Decay of solutions in nonsimple thermoelastic bars

International Journal of Engineering Science 48 (2010) 1233–1241 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci Decay of solutions in nonsimple thermoelastic bars Hugo D. Fernández Sare a, Jaime E. Muñoz Rivera a, Ramón Quintanilla b,* a b National Laboratory of Scientific Computations, LNCC/MCT, Rua Getúlio Vargas 333 Quitandinha, Petropolis, CEP 25651-070, RJ, Brazil Matemática Aplicada 2, UPC, C. Colón 11, 08222 Terrassa, Barcelona, Spain a r t i c l e i n f o Article history: Available online 8 June 2010 Dedicated to K.R. Rajagopal with great esteem on the occasion of his 60th birthday. a b s t r a c t In this paper we investigate the asymptotic behavior of the semigroup associated to the solutions of the initial boundary value problem for a one-dimensional nonsimple thermoelastic solids. We show that the semigroup is exponentially stable but is not analytic. Moreover we show the impossibility of time localization of the solutions. Ó 2010 Elsevier Ltd. All rights reserved. Keywords: Nonsimple thermoelastic bars Exponential decay Analyticity Localization of solutions 1. Introduction The material response of materials to stimuli depends in a relevant way on its internal structure, however classical elasticity does not consider the inner structure. Thus, it has been needed to develop some new mathematical models for continuum materials where this kind of effects were taken into account. Some of them are the porous elastic media, micropolar elastic solids, materials with microstrucutre, nonlocal continuum and nonsimple elastic solids. They are accounting for effects related to the size of defects and the microstructures. In the present paper we consider the last ones. We recall that from a mathematical point of view, these materials are characterized by the inclusion of higher order gradients of displacement in the basic postulates. They were introduced by Green and Rivlin [5], Mindlin [13] and Toupin [19,20]. More details on the subject can be found in the current books [4,6] on non-classical elasticity theories. The interest to introduce high order derivatives consists in the fact that the possible configurations of the materials are clarified more and more finely by the values of the successive higher gradients. We here will use the theory and the notation in the way developed by Iesan in his book [6]. When supply terms are not present the evolution equations are qu€ i ¼ T ji;i  Skji;kj ; qT 0 N_ ¼ Q i;i : The displacement of typical particles at the point x at time t is ui = ui(t, x); x 2 B (B is a bounded domain). The temperature in each point x and the time t is given by h = h(t, x). We denote by q the mass density, Tji, Skji the stress and the hyper-stress, N the entropy density, Qi the heat flux vector and T0 the uniform temperature at the reference configuration which is assumed strictly positive. The constitutive equations for an isotropic and centrosymmetric material become * Corresponding author. E-mail address: [email protected] (R. Quintanilla). 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.04.014 1234 H.D. Fernández Sare et al. / International Journal of Engineering Science 48 (2010) 1233–1241 T ij ¼ kerr dij þ 2l þ bhdij ; Sijk ¼ a1 2 qN ¼ ch  berr ; Q i ¼ Kh;i ; ðjrri djk þ 2jjrr dij þ jrrj dij Þ þ a2 ðjirr djk þ jjrr dik Þ þ 2a3 jrrk dij þ 2a4 jijk þ a5 ðjkji þ jkij Þ: Here eij ¼ 12 ðui;j þ uj;i Þ and jijk = uk,ji. To define a problem we need to impose initial and boundary conditions. Among the different boundary conditions we can assume (see [6], p. 256), we here restrict our attention to the ones of the kind ui ¼ 0 and ðui;j nj ¼ 0 or Srsi nr ns ¼ 0Þ and ðh ¼ 0 or Q i ni ¼ 0Þ: Here ni denotes the normal vector to the smooth boundary of the domain B. It is worth citing several papers on existence and decay results in this theory published in the recent years [7,8,15]. As the constitutive equations of nonsimple solids contain first and second order gradients, it is of interest understanding the relevance of the thermal dissipation in the global decay of the system. In particular, we would like to clarify the influence on the decay of the thermal dissipation, studying the longterm dynamics. This is the main aim of the present paper. In view of the known results for the classical theories we cannot expect to obtain a general result independent of the dimension. We recall that in classical or type III thermoelasticity (see [9,11,18,21]) the solutions of the one-dimensional problem decay to zero exponentially. However for dimension greater than one we only can expect for polynomial decay. In fact, for several geometries and isotropic materials there exist undamped isothermal solutions which do not decay with the time. We cannot look here for an uniform approach, because for nonsimple solids it seems a difficult problem. Our work wants to be a first step in the study of the time behavior of the nonsimple thermoelasticity and we will center our attention in the one-dimensional case. At the same time, it is worth noting that we can always see the one-dimensional case as a first approach to the case of thermoelastic bars by assuming that the cross-section is so small that we can consider the displacement and the temperature uniform for all the points in the same cross-section. This work continues the task developed in several recent papers where it has been tried to clarify the time decay of solutions for several non-classical thermomechnical situations [1– 3,12,14,17]. However, it is worth recalling that each one of these situations needs of a particular and different study and the arguments do not agree between them. As we emphasize the one-dimensional beam, it is suitable to set down the system of equations, the boundary conditions and the initial conditions in this particular case and with the easiest notation. We consider a beam composed by nonsimple thermoelastic continua that occupies the interval (0, L). The displacement and the temperature of typical particles at the point x at time t are u, h where u = u(t, x); h = h(t, x); x 2 (0, L). We denote by T, S the stress and the hyper-stress and Q the heat flux vector. Other functions as the mass density or the entropy use the same notation defined previously. In the absent of body forces the system of equations consists of the equation of motion qutt ¼ T x  Sxx ; ð1:1Þ the energy equation qT 0 N_ ¼ Q x ; ð1:2Þ and the constitutive equations T ¼ lux þ bh; S ¼ auxx ; ð1:3Þ qN ¼ bux þ ch; Q ¼ Khx : ð1:4Þ The material is assumed homogeneous and then the constitutive parameters l, b, a and c are assumed constants. q, c, l, K and a are assumed positive constants, the sign of b is not determined, but it is different from zero. The physical interpretation of the positivity of the mass density q and the heat capacity c is obvious and they are usually imposed in the mathematical and physical studies concerning thermoelasticity. Also the positive sign of the thermal conductivity K uses to be accepted. It is related to the defining property of a definite mechanical heat conductor. It is also known that when the thermal conductivity is negative we find an ill posed problem, that is a problem where the continuous dependence with respect initial data fails. Same problem happens in case that a is negative. Thus, to have a well posed problem we need to assume its positivity. In a more general context the positivity of a and l is related with the positivity of the internal energy and may be interpreted with the help of the theory of mechanical stability. If we substitute the constitutive equations into the motion equation and the energy equation, we obtain the system of field equations qutt  luxx þ auxxxx  bhx ¼ 0 in ð0; 1Þ  ð0; LÞ; cht  jhxx  buxt ¼ 0 in ð0; 1Þ  ð0; LÞ; ð1:5Þ KT 1 0 . where j ¼ The main aim of the papers is to study the problem proposed by the system (1.5), with the following initial conditions uð0; xÞ ¼ u0 ðxÞ; ut ð0; xÞ ¼ u1 ðxÞ; hð0; xÞ ¼ h0 ðxÞ; and associated with boundary conditions of the type x 2 ð0; LÞ ð1:6Þ H.D. Fernández Sare et al. / International Journal of Engineering Science 48 (2010) 1233–1241 uðt; 0Þ ¼ uðt; LÞ ¼ uxx ðt; 0Þ ¼ uxx ðt; LÞ ¼ hx ðt; 0Þ ¼ hx ðt; LÞ ¼ 0 in ð0; 1Þ; 1235 ð1:7Þ uðt; 0Þ ¼ uðt; LÞ ¼ uxx ðt; 0Þ ¼ uxx ðt; LÞ ¼ hðt; 0Þ ¼ hx ðt; LÞ ¼ 0 in ð0; 1Þ; ð1:8Þ uðt; 0Þ ¼ uðt; LÞ ¼ uxx ðt; 0Þ ¼ uxx ðt; LÞ ¼ hðt; 0Þ ¼ hðt; LÞ ¼ 0 in ð0; 1Þ; uðt; 0Þ ¼ uðt; LÞ ¼ ux ðt; 0Þ ¼ ux ðt; LÞ ¼ hx ðt; 0Þ ¼ hx ðt; LÞ ¼ 0 in ð0; 1Þ; ð1:10Þ ð1:11Þ uðt; 0Þ ¼ uðt; LÞ ¼ ux ðt; 0Þ ¼ ux ðt; LÞ ¼ hx ðt; 0Þ ¼ hðt; LÞ ¼ 0 in ð0; 1Þ ð1:13Þ uðt; 0Þ ¼ uðt; LÞ ¼ ux ðt; 0Þ ¼ ux ðt; LÞ ¼ hðt; 0Þ ¼ hðt; LÞ ¼ 0 in ð0; 1Þ: ð1:14Þ uðt; 0Þ ¼ uðt; LÞ ¼ uxx ðt; 0Þ ¼ uxx ðt; LÞ ¼ hx ðt; 0Þ ¼ hðt; LÞ ¼ 0 in ð0; 1Þ; uðt; 0Þ ¼ uðt; LÞ ¼ ux ðt; 0Þ ¼ ux ðt; LÞ ¼ hðt; 0Þ ¼ hx ðt; LÞ ¼ 0 in ð0; 1Þ; ð1:9Þ ð1:12Þ or It is worth noting that in the one-dimensional case the boundary conditions proposed previously agree with the ones proposed in the general case. That is ui,jnj = 0 becomes ux = 0, Srsinrns = 0 becomes uxx = 0 and Qini = 0 is hx = 0. Our purpose in this work is to investigate the stability and regularity of the solutions to system (1.5), with the initial conditions (1.6) and the boundary conditions (1.7) or ((1.8) or (1.9), . . . , or (1.14)). It is also suitable to pay attention to the mathematical aspect of the question we study here and to relate it with other similar problems. In the case of the nonsimple thermoelasicity we have the coupling of the usual heat conduction with a conservative equation with second order spatial derivative. The problem is to know if the dissipation mechanism determined by the heat conduction is so strong to guarantee a similar behavior as in the classical theory. It is worth noting that the combination of the conservative and the dissipative equations recall very much the problem determined in the case of the plate theory. The main difference is concerning the coupling terms. In our problem the coupling is weaker than the one of the plate theory. However, the exponential decay can be proved, but (of course) we cannot expect for the analyticity of the semigroup (as it will be proved). In contrast with the case of the plate theory [10], the lack of the analyticity is a consequence of the weakness of the coupling. In this paper we show that the solutions of our problem are exponentially stable. That is the decay is controlled by an negative exponential. This fact is relevant from the mechanical point of view. It implies that if we consider a thermoelastic perturbation, then after a small period of time the perturbations are so small that we can neglect. We also see that the solutions are not analytic functions. That is, in general we cannot guarantee that the orbits of the dynamical system defined by our system of equations are analytic functions with respect the time variable. Later we see if the decay is so fast to guarantee that the solutions vanish after a finite period of time. We will see that this is not possible. That is we will prove that the only solution which can be identically zero after a finite period of time is the null solution. This result is relevant in the sense that gives a good complement to the exponential stability of solutions. That is the solution decay in a fast way, but no so fast to guarantee the localization of the solutions. This paper is organized as follows. In Section 2, we state the frame where the problem is well posed. In Section 3 we prove the exponential decay of solutions and in Section 4 we prove the lack of the analyticity of the solutions. In the last section we show the impossibility of the localization of solutions. 2. Existence of solutions In this section we obtain an existence result for the solutions of the problem determined by (1.5) and (1.6) and associated with boundary conditions of the type (1.7), (1.8), (1.9), (1.10), (1.11), (1.12), (1.13) or (1.14). To this end we define the Hilbert spaces V 1 :¼ H2 ð0; LÞ \ H10 ð0; LÞ V 2 :¼ H20 ð0; LÞ;   Z L L1 :¼ w 2 L2 ð0; LÞ; wdx ¼ 0 ; Lj :¼ L2 ð0; LÞ; ð2:1Þ for j ¼ 2; 3; 4: 0 ð2:2Þ Therefore the phase space will be H ¼ Hij ¼ V i  L2 ð0; LÞ  Lj ; i ¼ 1; 2; j ¼ 1; 2; 3; 4: In these Hilbert space we define the inner product hU; U  iH ¼ 1 2 Z L 0   qv v  þ lux u x þ auxx uxx þ chh dx; where U = (u, v, h) and U* = (u*, v*, h*). Now, let us to introduce the operators 0 n 0 o B l @ ðÞ  qa @ xxxx ðÞ A ¼ Aij :¼ B @ q xx 0 I 0 0 q @ x ðÞ b @ ðÞ c x b j@ c xx ðÞ 1 C C: A ð2:3Þ 1236 H.D. Fernández Sare et al. / International Journal of Engineering Science 48 (2010) 1233–1241 Let us define M1 ¼ fh 2 H2 ; hx ð0Þ ¼ hx ðLÞ ¼ 0g; 2 M3 ¼ fh 2 H ; hx ð0Þ ¼ hðLÞ ¼ 0g; M2 ¼ fh 2 H2 ; hð0Þ ¼ hx ðLÞ ¼ 0g; ð2:4Þ M4 ¼ fh 2 H2 ; hð0Þ ¼ hðLÞ ¼ 0g; ð2:5Þ So the domain can be DðAij Þ ¼ Hij \ H4  V i  Mj : Using these notations, the initial-boundary value problem (1.5)–(1.7) can be rewritten as the following initial value problem d UðtÞ ¼ Aij UðtÞ; dt Uð0Þ ¼ U 0 ; i ¼ 1; 2 and 1; 2; 3; 4; where U(t) = (u, ut, h)0 and U0 = (u0, u1, h0)0 , and the prime is used to denote the transpose. In order to simplify the notation, we write A and H instead of Aij and Hij respectively for i = 1, 2 and j = 1, 2, 3, 4. Lemma 2.1. The operator A defined previously, is the infinitesimal generator of a C0-semigroup of contractions over H. Proof. Observe that DðAÞ ¼ H. We will show that A is a dissipative operator and that 0 2 qðAÞ, then our conclusion will follow using the well-known Lumer–Phillips theorem (see [16]). In fact RehAU; UiH ¼ j Z 0 L jhx j2 dx 6 0; therefore the operator A is dissipative. Finally, given F ¼ ðf ; h; qÞ 2 H, there exists a unique U = (u, v, h) in DðAÞ such that AU ¼ F in H. That is v ¼ f; luxx  auxxxx þ bhx ¼ qh; bv x þ jhxx ¼ cq: Therefore, there exists a unique solution of the problem AU ¼ F. Thus 0 2 qðAÞ. The proof is complete. h A direct consequence of this Lemma is the following result. Theorem 2.2. For any U 0 ¼ ðu0 ; u1 ; h0 Þ0 2 DðAÞ there exists a unique solution U(t) = (u, ut, h)0 of (1.5) and (1.6) satisfying ðu; ut ; hÞ0 2 C 1 ð½0; 1Þ; HÞ \ C 0 ð½0; 1Þ; DðAÞÞ; 3. Exponential stability Here we prove the exponential stability of the semigroup associated to the solutions of the system (1.5) and (1.6). To this end we shall use the following well-known result from semigroup theory (see e.g. [11, Theorem 1.3.2]). Theorem 3.1. A semigroup of contractions fetA gtP0 with infinitesimal generator A on a Hilbert space H with norm k  kH is exponentially stable if and only if iR  qðAÞ ð3:1Þ 9 C > 0 8 k 2 R : kðikI  AÞ1 kH 6 C: ð3:2Þ and To prove the exponential stability we use the following two Lemmas. Lemma 3.2. iIR = {ik; k 2 IR} is contained in qðAÞ. Proof. The operator A1 : H ! H is compact. In fact, consider (Fn) a bounded sequence in H and (Un) the sequence in DðAÞ such that F n ¼ AU n ; U n ¼ ðun ; v n ; hn Þ. Since A1 2 LðHÞ there exists a positive constant C such that kU n kH þ kAU n kH 6 C; 8n 2 N: ð3:3Þ m j From (3.3) we conclude that (un, vn, hn) is bounded in DðAÞ ¼ DðAij Þ. Since the embedding of H (0, L) in H (0, L), m > j, is compact, there exists a subsequence (um, vm, hm) and functions (u, v, h) such that H.D. Fernández Sare et al. / International Journal of Engineering Science 48 (2010) 1233–1241 1237 ðum ; v m ; hm Þ ! ðu; v ; hÞ in H ¼ Hij ; that is, the subsequence ðA1 F m Þ converges in H. Suppose that there exists k 2 IR, k – 0, such that ik is in the spectrum of A. Since A1 is compact, then ik must be a eigenvalue of A. Therefore, there is a vector U, U – 0, such that ðikI  AÞU ¼ 0 in H or equivalently iku  v ¼ 0; ð3:4Þ ikch  bv x  jhxx ¼ 0; ð3:6Þ ikqv  luxx þ auxxxx  bhx ¼ 0; ð3:5Þ Since hðikI  AÞU; UiH ¼ 0 we have Z L jhx j2 dx ¼ 0; 0 and then h = 0. By (3.4) and (3.6) we get u = v = 0. Thus we have a contraction and the proof is complete. h Lemma 3.3. The operator A defined in (2.3) satisfies lim sup kðikI  AÞ1 kLðHÞ < 1: jkj!1 Proof. Given k 2 IR and F ¼ ðf ; g; hÞ 2 H, there exists a unique U ¼ ðu; v ; hÞ 2 DðAÞ, such that ðikI  AÞU ¼ F, that is, iku  v ¼ f in V i ; ð3:7Þ 2 ikqv  luxx þ auxxxx  bhx ¼ qg in L ; ð3:8Þ ikch  bv x  jhxx ¼ ch in Li ; ð3:9Þ with i = 1, 2 and j = 1, 2, 3, 4. Also note that RehðikI  AÞU; UiH ¼ j Z L 0 jhx j2 dx ¼ RehF; UiH and then Z L jhx j2 dx 6 CkFkH kUkH ; 0 ð3:10Þ for a positive constant C. Multiplying (3.8) by u in L2(0, L) and using (3.7) we obtain l Z L 0 jux j2 dx þ a Z L 0 juxx j2 dx ¼ b Z 0 L  dx þ q hx u Z L 0 jv j2 dx þ q Z L 0 v f dx þ q Z L  dx: gu 0 which, using (3.10) implies that l 2 Z L 0 jux j2 dx þ a Z L 0 juxx j2 dx 6 qkv k2 þ CkUkH kFkH ; ð3:11Þ for a positive constant C. The next step is to estimate jjv jjL2 . To this end we define the functions /, x, z and y as solutions of the following problems:  /xx ¼ v in ½0; L with /x ð0Þ ¼ /x ðLÞ ¼ 0;  xxx ¼ h in ½0; L with xx ð0Þ ¼ xx ðLÞ ¼ 0;  zxx ¼ h in ½0; L with zð0Þ ¼ zðLÞ ¼ 0;  yxx ¼ g in ½0; L with yð0Þ ¼ yðLÞ ¼ 0: ð3:12Þ ð3:13Þ ð3:14Þ ð3:15Þ Note that, using Poincaré’s inequality, we have kzx kL2 6 C p khkL2 ; ð3:16Þ kyx kL2 6 C p kgkL2 ; ð3:17Þ where Cp > 0 is the Poincaré’s constant. Then, in order to estimate kv kL2 we multiply (3.9) by /x in L2(0,L) to obtain ikcðh; /x ÞL2  kðhxx ; /x ÞL2  bðv x ; /x ÞL2 ¼ cðh; /x ÞL2 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} :¼I1 :¼I2 :¼I3 :¼I4 : ð3:18Þ 1238 H.D. Fernández Sare et al. / International Journal of Engineering Science 48 (2010) 1233–1241 Now we estimate term by term. First, using (3.13) and (3.12) we obtain the identities I1 ¼ ikcðxxx ; /x ÞL2 ¼ ikcðxx ; /xx ÞL2 ¼ cðxx ; ikv ÞL2 ; so, we replace ikv given by (3.8). Then, using (3.13) and (3.15) we obtain c cl ca ca cb  x¼L þ ðh ; u Þ 2 þ khk2L2  cðxx ; yxx ÞL2 ðx ; lu  auxxxx þ bhx þ qgÞL2 ¼ ðh; ux ÞL2  ½hu q xx x¼0 q x xx L q q x xx q cl ca ca cb  x¼L þ ðh ; u Þ 2 þ khk2L2  cðh; yx ÞL2 : ¼ ðh; ux ÞL2  ½hu q xx x¼0 q x xx L q q I1 ¼ ð3:19Þ Also, using (3.12) we deduce and I3 ¼ bkv k2L2 : I2 ¼ kðhx ; v ÞL2 ð3:20Þ Finally, using (3.14) we have I4 ¼ cðzxx ; /x ÞL2 ¼ cðzx ; /xx ÞL2 ¼ cðzx ; v ÞL2 : ð3:21Þ Then, substituting (3.19)–(3.21) into (3.18) results ca bkv k2L2 ¼  q xx x¼L ½hu x¼0 þ cb q khk2L2 þ cl q ðh; ux ÞL2 þ ca q ðhx ; uxx ÞL2  cðh; yx ÞL2 þ kðhx ; v ÞL2  cðzx ; v ÞL2 ; which, using Sobolev’s embedding and inequalities (3.16), (3.17) and (3.10), we obtain 3kv k2L2 6 ejuxx ð0Þj2 þ ejuxx ðLÞj2 þ C e kUkH kFkH þ l 8 a kux k2L2 þ kuxx k2L2 : 4 ð3:22Þ for all e > 0 with Ce > 0. Here, note that in the case i = 1 the boundary terms in (3.22) do not appears. The problem is to estimate that terms in the case i = 2 and j = 1, 2, 3, 4. To overcome this problem we define the linear real function qðxÞ : ½0; L ! R given by 2 qðxÞ ¼  x þ 1; L for all x 2 ½0; L: ð3:23Þ So, multiplying (3.8) by q(x)ux in L2(0,L) results ikqðv ; qux ÞL2  lðuxx ; qux ÞL2 þ aðuxxxx ; qux ÞL2 bðhx ; qux ÞL2 ¼ qðg; qux ÞL2 : |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} :¼K 1 :¼K 2 ð3:24Þ :¼K 3 Using (3.7) we deduce that 1 1 K 1 ¼ qðv ; qv x  qf x ÞL2 ¼ qðv ; qv x ÞL2  qðv ; qf x ÞL2 : Taking the real part, using the definition of q(x) and the boundary conditions to i = 2, we obtain q 1 ReðK 1 Þ ¼  kv k2L2  qReðv ; qf x ÞL2 : L ð3:25Þ Analogously, taking the real part we have ReðK 2 Þ ¼  l L kux k2L2 : ð3:26Þ Finally, integrating by parts and using the properties of q(x) we have ReðK 3 Þ ¼ a 2 a 3a juxx ð0Þj2 þ juxx ðLÞj2  kuxx k2L2 : 2 L ð3:27Þ Then, substituting (3.25)–(3.27) into (3.24) we have a 2 a q l 3a 1 juxx ð0Þj2 þ juxx ðLÞj2 ¼ kv k2L2  kux k2L2 þ kuxx k2L2 þ qReðv ; qf x ÞL2 þ bReðhx ; qux ÞL2 þ Reðf 2 ; qux ÞL2 ; 2 L L L which implies h i juxx ð0Þj2 þ juxx ðLÞj2 6 C 1 kv k2L2 þ kux k2L2 þ kuxx k2L2 þ C 2 kUkH kFkH : Therefore, choosing e > 0 in (3.22) such that e 6 min   1 l a ; ; ; C 1 8C 1 4C 1 ð3:28Þ H.D. Fernández Sare et al. / International Journal of Engineering Science 48 (2010) 1233–1241 1239 we obtain, substituting (3.28) into (3.22), that 2kv k2L2 6 C e kUkH kFkH þ l 4 a kux k2L2 þ kuxx k2L2 : 2 ð3:29Þ So, combining (3.10), (3.11) and (3.29), there exists a constant M > 0 independent of k and F 2 H such that kUkH 6 CkFkH ; which implies condition (3.2). Thus, the proof is complete. h 4. A lack of analyticity The aim of this section is to show that the semigroup associated to system (1.5) is not analytic in general. To this end we use the following characterization of analytic semigroups. For the proof see [11]. Theorem 4.1. Let qðAÞ be the resolvent set of the linear operator A. Then, a semigroup of contractions fetA gtP0 in a Hilbert space H with norm k  kH is of analytic type if and only if and iR  qðAÞ ði :¼ pffiffiffiffiffiffiffi 1Þ ð4:1Þ lim sup kkðikI  AÞ1 kLðHÞ < 1: ð4:2Þ jkj!1 We first consider the case of boundary conditions (1.7), this is uðt; 0Þ ¼ uðt; LÞ ¼ uxx ðt; 0Þ ¼ uxx ðt; LÞ ¼ hx ðt; 0Þ ¼ hx ðt; LÞ ¼ 0 in ð0; 1Þ: The main result of this section is formulated in the following theorem Theorem 4.2. The semigroup SA11 ðtÞ, defined on the Hilbert space H11 , is not analytic. Proof. Applying Theorem 4.1, it is sufficient to show that there is a sequence (kn) of real numbers and a bounded sequence (Fn) in H11 such that kn ? 1 as n ? 1 and lim kkn ðikn I  A11 Þ1 kH11 ¼ 1: n!1     In fact, for each n 2 N, we consider F n ¼ 0; sin nLp x ; 0 and let us the vector Un = (un, vn, hn) belongs to DðA11 Þ be the unique solution of the resolvent equation ðikI  A11 ÞU n ¼ F n , with k 2 IR, or equivalently ikun  v n ¼ 0 in H2 \ H10 ð0; LÞ; ikqv n  lunxx þ aunxxxx  bhnx ¼ sin ikchn  bv nx  jhnxx ¼ 0 in L1 : np x L in L2 ð0; LÞ; Due to boundary conditions (1.7), the solutions of the above system are the form un ¼ An sin Then we obtain np x ; L hn ¼ Bn cos np x : L np 2 np 4 np  k2 qAn þ l An þ a An þ b Bn ¼ 1; L L L np np 2 An þ j Bn ¼ 0: ikcBn  ibk LqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL l þ aðnLp Þ2 and get We take k ¼ kn ¼ nqpL Bn ¼ L bpn 1 2 jqpn C L B : An ¼ @c þ i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 2 lL2 þ ap2 n2 b p n ; Since kU n k2H11 ¼ q ¼ 0 Z 0 aL 2b2 L Z L Z L Z L junx j2 dx þ a junxx j2 dx þ c jhn j2 dx P a junxx j2 dx 0 0 0 0 ! j2 q2 p2 n2 2 ; 8n 2 N; c þ 2 lL þ ap2 n2 jv n j2 dx þ l Z L ð4:3Þ 1240 H.D. Fernández Sare et al. / International Journal of Engineering Science 48 (2010) 1233–1241 we have lim kkn U n kH11 ¼ 1; n!1 which complete the proof of the Theorem. h Remark 4.3. In the case of boundary conditions (1.8), (1.9), . . . , (1.13) or (1.14), we also expect that the semigroup is not analytic. Note that, in these cases, we cannot apply the procedure used previously, because we do not have appropriate solutions (satisfying boundary conditions) to solve system (4.3). Explicitly speaking, the non-analyticity of the semigroup associated to system (1.5) with boundary conditions (1.8)–(1.14) is an open problem. 5. Impossibility of localization The aim of this section is to prove the impossibility of time localization of solutions of the system of the nonsimple thermoelasticity with the boundary conditions (1.7). To prove the impossibility of solutions we will show the uniqueness of solutions of the backward in time problem. Thus, it will be suitable to recall that the system of equations which govern the backward in time problem is: qutt  luxx  auxxxx  bhx ¼ 0 in ð0; 1Þ  ð0; LÞ; cht þ jhxx  buxt ¼ 0 in ð0; 1Þ  ð0; LÞ: ð5:1Þ uð0; Þ ¼ 0; ð5:2Þ We will study the problem determined by system (5.1), with boundary conditions (1.7) and with the null initial conditions ut ð0; :Þ ¼ 0; hð0; Þ ¼ 0: Lemma 5.1. Let us assume that the conditions hold. Let (u, h) be a solution of the problem determined by the system (5.1), the initial conditions (5.2) and the boundary conditions (1.7). Then u = h = 0. Proof. The energy conservation gives E1 ðtÞ ¼ 1 2 Z L 0 ðljux j2 þ ajuxx j2 þ qjut j2 þ ch2 Þdx ¼ Z t 0 Z L jh2x dx ds: 0 If we multiply the first equation of (5.1) by ut and the second one by h, we obtain E2 ðtÞ ¼ 1 2 Z L 0 ðljux j2 þ ajuxx j2 þ qjut j2  ch2 Þdx ¼ Z t 0 Z L 0 ð2but hx  jh2x Þdx ds: We need a third equality. To obtain it, we use the Lagrange identity method. For a fixed t 2 (0, T), we use the identities @ _ uð2t _ € ðsÞuð2t _ _ u €ð2t  sÞ; ðquðsÞ  sÞÞ ¼ qu  sÞ  quðsÞ @s @ _ _ ðchðsÞhð2t  sÞÞ ¼ chðsÞhð2t  sÞ  chðsÞhð2t  sÞ: @s From the field equations, the boundary conditions and the null initial conditions, we obtain that the following equality Z L 0 ðljux j2 þ ajuxx j2 þ ch2 Þdx ¼ Z L 0 qjut j2 dx: Thus, we have E2 ðtÞ ¼ Let Z L 0 ðljux j2 þ ajuxx j2 Þdx:  be a small, but positive constant. Let us consider E(t) = E2(t) + E1(t). We note that EðtÞ ¼ 1 2 Z 0 L ððqjut j2 þ ch2 Þ þ ð2 þ Þðljux j2 þ ajuxx j2 Þdx; is a positive function and it defines a measure on the solution. We take EðtÞ ¼ ð1  Þ Z 0 t Z 0 L jjhx j2 dx ds þ 2 Z 0 t Z ð5:3Þ  strictly less than one, but greater than zero. As L but hx dx ds; 0 we have: dE ¼ ð1  Þ dt Z L 0 jjhx j2 dx ds þ 2 Z 0 L but hx dx: ð5:4Þ 1241 H.D. Fernández Sare et al. / International Journal of Engineering Science 48 (2010) 1233–1241 The A-G inequality implies that Z L but hx dx 6 K 1 0 Z 0 L qut ut dx þ 1 Z 0 L jjhx j2 dx; where 1 is as small as we want, but positive and K1 can be calculated in terms of the constitutive coefficients and If we take 1 6 1  , there exists a positive constant C such that dE 6C dt Z L 0 ðqut ut þ ch2 Þdx: 1. ð5:5Þ We obtain that the estimate dE 6 CEðtÞ; dt ð5:6Þ is satisfied for every t P 0. This inequality implies that E(t) 6 E(0)exp(C t) for every t greater than zero. As we assume null initial conditions we see that E(t)  0 for every t P 0. If we take into account the definition of E(t), it follows that u  0, h  0 for every t P 0 and then in view of the initial conditions, it follows that the only solution to our problem is the null solution. Thus, we can state the following: h Theorem 5.2. 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