ON THE UNIQUENESS AND CONTINUOUS DEPENDENCE IN THE LINEAR THEORY OF THERMO-MICROSTRETCH ELASTICITY BACKWARD IN TIME

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII ALI CUZA DIN IAŞI (SN) MATEMATICĂ, Tomul LIX, 3, f ON THE UNIQUENESS AND CONTINUOUS DEPENDENCE IN THE LINEAR THEORY OF THERMO-MICROSTRETCH ELASTICITY BACKWARD IN TIME BY EMILIAN BULGARIU Abstract We study the uniqueness and the continuous dependence problems for the thermo-microstretch elastic processes backward in time The data are given for the final time t = and we want to study the solution at the previous moments We transform the problem in a boundary-initial value problem by an appropriate change of variables The uniqueness theorems presented in this article extend in a particular case the uniqueness theorem of Passarella and Tibullo () and we also discuss a different class of problems than the one considered by them We find some estimates that prove the continuous dependence of solution with respect to the final data Mathematics Subject Classification : 74B99, 74H5, 35B3 Key words: thermo-microstretch elasticity, backward in time process, uniqueness theorem, continuous dependence Introduction The theory of thermo-microstretch elastic solids was introduced by Eringen 6 The particles of the solid with microstretch can expand and contract independent of translations and the rotations which they execute In the class of nonstandard problems, the backward in time problems have been initially considered by Serrin, who studied the uniqueness of solutions for the Navier-Stokes equations backward in time An important study was made by Ames and Straughan for this class of nonstandard and improperly posed problems Ciarletta 5 studied the uniqueness and continuous dependence problems for the thermoelastic processes backward

2 34 EMILIAN BULGARIU in time At this hour, the nonstandard problems are intensively studied and we mention only some of the researchers in this domain: Ames,, Payne,7, Knops 7, Ciarletta 5,4, Chiriţă 3,4, Quintanilla 9, Straughan, 9 and Passarella 8 In this article we consider the boundary final value problem for the linear theory of thermo-microstretch elastic solids The data are given for the final time t = and we want to study the solution at the previous moments By an appropriate change of variable, we transform this problem into a boundary initial value problem Using the Lagrange-Brun identities, we deduce some preliminary results that combined with a method based on Gronwall s inequality will be the principal ingredients in obtaining the uniqueness and the continuous dependence results Passarella and Tibullo 8 have demonstrated the uniqueness of solutions for the backward in time problem of the linear theory of thermomicrostretch elastic materials and the impossibility of the localization in time of the solution of the corresponding forward in time problem Our results concerning the uniqueness of solution extend in a particular case the uniqueness theorem of Passarella and Tibullo 8 and we also discuss a different class of problems than the one considered by them Some estimates that prove the continuous dependence of solution with respect to the final data are obtained The boundary final value problem In this article, we shall denote by R 3 the domain occupied by an anisotropic and inhomogeneous thermo-microstretch elastic solid, whose boundary surface is The standard convention of summation over repeated suffixes is adopted and a subscript comma denotes the spatial partial differentiation with respect to the corresponding cartesian coordinate and a superposed dot denotes differentiation with respect to time Greek subscripts vary over {,} and Latin subscripts vary over {,,3} We consider the boundary final value problems on the time interval ( T,,T > and T may be infinite The fundamental system of field equations consists 6 of the geometric relations () e ij = u j,i +ε jik ϕ k, κ ij = ϕ j,i, γ i = ψ,i on ( T,, the constitutive equations t ij = A ijrs e rs +B ijrs κ rs +D ijr γ r +A ij ψ β ij θ,

3 3 BACKWARD IN TIME THERMO-MICROSTRETCH ELASTICITY 34 () m ij = B rsij e rs +C ijrs κ rs +E ijr γ r +B ij ψ C ij θ, 3π i = D rsi e rs +E rsi κ rs +D ij γ j +d i ψ ξ i θ, 3σ = A rs e rs +B rs κ rs +d i γ i +mψ ζθ, ρη = β rs e rs +C rs κ rs +ξ i γ i +ζψ +aθ, q i = k ij θ,j in ( T,, the equations of motion (3) t ji,j +f i = ρü i, m ji,j +ε irs t rs +g i = I ij ϕ j, π i,i σ +h = J ψ in ( T,) and the energy equation (4) ρt η = q i,i +r in ( T,) In the above equations we have used the following notations: t ij is the stress tensor, m ij is the couple stress tensor, π i is the microstress vector, σ is the scalar microstress function, η is the specific entropy, ρ is the mass density (in the reference configuration), f i is the body force, g i is the body couple, h is the (scalar) body load, r is the heat source density, q i is the heat flux vector, I ij is the microinertia tensor, J is the microstretch inertia and ε ijk is the alternating symbol The variables of this theory are: u i the components of the displacement vector, ϕ i the components of the microrotation vector, ψ the microstretch function and θ the variation of temperature from the uniform reference absolute temperature T We assume the following symmetries for the constitutive coefficients and the microinertia tensor (5) A ijrs = A rsij, C ijrs = C rsij, D ij = D ji, k ij = k ji, I ij = I ji Considering (), the energy equation can be rewritten in the following form (6) β ij ė ij C ij κ ij ξ i γ i ζ ψ + T q i,i + T r = a θ in ( T,) We consider the boundary-final value problem (P) defined by relations

4 34 EMILIAN BULGARIU 4 () (3) and (6), the final conditions (7) u i (x,) = u i (x), u i(x,) = u i (x), ϕ i(x,) = ϕ i (x), ϕ i (x,) = ϕ i (x), ψ(x,) = ψ (x), ψ(x,) = ψ (x), θ(x,) = θ (x), x and the boundary conditions (8) u i (x,t) = ũ i (x,t) on Σ ( T,, t i (x,t) t ji (x,t)n j (x) = t i (x,t) on Σ ( T,, ϕ i (x,t) = ϕ i (x,t) on Σ 3 ( T,, m i (x,t) m ji (x,t)n j (x) = m i (x,t) on Σ 4 ( T,, ψ(x,t) = ψ(x,t) on Σ 5 ( T,, π(x,t) π i (x,t)n i (x) = π(x,t) on Σ 6 ( T,, θ(x,t) = θ(x,t) on Σ 7 ( T,, q(x,t) q i (x,t)n i (x) = q(x,t) on Σ 8 ( T,, whereu i, u i,ϕ i, ϕ i,ψ, ψ,θ,ũ i, t i, ϕ i, m i, ψ, π, θ and q areprescribedfunctions, n i are the components of the outward unit normal vector to the boundary surface and Σ i,i =,,,8 are subsurfaces of, such that Σ Σ = Σ 3 Σ 4 = Σ 5 Σ 6 = Σ 7 Σ 8 = and Σ Σ = Σ 3 Σ 4 = Σ 5 Σ 6 = Σ 7 Σ 8 = We assume that ρ,i ij,j and the constitutive coefficients are continuous and bounded fields on, the constitutive coefficients are continuously differentiable functions on and (9) ρ(x) ρ >, J(x) J >, I(x) I >, where I(x) denotes the minimum eigenvalue of I ij (x) and ρ,j,i are constants We say that the internal energy density per unit of volume is a positive semidefinite quadratic form if () W(κ), for all κ = {e ij,κ ij,γ i,ψ}, where () W(κ) = A ijrse ij e rs +C ijrs κ ij κ rs +D ij γ i γ j +mψ +B ijrs e ij κ rs +D ijr e ij γ r +E ijr κ ij γ r +A ij e ij ψ +B ij κ ij ψ +d i γ i ψ

5 5 BACKWARD IN TIME THERMO-MICROSTRETCH ELASTICITY 343 We denote by W(t) = W({e ij (x,t),κ ij (x,t),γ i (x,t),ψ(x,t)}) We define () E(τ,τ )=A ijrs e ij (τ )e rs (τ )+C ijrs κ ij (τ )κ rs (τ )+D ij γ i (τ )γ j (τ ) +mψ(τ )ψ(τ )+B ijrs e ij (τ )κ rs (τ )+e ij (τ )κ rs (τ ) +D ijr e ij (τ )γ r (τ )+e ij (τ )γ r (τ )+A ij e ij (τ )ψ(τ )+e ij (τ )ψ(τ ) +E ijr κ ij (τ )γ r (τ )+κ ij (τ )γ r (τ )+B ij κ ij (τ )ψ(τ )+κ ij (τ )ψ(τ ) +d i γ i (τ )ψ(τ )+γ i (τ )ψ(τ ), where, for convenience, the dependence on x was suppressed We can remark that E(t,t) = W(t) If k ij is a positive definite tensors, we have (3) k m θ,i θ,i k ij θ,i θ,j k M θ,i θ,i, where k m and k M are positive constants, related to the minimum and the maximum eigenvalue (conductivity moduli) for k ij By using the relations (), () and (), we obtain (4) t ij ė ij +m ij κ ij +3π i γ i +3σ ψ = Ẇ β ijė ij θ C ij κ ij θ ξ i γ i θ ζ ψθ For future convenience, we set (5) δ = sup β ij β ij + C ij C ij + ξ i ξ i >, ρ I 3J (6) δ = sup ρ β ji,j β ki,k + I (ε jik β kj +C ki,k )(ε jil β lj +C li,l ) + ξ i,i ξ i,i + ζ > 3J 3J 3 The transformed boundary-initial value problem The aim of this article is to study the uniqueness and the continuous dependence of the solutions of the boundary-final value problem (P) with respect to the final data To do this, we transform (P) into a boundaryinitial value problem using the change of variables: t t Thus, the boundary-initial value problem (P) is defined by the geometric equations (3) e ij = u j,i +ε jik ϕ k, κ ij = ϕ j,i, γ i = ψ,i on,t),

6 344 EMILIAN BULGARIU 6 the constitutive equations (3) t ij = A ijrs e rs +B ijrs κ rs +D ijr γ r +A ij ψ β ij θ, m ij = B rsij e rs +C ijrs κ rs +E ijr γ r +B ij ψ C ij θ, 3π i = D rsi e rs +E rsi κ rs +D ij γ j +d i ψ ξ i θ, 3σ = A rs e rs +B rs κ rs +d i γ i +mψ ζθ, ρη = β rs e rs +C rs κ rs +ξ i γ i +ζψ +aθ, q i = k ij θ,j in,t), the equations of motion (33) t ji,j +f i = ρü i, m ji,j +ε irs t rs +g i = I ij ϕ j, π i,i σ +h = J ψ in (,T), the energy equation (34) β ij ė ij +C ij κ ij +ξ i γ i +ζ ψ + T q i,i + T r = a θ in (,T), with the initial conditions (35) u i (x,)=u i (x), u i(x,)= u i (x),ϕ i(x,)=ϕ i (x), ϕ i(x,) = ϕ i (x), ψ(x,) = ψ (x), ψ(x,) = ψ (x), θ(x,) = θ (x), x and the boundary conditions (36) u i (x,t)=ũ i (x,t) on Σ,T), t i (x,t)= t i (x,t) on Σ,T), ϕ i (x,t)= ϕ i (x,t) on Σ 3,T), m i (x,t)= m i (x,t) on Σ 4,T), ψ(x,t)= ψ(x,t) on Σ 5,T), π(x,t)= π(x,t) on Σ 6,T), θ(x,t)= θ(x,t) on Σ 7,T), q(x,t)= q(x,t) on Σ 8,T) By a solution of the boundary-initial value problem (P) we mean an ordered array ϖ = u i,ϕ i,ψ,e ij,κ ij,γ i,t ij,m ij,π i,σ,θ,θ,i,q i with its components continuous on, T) and satisfying equations (3) (36) In the rest of this section we establish some auxiliary identities concerning the solutions of the boundary-initial value problem (P) with the externalgivendatad = f i,g i,h,r,u i, u i,ϕ i, ϕ i,ψ, ψ,θ,ũ i, t i, ϕ i, m i, ψ, π, θ, q

7 7 BACKWARD IN TIME THERMO-MICROSTRETCH ELASTICITY 345 Lemma 3 Let ϖ be a solution of the boundary-initial value problem (P) corresponding to the external given data D Then, for all t,t), we have (37) = + + ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t)+w(t)+aθ (t) dv t t T k ij θ,i (s)θ,j (s)dvds ρ u i () u i ()+I ij ϕ i () ϕ j ()+3J ψ ()+W()+aθ () dv f i (s) u i (s)+g i (s) ϕ i (s)+3h(s) ψ(s) r(s)θ(s) dvds T t t i (s) u i (s)+m i (s) ϕ i (s)+3π(s) ψ(s) q(s)θ(s) dads T Proof We deduce from (5), (4), (3) and (33) that (38) ρ u i (s) u i (s)+i ij ϕ i (s) ϕ j (s)+3jψ s (s)+w(s) = f i (s) u i (s) +g i (s) ϕ i (s)+3h(s) ψ(s)+ t ji (s) u i (s)+m ji (s) ϕ i (s)+3π j (s) ψ(s) +β ij ė ij (s)θ(s)+c ij κ ij (s)θ(s)+ξ i γ i (s)θ(s)+ζ ψ(s)θ(s),j If we integrate the above relation over,t,t,t) and then use the divergence theorem and equivalences from (8), we have (39) ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t)+w(t)dv = ρ u i () u i ()+I ij ϕ i () ϕ j ()+3J ψ ()+W()dv t + f i (s) u i (s)+g i (s) ϕ i (s)+3h(s) ψ(s)dvds t + t i (s) u i (s)+m i (s) ϕ i (s)+3π(s) ψ(s)dads t + β ij ė ij (s)θ(s)+c ij κ ij (s)θ(s)+ξ i γ i (s)θ(s)+ζ ψ(s)θ(s)dvds

8 346 EMILIAN BULGARIU 8 Further, by using the relations (3) and (34) we deduce (3) aθ (s) = β ij ė ij (s)θ(s) C ij κ ij (s)θ(s) ξ i γ i (s)θ(s) s ζ ψ(s)θ(s) q i (s)θ(s) + k ij θ,i (s)θ,j (s) r(s)θ(s) T,i T T which by integrating over,t,t,t) and by using the divergence theorem and the relation (8), gives us the identity (3) t aθ (t)dv k ij θ,i (s)θ,j (s)dvds = aθ ()dv T t t r(s)θ(s)dvds q(s)θ(s)dads T T t β ij ė ij (s)θ(s)+c ij κ ij (s)θ(s)+ξ i γ i (s)θ(s)+ζ ψ(s)θ(s)dvds Summing the relation (39) with (3) we obtain the identity (37), and so the proof is complete Lemma 3 Let ϖ be a solution of the boundary-initial value problem (P) corresponding to the external given data D Then, for all t, T ), we have { ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3jψ (t) W(t)+aθ (t) } dv { = ρ ui () u i (t)+i ij ϕ i () ϕ j (t)+3j ψ() ψ(t) (3) E(,t) +aθ()θ(t) } dv t { + f i (t s) u i (t+s) f i (t+s) u i (t s) +g i (t s) ϕ i (t+s) g i (t+s) ϕ i (t s) +3 h(t s) ψ(t+s) h(t+s) ψ(t s) + } r(t s)θ(t+s) r(t+s)θ(t s) dvds T t { + t i (t s) u i (t+s) t i (t+s) u i (t s)

9 9 BACKWARD IN TIME THERMO-MICROSTRETCH ELASTICITY 347 +m i (t s) ϕ i (t+s) m i (t+s) ϕ i (t s) +3 π(t s) ψ(t+s) π(t+s) ψ(t s) + } q(t s)θ(t+s) q(t+s)θ(t s) dads T Proof Considering t, T ), from (5) we have (33) ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t) = ρ u i () u i (t)+i ij ϕ i () ϕ j (t)+3j ψ() ψ(t) t { + ρ u i (t+s)ü i (t s) u i (t s)ü i (t+s) +I ij ϕ i (t+s) ϕ j (t s) ϕ i (t s) ϕ j (t+s) } + 3J ψ(t+s) ψ(t s) ψ(t s) ψ(t+s) ds Further, using the relations (5), () and (3) (34), we deduce that ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t) W(t)+aθ (t) (34) = ρ u i () u i (t)+i ij ϕ i () ϕ j (t)+3j ψ() ψ(t) E(, t) + aθ()θ(t) t { + f i (t s) u i (t+s) f i (t+s) u i (t s) +g i (t s) ϕ i (t+s) g i (t+s) ϕ i (t s) +3 h(t s) ψ(t+s) h(t+s) ψ(t s) + } r(t s)θ(t+s) r(t+s)θ(t s) ds T t { + t ji (t s) u i (t+s) t ji (t+s) u i (t s) +m ji (t s) ϕ i (t+s) m ji (t+s) ϕ i (t s) +3 π j (t s) ψ(t+s) π j (t+s) ψ(t s) + } q j (t s)θ(t+s) q j (t+s)θ(t s) T,j ds, t, T ) To complete the proof of this lemma, we integrate relation (34) over, use the divergence theorem and the equivalences from (8)

10 348 EMILIAN BULGARIU The identities obtained in this section constitute the essential ingredients in deducing the uniqueness and continuous dependence results in the next three sections, for the boundary-initial value problem (P) with respect to the external given data D 4 Uniqueness results The aim of this section is to establish the uniqueness of the solution of the boundary-initial value problem(p) Passarella and Tibullo8 have demonstrated a uniqueness result for the problem we have considered The theorems given in this section extend their theorem in the particular case when meas Σ 8 = In what follows, we assume that the symmetry relations (5) are satisfied We will use the following hypotheses: (H ) the relation (9) holds true and k ij is a positive definite tensor (ie the relation (3) holds true); (H ) W is a positive semidefinite quadratic form; (H 3 ) a(x) a <, where a is a constant These assumptions are reasonable and characterize a certain state supported by the thermo-microstretch elastic body Theorem 4 Assume that (H ) and (H ) hold true Then the boundary-initial value problem (P) has at most one solution Proof We consider ϖ (α) = u (α) i,ϕ (α) i,ψ (α),e (α) ij,κ(α) ij,γ(α) i,t (α) ij,m(α) ij, π (α) i,σ (α),θ (α), θ (α),i,q (α) i (α =,) two solutions of the boundary-initial value problem (P) corresponding to the same external given data D The difference ϖ = ϖ () ϖ () = u i,ϕ i,ψ,e ij,κ ij,γ i,t ij,m ij,π i,σ,θ,θ,i,q i is a solution for the boundary-initial value problem (P) corresponding to the null external given data Relations (37) and (3), in the context of this theorem, provides us ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t) dv (4) t = T k ij θ,i (s)θ,j (s)dvds, t, T )

11 BACKWARD IN TIME THERMO-MICROSTRETCH ELASTICITY 349 The assumption that meas Σ 8 = and the homogeneous external given data implies that θ(x,t) = on,t) and hence we obtain (4) t β ij ė ij (s)θ(s)+c ij κ ij (s)θ(s)+ξ i γ i (s)θ(s)+ζ ψ(s)θ(s) dvds t { = β ij θ(s),i u j (s) ε jik β ij ϕ k (s)θ(s)+c ij θ(s),i ϕ j (s) +ξ i θ(s),i ψ(s) ζ ψ(s)θ(s) } dvds The Poincaré inequality (see 5) gives us (43) θ,i (t)θ,i (t)dv λ θ (t)dv, where λ > is the minimum eigenvalue of the clamped membrane problem On the other hand, by using Schwarz s inequality, arithmetic-geometric mean inequality and the relations (5)-(6), we deduce (44) t β ij ė ij (s)θ(s)+c ij κ ij (s)θ(s)+ξ i γ i (s)θ(s)+ζ ψ(s)θ(s) dvds t { ε +ε ρ u i (s) u i (s)+i ij ϕ i (s) ϕ j (s)+3j ε ε ψ (s) } +ε δ θ (s)+ε δ θ,i (s)θ,i (s) dvds, ε,ε > Supposing that hypothesis (H ) holds true and considering the relations (9), (3), (43) and (44), we obtain (45) t ε +ε ε ε t +µ β ij ė ij (s)θ(s)+c ij κ ij (s)θ(s)+ξ i γ i (s)θ(s)+ζ ψ(s)θ(s) dvds t ρ u i (s) u i (s)+i ij ϕ i (s) ϕ j (s)+3jψ (s) dvds T k ij θ,i (s)θ,j (s)dvds, ε,ε >,t,t),

12 35 EMILIAN BULGARIU where µ = T k m (ε λ δ +ε δ ) and λ is defined by (43) Moreover, from (4) and (45), we deduce (46) t β ij ė ij (s)θ(s)+c ij κ ij (s)θ(s)+ξ i γ i (s)θ(s)+ζ ψ(s)θ(s) dvds ε +ε t ρ u i (s) u i (s)+i ij ϕ i (s) ϕ j (s)+3j ψ (s) dvds ε ε +µ ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t) dv, t, T Assuming that hypothesis (H ) holds true, we can conclude from relations (39) and (46), with null external given data that (47) ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t)+w(t) dv ε +ε t ρ u i (s) u i (s)+i ij ϕ i (s) ϕ j (s)+3j ψ (s) ε ε +µ ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t) dv, t, T ) dvds ) Choosing the parameters ε,ε sufficiently small (eg ε = λkm and 3δ T ε = km 3δ T ) we have µ > and hence, from () and (47), we deduce ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3jψ (t) dv ε +ε (48) t ρ u i (s) u i (s)+i ij ϕ i (s) ϕ j (s)+3j ψ (s) ( µ)ε ε dvds, t By means of Gronwall s inequality, the above relation gives us (49), T ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t) dv =, t, T ) Considering the relation (9), we have that all the terms in the integral are positive and hence, we deduce that (4) u i (x,t) =, ϕ i (x,t) =, ψ(x,t) = in, T ) )

13 3 BACKWARD IN TIME THERMO-MICROSTRETCH ELASTICITY 35 and by taking into account the null initial conditions for ϖ, the relations (3) and (4) and the assumption that θ(x,t) = on,t), we get (4) u i (x,t) =, ϕ i (x,t) =, ψ(x,t) =, θ(x,t) = in, T ) For the case when T =, the relations (4) give us the uniqueness of solutions of the boundary-initial value problem (P) If T <, we repeat the procedure of proof on the time interval T,T) and we get the relations (4) for T, 3T ) 4 Continuing this algorithm for extension of the set on witch we have the uniqueness of solution (P), we obtain the validity of relations (4) over,t) Theorem 4 Assume that (H ) and (H 3 ) hold true Then the boundary-initial value problem (P) has at most one solution Proof The relations (4)-(46) from the proof of the previous theorem remain valid because they are obtained under the only hypothesis (H ) From relation (3) with null external given data, we obtain that (4) t T k ij θ,i (s)θ,j (s)dvds t +ξ i γ i (s)θ(s)+ζ ψ(s)θ(s) dvds, t,t) βij ė ij (s)θ(s)+c ij κ ij (s)θ(s) Further, using relations (4) and (45), the above inequality becomes (43) t +µ t t s k ij θ,i (s)θ,j (s)dvds ε +ε T ε ε k ij θ,i (s)θ,j (s)dvds, ε,ε >, t T k ij θ,i (z)θ,j (z)dvdzds T, T ) If we choose parameters ε,ε sufficiently small (eg ε = λkm 3δ T and ε = k m 3δ T ) such that µ >, from relation (43) we have that (44) t t s k ij θ,i (s)θ,j (s)dvds ε +ε T µ ε ε k ij θ,i (z)θ,j (z)dvdzds, ε,ε >, t T, T )

14 35 EMILIAN BULGARIU 4 By applying Gronwall s lemma to the above relation, we deduce (45) t k ij θ,i (s)θ,j (s)dvds =, t, T ) T Having that k ij is a positivedefinitetensor, theassumptionmeas Σ 8 = and the null external given data, from relations (9), (4) and (45) we have that (46) u i (x,t) =, ϕ i (x,t) =, ψ(x,t) =, θ(x,t) = in, T ) Finally, we extend relations (46) to,t) using the same procedure as at the end of the proof of the previous theorem and so, for hypotheses (H ) and (H 3 ), we have the uniqueness of solutions of the boundary-initial value problem (P) Remark 4 If we assume that (H ), (H ) and (H 3 ) hold true, we obtain the uniqueness of the solution of the boundary-initial value problem (P) in,t), without any procedure of extension We can now compare our results with the uniqueness theorem of Passarella and Tibullo 8: - in Theorem 4 we extend their result in a particular case: W is considered a positive semidefinite quadratic form and we had no condition for a, but we supplementary imposed that meas Σ 8 = ; - in Theorem 4 we discussed a different class of problems than the one considered by Passarella and Tibullo because we don t impose any restriction on W and the condition considered for a is complementary to the one they considered 5 Continuous dependence with respect to the final data In this section, we obtain the continuous dependence with respect to the final data in the context of the hypotheses (H ), (H ) and (H 3 ) This result is obtained without imposing any constraining restriction upon the solution Theorem 5 If ϖ = u i,ϕ i,ψ,e ij,κ ij,γ i,t ij,m ij,π i,σ,θ,θ,i,q i is a solution of the boundary-initial value problem (P) corresponding to the external given data D =,,,,u i, u i,ϕ i, ϕ i,ψ, ψ,θ,,,,,,,,

15 5 BACKWARD IN TIME THERMO-MICROSTRETCH ELASTICITY 353 assuming that the hypotheses (H ),(H ) and (H 3 ) hold true, we have the estimate (5) E(t)+ t T k ij θ,i (s)θ,j (s)dvds E()exp(Mt), t,t) where M > is a constant which depends on T,k m,λ and δ defined in the previous sections and (5) E(t)= ρ u i (t) u i (t)+i ij ϕ i (t) ϕ j (t)+3j ψ (t)+w(t) aθ (t)dv Proof To study the continuous dependence with respect to the initial data means that the solution of the boundary-initial values problem (P) do not change significantly when we introduce small perturbations of initial data Because the boundary-initial values problem (P) is linear, then the continuous dependence with respect to the initial data is equivalent to the stability of the null solution Therefore, we consider ϖ = u i,ϕ i,ψ,e ij,κ ij,γ i,t ij,m ij,π i,σ,θ,θ,i,q i a solution of the boundaryinitial value problem (P) corresponding to the external given data D =,,,,u i, u i,ϕ i, ϕ i,ψ, ψ,θ,,,,,,, Combining (39) and (3) in the context of the external given data D with (5), we have (53) E(t)+ t T k ij θ,i (s)θ,j (s)dvds = E()+4 t βij ė ij (s)θ(s) +C ij κ ij (s)θ(s)+ξ i γ i (s)θ(s)+ζ ψ(s)θ(s) dvds, t,t) Using Schwarz s inequality, the arithmetic-geometric mean inequality, the divergence theorem combined with relations (5), (9), (3), (5) (6) and (43), we obtain a convenient approximation for the integral in the right-hand side of (53) (54) 4 t ε +ε ε ε β ij ė ij (s)θ(s)+c ij κ ij (s)θ(s)+ξ i γ i (s)θ(s)+ζ ψ(s)θ(s)dvds t ρ u i (s) u i (s)+i ij ϕ i (s) ϕ j (s)+3j ψ (s) dvds + T ( ε δ λ +ε δ ) t k m T k ij θ,i (s)θ,j (s)dvds

16 354 EMILIAN BULGARIU 6 for all ε,ε >,t,t), and λ defined by (43) To simplify the above relation, we choose ε,ε and M to be (55) ε = k mλ 4T δ, ε = k m 4T δ, M = 8T (δ +λδ ) k m λ and thus, substituting relation (54) in the integral in the right-hand side of (53), we get (56) t E(t)+ k ij θ,i (s)θ,j (s)dvds E() T t +M ρ u i (s) u i (s)+i ij ϕ i (s) ϕ j (s)+3j ψ (s) dvds, t,t) Because we work under the hypotheses (H ), (H ) and (H 3 ), we can write relation (56) in the form (57) E(t)+ +M t t k ij θ,i (s)θ,j (s)dvds E() T s k ij θ,i (z)θ,j (z)dvdz T { E(s)+ } ds, t,t) Wecaneasily observethatweareinthehypothesesofthegronwall s lemma and so the estimate (5) holds true and hence the proof is complete Acknowledgments The author acknowledges support from the Romanian Ministry of Education and Research through CNCSIS -UEFISCSU, project PN II-RU TE code 84, no 86/37 REFERENCES Ames, KA; Payne, LE Stabilizing solutions of the equations of dynamical linear thermoelasticity backward in time, Stability Appl Anal Contin Media, (99), 43 6 Ames, KA; Straughan, B Non-Standard and Improperly Posed Problems, Mathematics in Science and Engineering Series, Academic Press, 94, 997

17 7 BACKWARD IN TIME THERMO-MICROSTRETCH ELASTICITY Chiriţă, S Uniqueness and continuous dependence of solutions to the incompressible micropolar flows forward and backward in time, Internat J Engrg Sci, 39 (), Chiriţă, S; Ciarletta, M Spatial behavior in dynamical thermoelasticity backward in time, The Fourth International Congress on Thermal Stresses THERMAL STRESSES, Osaka, Japan, Ciarletta, M On the uniqueness and continuous dependence of solutions in dynamical thermoelasticity backward in time, J Thermal Stresses, 5 (), Eringen, AC Theory of thermomicrostretch elastic solids, Internat J Engrg Sci, 8 (99), Knops, RJ; Payne, LE On the uniqueness and continuous data dependence in dynamical problems of linear thermoelasticity, Internat J Solids Structs, 6 (97), Passarella, F; Tibullo, V Some results in linear theory of thermoelasticity backward in time for microstretch materials, J Therm Stresses, 33 (), Quintanilla, R; Straughan, B Energy bounds for some non-standard problems in thermoelasticity, Proc R Soc Lond Ser A Math Phys Eng Sci, 46 (5), 47 6 Serrin, J The initial value problem for the Navier-Stokes equations, 963 Nonlinear Problems (Proc Sympos, Madison, Wis, 96), pp Univ of Wisconsin Press, Madison, Wis Received: 7IV Revised: VI Accepted: 5VI Faculty of Mathematics, Al I Cuza University, Iaşi, Carol I, no, 756-Iaşi, ROMANIA

18

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