Existence and Uniqueness of Solutions to the

Existence and Uniqueness of Solutions to the Equations of Incremental Thermoelasticity with Voids1 F. Martinez‡, R. Quintanilla† Departament Matemàtica Aplicada II, Universitat Politècnica de Catalunya Abstract. The theory of elastic materials with voids describes the behaviour of porous solids in which the skeletal or matrix material is elastic and the intersticies are void of material. In this paper we study the small deformations imposed on a large nonlinear general deformation in the theory of thermoelasticity with voids. We apply the theory of semigroups of linear operators to obtain some results on existence, uniqueness and continuous dependence respect initial conditions. The method apply for the zero Dirichlet boundary conditions. In the last section we extend the uniqueness and continuous dependence results for a mixed boundary condition problem. To this end we use the energy methods.

1 Introduction Goodman and Cowin [1] have introduced the concept of a distributed body as a continuum model for granular and porous bodies. Using this concept Nunziato and Cowin [2] have presented a theory for the behaviour of porous solids in which the skeletal or matrix material is elastic and the intersticies are void of material. Cowin and Nunziato [3] have obtained some results in the linear theory. Others results in linear theory without thermal effects have been obtained by Puri and Cowin [4]. Chandrasekharaiah and Cowin [5] have obtained the complete solution for linear, isothermal, homogeneous elastic materials and Ie¸san and Quintanilla [6] have established decay estimates and energy bounds for porous elastic cylinders. A recent good survey on this theory can be found in [7]. In [8] Ie¸san has studied a linear theory of thermoelastic materials with voids and have established uniqueness, reciprocal and variational theorems. Rusu [9] has established existence and uniqueness in linear thermoelasticity of materials with voids. Since Ie¸san [10] introduced the incremental equations for a thermoelastic material many papers have been devoted to study several aspects of this problem. Navarro & Quintanilla [11] obtained an existence theorem and recently it has been extended for unbounded domains and more relaxed assumptions on the constitutive equations [12]. Quintanilla & Williams [13] have also stated and studied the incremental equations in isothermal viscoelasticity. Many others results on incremental thermoelasticity can be found [14-17]. In [18] Ie¸san has presented the incremental equations for thermoelastic materials with voids. 1 Trends

in Applications of Mathematics to Mechanics, M.M. Marques and J.F. Rodrigues (eds.), 45-56, Pitman Monographs and Surveys in Pure and Applied Mathematics, 77, 1995. ‡F.I.B., Pau Gargallo 5, 08028 Barcelona, Spain †E.T.S.I.T., Colom 11, 08222 Terrassa, Barcelona, Spain

1

2

INCREMENTAL THERMOELASTICITY WITH VOIDS

The present paper applies semigroup theory to the problem of existence and uniqueness of solutions for standard initial boundary value problem of the theory established in [18].

2 Notation and basic equations We refer the motion to the reference configuration Ω0 and a fixed system of rectangular Cartesian axes Oxk (k = 1, 2, 3). We shall employ the usual summation and differentiation conventions. We call ∂Ω0 the boundary of Ω0 . As usual, letters in boldface stand for tensors of order p > 1, and, if v has order p, we write vi1 ...ip for its components in the Cartesian coordinate frame. In all what follows, we use a superposed dot to denote partial differentiation with respect to time. We consider two others states: the primary state Ω and the secondary state ∗ Ω and we introduce incremental quantities (see [18]) associated with a difference of a motion between the secondary and the primary states. Thus, if the point X = [X A ] in Ω0 moves to x = [xi ] in Ω and to y = [yi ] in Ω∗ , then u = [ui ] = [yi −xi ] is the incremental displacement, if T and T ∗ are the absolute temperature associated with Ω and Ω∗ then θ = T ∗ − T is the incremental temperature and if ν and ν ∗ are the volume fraction field in Ω and Ω∗ respectively then ϕ = ν ∗ − ν is the incremental volume fraction. We study the theory of perturbed thermoelastic deformations referred to the configuration Ω0 . The field equations of the theory consists of the equations of motion P Ai,A + ρ0 F i = ρ0 u ï , (1) the equation of equilibrated forces M A,A + χ + ρ0 L = ρ0 κϕ, ¨

(2)

T γ˙ + ρ0 θη˙ = ΦA,A + ρ0 S,

(3)

the equation of energy and the constitutive equations P Ki = AKijN uj,N + DKiM ϕ,M + B Ki ϕ − β Ki θ, γ = β N j uj,N + AK ϕ,K + Bϕ + Aθ, χ = −B N j uj,N − B K ϕ,K − ξϕ + Bθ, M K = DN iK ui,N + AKN ϕ,N + B K ϕ − AK θ, ΦA = RAjN uj,N + LAK ϕ,K + E A ϕ + DA θ + K AM θ,M ,

(4)

in Ω0 × [0, t1 ]. We have employed the following notation: P Ki is the incremental first Piola-Kirchhoff stress tensor measured per unit area in Ω0 ; γ is the incremental entropy measured per unit volume in Ω0 ; χ is the incremental intrinsic equilibrated body force measured per unit volume in Ω0 ; M K is the incremental equilibrated stress measured per unit area of the X A planes in Ω0 ; ΦA is the incremental heat flux vector measured per unit area of the X A planes in Ω0 ; F i is the incremental body force vector; L is the incremental extrinsic equilibrated force; S is the incremental heat supply; κ is the equilibrated inertia and ρ0 is


3

the density in the reference configuration. The thermoelastic coefficients AKijN , DKiN , RKiN , B Ki , β Ki , AKL , K AM , LAM , AK , B K , DK , E K , A, B, ξ are functions of the deformation, temperature and volumen fraction of the body in the primary state. The constitutive coefficients have the symmetry property AKijN = AN jiK ,

ALN = AN L ,

K LN = K N L .

Substituting (4) in (1) − (3) we obtain: 1 u ï = [AKijN uj,N + DKiM ϕ,M + B Ki ϕ − β Ki θ],K + F i ρ0

(5)

in Ω0 × [0, t1 ],

1 θ˙ = [RAjN uj,N + LAK ϕ,K + E A ϕ + DA θ + K AM θ,M ],A AT 1 β N j u˙ j,N + AK ϕ˙ ,K + B ϕ˙ − A 1 ˙ ˙ + Aθ ˙ − ρ0 η˙ θ + ρ0 S β N j uj,N + A˙ K ϕ,K + Bϕ − A AT AT

(6)

in Ω0 × [0, t1 ], (7)

1 [DKiN ui,K + AN K ϕ,K + B N ϕ − AN θ],N ρ0 κ L 1 − B N j uj,N − B K ϕ,K − ξϕ + Bθ + in Ω0 × [0, t1 ]. + ρ0 κ κ We consider the boundary conditions ˜ (X, t), P Ki (X, t)nK = P˜ i (X, t), M K (X, t)nK = M ˜ ΦK (X, t)nK = Φ(X, t) on ∂Ω1 × [0, t1 ], ϕ¨ =

˜ ui (X, t) = u ˜i (X, t), ϕ(X, t) = ϕ(X, ˜ t), θ(X, t) = θ(X, t)

(8)

on ∂Ω2 ×[0, t1 ], (9)

where ∂Ω1 ∪ ∂Ω2 = ∂Ω0 , ∂Ω1 ∩ ∂Ω2 = ∅ and nK is the outward normal to ∂Ω1 , and the initial data ˙ u(X, 0) = u0 (X), u(X, 0) = v0 (X), ϕ(X, 0) = ϕ0 (X), ϕ(X, ˙ 0) = ψ 0 (X), θ(X, 0) = θ0 (X)

on Ω0 ,

(10)

where u0 (X), v0 (X), ϕ0 (X), ψ 0 (X), θ0 (X) are given functions. We shall assume that the constitutive fields are continuously differentiable with values that are, for fixed t, Lebesgue measurable and essentially bounded. We also make the following assumptions (a) 0 < ρ1 ≤ ess inf X∈Ω0 ρ0 (X) ≤ ρ2 , (b) 0 < κ1 ≤ ess inf X∈Ω0 κ(X) ≤ κ2 , (c) ess inf X∈Ω0 A(X, t) ≥ A0 > 0, Z AKijN ui,K uj,N + AKN ϕ,K ϕ,N + ξϕ2 + 2DKiN ui,K ϕ,N + 2B Ki ui,K ϕ (d) Ω0 Z (ui,K ui,K + ϕ,K ϕ,K + ϕ2 ) dV + 2B K ϕ,K ϕ dV ≥ δ Ω0

1 (e) T

Z

Ω0

∞ ∀u ∈ C∞ 0 (Ω0 ), ∀ϕ ∈ C0 (Ω0 ), Z θ,K θ,K dV K AN θ,A θ,N dV ≥ K Ω0

∀θ ∈ C0∞ (Ω0 ).

(11)

4


The thermomechanical interpretation of (a), (b) and (c) is obvious. The estimate (e) is related to the well-known property of a definite heat conductor. Assumption (d) is usual in the study of the well-posed problems of elasticity (see [19]).

3 Existence result In this Section we work the particular case of boundary conditions of (9): u(X, t) = 0, ϕ(X, t) = 0, θ(X, t) = 0 on ∂Ω0 × [0, t1 ],

(9’)

We whish to transform (6)-(8), (9’), (10) to an abstract problem on a Hilbert ˙ space. Let us consider the element ω = (u, v, ϕ, ψ, θ) where u(X, t) = v(X, t) and ϕ(X, ˙ t) = ψ(X, t). We shall denote by X the space W01,2 (Ω0 ) × L2 (Ω0 ) × W01,2 (Ω0 ) × L2 (Ω0 ) × L2 (Ω0 ), where L2 (Ω0 ) = [L2 (Ω0 )]3 and W01,2 (Ω0 ) = [W01,2 (Ω0 )]3 being W01,2 (Ω0 ) the well know Sobolev space [20]. To tranform our problem (6)-(8), (9’), (10) to an initial value problem in the Hilbert space X we define the operators:

Bu =

1 [AKijN uj,N ],K , ρ0

Cϕ =

1 [DKiM ϕ,M + B Ki ϕ],K , ρ0

Dθ = −

1 [β Ki θ],K , ρ0

1 [DKiA ui,K ],A − B N j uj,N , ρ0 κ 1 Fϕ = [ALK ϕ,K + B L ϕ],L − B K ϕ,K − ξϕ , ρ0 κ 1 −1 1 [RAjN uj,N ],A − β˙ N j uj,N , Gθ = [AK θ],K − Bθ , Hu = ρ0 κ AT A Eu =

1 Kv = − β N j v j,N , A Mψ = −

1 A˙ K B˙ [LAK ϕ,K + E A ϕ],A − ϕ,K − ϕ, AT A A A˙ 1 ρ0 η˙ Nθ = [DA θ + K AM θ,M ],A − θ − θ, (12) AT A AT

Lϕ =

B AK ψ ,K − ψ, A A

and let A be the operator on X defined by 0 B  A=0  E H 

Id 0 0 0 K

0 C 0 F L

0 0 Id 0 M

 0 D  0 ,  G N

with domain  u v   D = D(A) = {(u, v, ϕ, ψ, θ) ∈ X | A  ϕ  ∈ X }.   ψ θ 

(13)


5

For later use, we remark that D(A) is dense in X . Now, we can write our problem as an abstract evolutionary equation    0  u(t) u(t) F v(t)   v(t)   d       0  ϕ(t)  = A  ϕ(t)  +  dt      L  ψ(t) ψ(t) κ ρ0 S θ(t) θ(t) AT 



  ,  

   u0 u(0)  v(0)   v0       ϕ(0)  =  ϕ0  , t ∈ [0, t1 ]. (14)     ψ0 ψ(0) θ0 θ(0) 

To prove the existence and uniqueness result for the equation (14), we define the inner product in X ˜ θ)i ˜ t ˜ , ϕ, h(u, v,ϕ, ψ, θ), (˜ u, v ˜ ψ, Z 1 AKijN ui,K u ˜j,N + AKN ϕ,K ϕ˜,N + ξϕϕ˜ + DKiN (˜ ui,K ϕ,N + ui,K ϕ˜,N ) = 2 Ω0 + B K (ϕϕ ˜ ,K + ϕϕ˜,K ) + B Ki (˜ ui,K ϕ + ui,K ϕ) ˜ + ρ0 v i v˜i + ρ0 κψ ψ˜ + Aθθ˜ dV. (15) By the assumptions on the constitutive functions we see that the norm induced by this product is equivalent to the original norm in X . Lemma 1. For each t ∈ [0, t1 ] the operator A is the generator of a quasicontractive semigroup. Proof. First, we prove that there exists a constant C1 > 0 such that hAω, ωit ≤ C1 hω, ωit . Using the Green-Gauss formula and the boundary conditions we have Z 1 K AM T ,A − DM T hAω, ωit = θθ,M dV 2 Ω0 T2 Z Z 1 RAiN K AM 1 ui,N θ,A dV − θ,A θ,M dV − 2 Ω0 T 2 Ω0 T Z Z ˙ 2 1 E A T ,A − BT LAM 1 θϕ dV − θ,A ϕ,M dV + 2 2 Ω0 T 2 Ω0 T Z Z 1 1 RAiK T ,A − β˙ Ki T 2 EA + ui,K θ dV − θ,A ϕdV 2 2 Ω0 T 2 Ω0 T Z Z ˙ 2 − ρ0 T η˙ 1 DA T ,A − AT LAM T ,A − A˙ M T 2 1 2 θ dV + θϕ,M dV. + 2 Ω0 T2 2 Ω0 T2 (16) By Cauchy-Schwarz and arithmetic-geometric mean inequalities we obtain Z 1 m1 ǫ5 hAω, ωit ≤ ui,K ui,K dV + 2 2ǫ1 2 Ω0 Z Z ǫ2 ǫ3 ǫ4 1 ǫ1 1 m4 ǫ8 + + + −K + θ,K θ,K dV + + ϕ,N ϕ,N dV 2 2 2 2 2 2 2ǫ4 2 Ω0 Ω0 Z Z m5 m8 ǫ7 1 m3 m7 1 m2 2 + + + m6 + θ dV + + + ϕ2 dV, 2 2ǫ2 2ǫ5 2ǫ8 2 Ω0 2 2ǫ3 2ǫ7 Ω0 (17)

6


where ǫi > 0 are arbitrary constants, and mi = max[0,t1 ] ess supX∈Ω0 Mi with (K T −D T )(K T −D T ) RAiN M12 = 9RAiN , M22 = AM ,A M T 4 N M ,N M , M32 = E ATE2 A , M42 = T2

˙ 2 −ρ0 T η) (R T −β˙ T 2 )(R T −β˙ T 2 ) (D T −AT ˙ 2 9LAM LAM , M52 = AiK ,A Ki T 4 N iK ,N Ki , M62 = A ,A T 4 , T2 2 2 2 2 ˙ ˙ ˙ (L T −A T )(L T −A T ) (E T −BT ) , and M82 = AM ,A M T 4 N M ,N M . If we choose M72 = A ,AT 4 ǫ1 + ǫ2 + ǫ3 + ǫ4 − 2K ≤ 0 then we have

hAω, ωit ≤ C1 hω, ωit . Now, we show that the condition Rang(λId − A) = X is satisfied. ˆ θ) ˆ ∈ X , we must show that the system of equations ˆ , ϕ, (ˆ u, v ˆ ψ,

(18) Let

ˆ, λu − v = u ˆ, λv − Bu − Cϕ − Dθ = v λϕ − ψ = ϕ, ˆ ˆ λψ − Eu − Fϕ − Gθ = ψ, ˆ (λ − N )θ − Hu − Kv − Lϕ − Mψ = θ,

(19)

has a solution in D. Introducing the first and third equations into the others and dividing the fifth equation by λ we obtain the system        λ2 Id − B ˆ + λˆ v u −C −D u u   ψˆ + λϕˆ λ2 Id − F −G  ϕ =  A˜λ  ϕ  =  −E ˆ . H L N  θ − Kˆ u − M ϕ ˆ θ θ − − K − − M Id − λ λ λ λ (20) R ˜ ˜ ˜i + Aθθ + ρ0 κϕϕ) ˜ dV and consider the biLet h(u, ϕ, θ), (˜ u, ϕ, ˜ θ)i = Ω0 (ρ0 ui u   u ˜ ˜ ˜  u, ϕ, ˜ θ)i. In the same way that linear form B λ [(u, ϕ, θ), (˜ u, ϕ, ˜ θ)] = hAλ ϕ  , (˜ θ (16), (17) we get Z 2 ρ0 λ ui ui + AKijN ui,K uj,N + AKN ϕ,K ϕ,N B λ [(u, ϕ, θ), (u, ϕ, θ)] = Ω0 + 2B K ϕϕ,K + 2DKiN ui,K ϕ,N + 2B Ki ui,K ϕ + (ρ0 κλ2 + ξ)ϕ2 dV Z n 1 ˙ + ρ0 η˙ + A˙ + A θ2 + θ β˙ iK ui,K + A˙ K ϕ,K + Bϕ λ Ω0 T θ o + RAiN ui,N + DA θ + K AM θ,M + E A ϕ + LAM ϕ,M dV ,A T Z Z Z m1 ǫ5 2 2 2 ρ0 κλ ϕ dV + δ − ρ0 λ ui ui dV + − ui,K ui,K dV ≥ 2λǫ1 2λ Ω0 Ω0 Ω0 Z ǫ7 m5 m8 m6 m2 − θ2 dV − − − + A0 − 2λǫ2 2λǫ5 2λǫ8 λ 2λ Ω0 Z m3 m7 + δ− − ϕ2 dV 2λǫ3 2λǫ7 Ω0 Z 1 θ,K θ,K dV (ǫ1 + ǫ2 + ǫ3 + ǫ4 − 2K) − 2λ Ω0 Z ǫ8 m4 ϕ,N ϕ,N dV. (21) − + δ− 2λǫ4 2λ Ω0


7

Now, we may take ǫ1 + ǫ2 + ǫ3 + ǫ4 < 2K and λ so large such that B λ [(u, ϕ, θ), (u, ϕ, θ)] ≥ C3 k(u, ϕ, θ)k2 W1,2 ×W 1,2 ×W 1,2 . 0

0

0

Thus B λ determines a norm which is equivalent to the usual one on W01,2 × v + λˆ u, ψˆ + λϕ, ˆ (θˆ − Kˆ u− W01,2 × W01,2 . It is a simple calculation to see that (ˆ −1,2 −1,2 −1,2 Mϕ)/λ) ˆ ∈W ×W ×W . Hence Riesz’s theorem implies that there exists a unique solution (u, ϕ, θ) ∈ W01,2 × W01,2 × W01,2 to the system (20). From first and third equations of (19) we see that v ∈ W01,2 and ψ ∈ W01,2 . Using the Lumer-Phillips corollary to the Hille-Yosida theorem, the Lemma is proved. Lemma 2. There exists a positive contant C2 such that kωkt ≤ exp(C2 t1 ), kωks

(22)

holds for each ω ∈ X and each t, s ∈ [0, t1 ]. Proof. We define for each t, h(t) ≡ k · k2t , where k · kt is the norm induced by the inner product (15). If we derive h(t), we obtain Z 1 ˙ ˙ 2 h(t) = A˙ KijN ui,K uj,N + A˙ KN ϕ,K ϕ,N + ξϕ 2 Ω0 ˙ 2 dV. + 2D˙ KiN ui,K ϕ,N + 2B˙ Ki ui,K ϕ + 2B˙ K ϕ,K ϕ + Aθ After use of the Cauchy-Schwarz inequality we obtain Z Z Z a1 ˙ ui,K uj,N dV ≤ AKijN ui,K uj,N dV ≤ a1 AKijN ui,K uj,N dV, δ Ω0 Ω0 Ω0 Z Z Z a2 ˙ AKN ϕ,K ϕ,N dV, ϕ,K ϕ,N dV ≤ AKN ϕ,K ϕ,N dV ≤ a2 δ Ω0 Ω0 Ω0 Z Z Z a3 2 2 ˙ ϕ dV ≤ ξϕ dV ≤ a3 ξϕ2 dV, δ Ω0 Ω0 Ω0 Z ǫ a Z Z 1 9 4 2 ˙ B K ϕϕ,K dV ≤ ξϕ dV + AKN ϕ,K ϕ,N dV, 2δ Ω0 2ǫ9 δ Ω0 Ω0 Z Z ǫ a Z 1 10 5 ˙ DKiN ui,K ϕ,N dV ≤ AKijN ui,K uj,N dV + AKN ϕ,K ϕ,N dV, 2δ 2ǫ10 δ Ω0 Ω0 Ω0 Z Z ǫ Z a6 11 ˙ B Ki ui,K ϕ dV ≤ AKijN ui,K uj,N dV + ξϕ2 dV, 2δ 2ǫ δ 11 Ω0 Ω0 Ω0

where ˙ a1 = max ess sup A˙ KijN A˙ KijN , a2 = max ess sup A˙ KN A˙ KN , a3 = max ess sup ξ, [0,t1 ]

a4

=

[0,t1 ]

X∈Ω0

max ess sup B˙ K B˙ K , [0,t1 ]

a5

=

X∈Ω0

a6 = max ess sup B˙ Ki B˙ Ki . Then [0,t1 ]

[0,t1 ]

X∈Ω0

9 max ess sup D˙ KiN [0,t1 ]

X∈Ω0

D˙ KiN ,

X∈Ω0

X∈Ω0

˙ h(t) ≤ 2C2 h(t),

(23)

8


an integration of (23) betwen s and t yields h(t) ≤ h(s) exp(2C2 |t − s|) ≤ h(s) exp(2C2 t1 ), then

kωkt ≤ exp(C2 t1 ), kωks

which is the desired result. Theorem 1. The family of operators {A(t), t ∈ [0, t1 ]} with A defined by (12), (13) is stable in the sense of Kato with stability constants Γ = exp(2C2 t1 ) and C1 . Theorem 2. Assume that the conditions (a) − (e) are satisfied. We suppose that F ∈ C 1 ([0, t1 ], L2 (Ω0 )) ∩ C 0 ([0, t1 ], W01,2 (Ω0 ) ∩ W2,2 (Ω0 )), ρ0 S ∈ C 1 ([0, t1 ], L2 (Ω0 )) ∩ C 0 ([0, t1 ], W 2,2 (Ω0 )) AT L ∈ C 1 ([0, t1 ], L2 (Ω0 )) ∩ C 0 ([0, t1 ], W01,2 (Ω0 ) ∩ W 2,2 (Ω0 )). κ Then for any (u0 , v0 , ϕ0 , ψ 0 , θ0 ) ∈ D exists a unique solution (u(t), v(t), ϕ(t), ψ(t), θ(t)) ∈ C 1 ([0, t1 ], X ) ∩ C 0 ([0, t1 ], D) which satisfies (14). Remark 1: Since A is a generator of a quasi-contractive semigroup we have the following estimate for the solutions k(u(t), v(t),ϕ(t), ψ(t), θ(t))k ≤ ΓeC1 t k(u0 , v0 , ϕ0 , ψ0 , θ0 )k Z t L(τ ) ρ0 S(τ ) + k(0, F(τ ), 0, , )kdτ t ∈ [0, t1 ]. κ A(τ )T (τ ) 0 Remark 2: The last theorem and the previous remark allows to affirm that, whenever the assumptions on the constitutive coefficients are satisfied, the problem (6) − (8), (9′ ), (10) of the incremental thermoelasticity with voids is a wellposed problem.

4 Uniqueness and continuous dependence results In this Section we are going to obtain an uniqueness and continuous dependence results for the general problem (6)-(10). ¯ , ϕ, Let u ¯ θ¯ the difference of two solutions of the system (6)-(8) corresponding to two different systems of initial data, supply terms and equal boundary conditions. ¯ 0 (X), v ¯ 0 (X), ϕ¯0 (X), ψ¯0 (X), θ¯0 (X) the diffence between two different systems Let u


9

¯ L, ¯ S¯ the diffence between two different systems of supply of initial data and F, terms. In order to obtain the results of this Section is sufficient to consider the problem ¨¯i = u

1 ¯ ,K + F¯ i [AKijN u ¯j,N + DKiM ϕ¯,M + B Ki ϕ¯ − β Ki θ] ρ0

in Ω0 × [0, t1 ],

1 [RAjN u ¯j,N + LAK ϕ¯,K + E A ϕ¯ + DA θ¯ + K AM θ¯,M ],A θ¯˙ = AT 1 − βNj u ¯˙ j,N + AK ϕ¯˙ ,K + B ϕ¯˙ A ρ0 η˙ ρ0 S¯ 1 ˙ βNj u ¯j,N + A˙ K ϕ¯,K + B˙ ϕ¯ + A˙ θ¯ − θ¯ + − A AT AT ¨¯ = ϕ

1 ¯ ,N [DKiN u ¯i,K + AN K ϕ¯,K + B N ϕ¯ − AN θ] ρ0 κ ¯ L 1 − BNj u ¯j,N − B K ϕ¯,K − ξ ϕ¯ + B θ¯ + + ρ0 κ κ

in Ω0 × [0, t1 ],

in Ω0 × [0, t1 ],

with boundary conditions ¯ K (X, t)nK = 0, Φ ¯ K (X, t)nK = 0 on ∂Ω1 × [0, t1 ], P¯ Ki (X, t)nK = 0, M ¯ ¯ (X, t) = 0, ϕ(X, u ¯ t) = 0, θ(X, t) = 0 on ∂Ω2 × [0, t1 ], and the initial data ¯ (X, 0) = u ¯ 0 (X), u ¯˙ (X, 0) = v ¯ 0 (X), ϕ(X, u ¯ 0) = ϕ¯0 (X), ¯ ϕ(X, ¯˙ 0) = ψ¯0 (X), θ(X, 0) = θ¯0 (X)

on Ω0 .

We consider the function Z 1 E(t) = AKijN u ¯i,K u ¯j,N + AKN ϕ¯,K ϕ¯,N + ξ ϕ¯2 + 2DKiN u ¯i,K ϕ¯,N 2 Ω0 + 2B K ϕ¯ϕ¯,K + 2B Ki u ¯i,K ϕ¯ + ρ0 v¯i v¯i + ρ0 κψ¯2 + Aθ¯2 dV.

This function is a measure of the distance between these two solutions. Now, if we derive respect to the time and using the Green-Gauss formula we obtain Z h RAiN K AM ¯ ¯ K AM T ,A − DM T ¯¯ ˙ θθ,M − u ¯i,N θ¯,A − θ,A θ,M E(t) = 2 T T T Ω0 2 ˙ ˙ 2 E A T ,A − BT ¯ϕ¯ − LAM θ¯,A ϕ¯,M + RAiK T ,A − β Ki T u + θ ¯i,K θ¯ T2 T T2 i 2 ˙ 2 − ρ0 T η˙ ˙ DA T ,A − AT EA ¯ ¯2 + LAM T ,A − AM T θ¯ϕ¯,M dV θ,A ϕ¯ + θ − T2 T2 Z T 1 A˙ KijN u ¯i,K u ¯j,N + A˙ KN ϕ¯,K ϕ¯,N + ξ˙ϕ¯2 + 2 Ω0 + 2D˙ KiN u ¯i,K ϕ¯,N + 2B˙ Ki u ¯i,K ϕ¯ + 2B˙ K ϕ¯,K ϕ¯ + A˙ θ¯2 dV Z Z h h ¯i θ¯ i ¯ ¯ ψ¯ + S¯ θ dV. ¯ ¯ ¯ ρ0 F¯ i v¯i + L + P Ki v¯i + M K ψ + ΦK nK dA + T T ∂Ω0 Ω0

10


The calculations presented in Lemmas 1 and 2 and the boundary conditions allows to deduce Z h ¯i ¯ ψ¯ + S¯ θ dV. ˙ ρ0 F¯ i v¯i + L E(t) ≤ (C1 + 2C2 )E(t) + T Ω0 Thus, using the Hlder inequality, we obtain ˙ E(t) ≤ 2K1 E(t) + 2K2 E 1/2 (t)g(t), where g(t) =

R

Ω0

ρ0 F¯ i F¯ i +

ρ0 ¯ 2 κ L

+

ρ0 2 ¯ 2 AT 2 S

dV

1/2

.

If we fix s ∈ [0, t1 ] and integrate over [0, τ ], τ ∈ [0, s], we obtain E(τ ) ≤ E(0) + 2K1

Z

τ

Z

E(t)dt + 2K2 0

τ

g(t)E 1/2 (t)dt. 0

To obtain our results we need the following result [21], Lemma 3. Assume that the nonnegative functions f (t) ∈ L∞ [0, s] and g(t) ∈ L1 [0, s] satisfy the inequality 2

2 2

f (τ ) ≤ M f (0) +

Z

τ 0

(2α + 4βτ )f 2 (t) + 2N g(t)f (t) dt,

τ ∈ [0, s]

where α, β, M and N are nonnegative constant. Then f (s) ≤ e

ηs+βs2

M f (0) + N

Z

s 0

g(t)dt ,

where η = α + β/α. Applying the Lemma 3 with f (t) = E 1/2 (t) we have E

1/2

h

(s) ≤ E

1/2

(0) + K2

Z

s 0

i g(t)dt eK1 s .

(24)

Remark 3: Inequality (24) is the same that the one obtained in the remark 1 for the more restricted problem studied at previous Section. This inequality gives the continuous dependence result on initial and external parameters. In particular, we may obtain Theorem 3. Assume that the assumptions (11) are satisfied, then the initial boundary problem (6)-(10) has at most one solution. Proof. Suppose that two solutions exist, then by (24), we have E(t) = 0 for their difference, which implies, jointly with (11), θ¯ = 0,

ϕ¯ = 0,

v¯i = 0.

From the last equality we deduce that u ¯i is constant and from the initial conditions we have u ¯i = 0.


11

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