the system of external data 0 MF , a , b , d N having a bounded support and satisfyi i i i ing relations (5.2) and (5.3). ¹hen continuous dependence upon the external data is described by the following estimate:

L

t s

P0P0 P©BÒe~j (q)ui(jdi!dQ i) da dq ds t # P0P©BÒe~j (s)uidi da ds, 0)t)¹.

For later convenience we introduce the notation

t

P0J(0, s) ds)P0¼(s) ds,

0)t)¹.

(5.13)

On this basis and by using estimate (5.9), we conclude that the solutions depend continuously on the external data in the measure provided by I*(r, t).

S. Chirit, a\ , L. Nappa / International Journal of Non-Linear Mechanics 34 (1999) 159–167

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