J. Differential Equations 255 (2013) 2267–2290 Contents lists available at SciVerse ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Decay rates for Bresse system with arbitrary nonlinear localized damping Wenden Charles a,1 , J.A. Soriano b,2 , Flávio A. Falcão Nascimento c,∗,3 , J.H. Rodrigues b,4 a b c Center of Exact Sciences and Technology, Federal University of Acre, 69920-900, Rio Branco, AC, Brazil Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil Faculty of Philosophy Dom Aureliano Matos, State University of Ceará, 62.930-000, Limoeiro do Norte, CE, Brazil a r t i c l e i n f o Article history: Received 30 November 2012 Revised 25 May 2013 Available online 2 July 2013 a b s t r a c t In this paper we consider a vibrating system of Bresse type, in a one-dimensional bounded domain with nonlinear localized damping mechanisms acting in all the three wave equations. We obtain some rates of decay for its solutions with no restrictions around the coefficients as well as the condition of equal-speed wave propagation. A new result concerning an internal observability for the conservative system was also proved in order to reach the asymptotics above. © 2013 Elsevier Inc. All rights reserved. 1. Introduction Let L > 0 be given. In this article, we are going to consider the Bresse system subject to dissipative mechanisms as follows: ⎧ ⎨ ρ1 ϕtt − κ (ϕx + ψ + lw )x − κ0l[ w x − lϕ ] + α1 (x) g 1 (ϕt ) = 0, ρ ψ − bψxx + κ (ϕx + ψ + lw ) + α2 (x) g2 (ψt ) = 0, ⎩ 2 tt ρ1 w tt − κ0 [ w x − lϕ ]x + κ l(ϕx + ψ + lw ) + α3 (x) g3 ( w t ) = 0 * Corresponding author. E-mail addresses: wenden@ufac.br (W. Charles), jaspalomino@uem.br (J.A. Soriano), flavio.falcao@uece.br (F.A. Falcão Nascimento), jh.rodrigues@ymail.com (J.H. Rodrigues). 1 Research supported by the Brazilian agency: CNPq, Ed. No. 70/2009 MCT/CNPq 145291/2010-3. 2 Research supported by the Brazilian agency: CNPq, Grant 305209/2011-6. 3 Research supported by the Brazilian agency: CNPq, Ed. No. 70/2009 MCT/CNPq 141878/2010-0. 4 Research supported by the Brazilian agency: CAPES. 0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2013.06.014 (1.1) 2268 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 in (0, L ) × R+ , where R+ = (0, +∞), with Dirichlet boundary conditions ϕ (0, t ) = ϕ ( L , t ) = ψ(0, t ) = ψ( L , t ) = w (0, t ) = w ( L , t ) = 0, t ∈ R+ , (1.2) and initial conditions ϕ (·, 0) = ϕ0 , ψt (·, 0) = ψ1 , ϕt (·, 0) = ϕ1 , w (·, 0) = w 0 , ψ(·, 0) = ψ0 , w t (·, 0) = w 1 . (1.3) The positive constants ρ1 , ρ2 , b, l, κ0 , κ are related to composition of the material. By w, ϕ , and ψ we are denoting, respectively, the longitudinal, vertical and shear angle displacements, and {ϕ , ψ, w } is a sought solution of (1.1)–(1.3). Elastic structures of the arches type are objects of study in many areas like mathematics, physics and engineering. For more details, the interested reader can visit the works [11,12] and references therein. As we shall see, the nonnegative localizing functions αi will be supposed to belong to L ∞ (0, L ) while the functions g i will be supposed to be continuous and monotone. Remark 1.1 (Timoshenko system). If l → 0, this model reduces to the well-known Timoshenko beam equations. Related to the objectives of this paper there are few results in the literature. We would like to mention the work of Liu and Rao [11] where they studied the asymptotic behavior of Bresse system, also known as the circular arch problem (see Fig. 1), in the context of linear thermoelasticity and proved that the exponential decay of its solutions occurs if and only if the following hypothesis of equal-speed waves of propagation holds ρ1 κ = , and κ = κ0 . ρ2 b (1.4) The relation (1.4) has been used in many works in order to establish exponential decay rates, see for instance [12,2,7,8]. However, in the case of different wave speed of propagation, i.e. when (1.4) fails, the same works mentioned before present only polynomial rates of decay. This assumption, on the other hand, is a mathematical hypothesis which is not realistic from physical point of view. In fact, it never happens since the Poison ratio is less than 12 , i.e. ν ∈ (0, 12 ). It is worth to mention that in all these papers, a linear damping mechanism has been considered. Our main goal is to obtain rates of decay for the nonlinear localized damped system (1.1)–(1.3) with no restrictions or relation between the coefficients, as well as the hypothesis of equal speed of propagation for the waves. Furthermore, an important internal observability inequality for the conservative system is also proved in order to achieve this aim. As far as we are concerned, this is the first work which establishes this internal observability. Multipliers technique was used in order to state this last result. Soriano and collaborators in the unpublished result Asymptotic stability for Bresse systems have also obtained some rates of decay for Bresse system with a full damping acting on the shear angle displacement equation and with localized damping mechanisms in the other equations. In the same work, an internal observability for the respective conservative system was also derived under the assumption of different speeds of propagation for the waves. Although, we believe that the observability established in the present paper seems to improve that one mentioned before, since the damping regions present less restrictions. Concerning our asymptotic stability result, we would like to mention the works of Cavalcanti et al. [5,4,6] where the authors made use of a method first introduced by Lasiecka and Tataru [10] and quite used lately. It is worth to emphasize the work [6] in which computations were well adapted in order to provide the desired decay rates for our problem. This method provides rates of decay which depend on a solution of an ordinary differential equation. W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 2269 This paper is structured as follows. Section 2 is devoted to the assumptions, preliminary results and the existence of solutions to the system (1.1)–(1.3) by using nonlinear semigroup theory. Section 3 brings a new and important observability result concerning the conservative system. Finally, Section 4 is dedicated to present the results concerning the rates of decay for the solutions of the nonlinear damped problem. 2. Assumptions, preliminary results and existence Let us consider the Bresse system (1.1)–(1.3). The following assumptions around the parameters of the problem are made: Assumption 2.1. The feedback function g i , for each i = 1, 2, 3, is continuous and monotone increasing, and, in addition, satisfies the following conditions: (i) g i (s)s > 0 for s = 0, (ii) ki s  g i (s)  K i s for |s| > 1, where ki and K i are positive constants, and ki  K i . Assumption 2.2. Assume that αi ∈ L ∞ (0, L ) are nonnegative functions such that αi (x)  αi > 0 in I i , i = 1, 2, 3, and Ĩ := 3  I i = ∅. (2.5) i =1 Remark 2.1. As we can see, the localizing functions allow us to consider dampings mechanisms acting in an arbitrarily small region of the beam. If {ϕ , ψ, w } is a solution of (1.1)–(1.3) then the energy of system related to this solution will be denoted by E (t ), with t nonnegative, and given by E (t ) = 1 L 2   ρ1 |ϕt |2 + ρ2 |ψt |2 + ρ1 | w t |2 + b|ψx |2 + κ0 | w x − lϕ |2 + κ |ϕx + ψ + lw |2 (x, t ) dx. 0 (2.6) We can prove that the system (1.1)–(1.3) is dissipative as stated below: Lemma 2.1. The energy functional E defined by (2.6), satisfies: dE (t ) dt L =− α1 (x) g1 (ϕt )ϕt + α2 (x) g2 (ψt )ψt + α3 (x) g3 ( w t ) w t dx  0, ∀t  0. (2.7) 0 Next, we are going to discuss the existence, uniqueness and smoothness of the solutions of (1.1)–(1.3). In order to do this, we will make use of nonlinear semigroups theory widely treated by Barbu in [1] and also by Brézis in [3]. First, let us consider the Hilbert space H := H 01 (0, L ) × L 2 (0, L ) × H 01 (0, L ) × L 2 (0, L ) × H 01 (0, L ) × L 2 (0, L ) endowed with the norm 2270 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 2 U 2H := {ϕ , Φ, ψ, Ψ, w , W} H L ρ1 |Φ|2 + ρ2 |Ψ |2 + ρ1 |W|2 + b|ψx |2 + k0 | w x − lϕ |2 + k|ϕx + ψ + lw |2 dx, ≡ 0 which is equivalent to the usual norm of H. If we denote V (t ) = {ϕ , ϕt , ψ, ψt , w , w t } then the initial boundary value problem (1.1)–(1.3) can be rewritten as a first order problem as follows: ⎧ ⎨ dV (t ) + A V (t ) = 0, ⎩ dt V (0) = V 0 (2.8) where V 0 = {ϕ0 , ϕ1 , ψ0 , ψ1 , w 0 , w 1 } and the operator A = D (A) ⊂ H → H is given by A = −( A 1 + A 2 ) with component operators defined by  3 D ( A 1 ) = H 01 (0, L ) ∩ H 2 (0, L ) × H 01 (0, L ) D ( A2) = H and and ⎛ 0 1 2 ⎜ ρκ ∂x2 − κρ0 l I 0 1 ⎜ 1 ⎜ 0 0 ⎜ A1 = ⎜ κ − ∂ 0 ⎜ ρ2 x ⎜ ⎝ 0 0 − (κ +ρ1κ0 )l ∂x ⎛0 ⎜0 ⎜ ⎜0 ⎜ A2 = ⎜ ⎜0 ⎜ ⎝0 0 0 0⎞ 0 0 0 κ ρ1 ∂x 0 0 1 0 b 2 κ ρ2 ∂x − ρ2 I 0 − ρκ2l I 0 0 0 − ρκ1l I 0 κ0 2 κ l2 ρ1 ∂x − ρ1 I (κ +κ0 )l ρ1 ∂x 1 0 0 0 0 − αρ1 (1x) g 1 (.) 0 0 0 0 0 0 0 0 0 0 0 0 0 − ρ2 g 2 (.) 0 2 0 0 0 0 0 α (x) 0 0⎟ ⎟ 0⎟ ⎟ ⎟, ⎟ 1⎠ 0⎟ 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠ α (x) 0 − ρ3 g 3 (.) 1 Note that, in this case we have D (A) = D ( A 1 ). Theorem 2.2. Let Assumptions 2.1 and 2.2 hold. For any U 0 ∈ D (A) there will exist a unique strong solution for (2.8). Furthermore, if U 0 ∈ H then (2.8) will admit a unique weak solution. Proof. We endeavor to prove that A = −( A 1 + A 2 ) is maximal monotone operator of H. At this moment we shall divide our proof into two parts. First, we will use Corollary 1.1 in Barbu [1] in order to conclude that: (i) The operator − A 1 is maximal monotone. Second, we will use Theorem 3.1 in Brézis [3] in order to conclude that: (ii) − A 2 is monotone, hemicontinuous, and a bounded operator. Proof of (i). Our strategy here is to show that − A 1 is monotone and R( I − A 1 ) = H. So, by using Proposition 2.2 in Brézis [3] the result will follow. Indeed, the monotone property follows from the fact that W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 (− A 1 U , U )H = 0, 2271 ∀U ∈ D (A). Now, let F = { F 1 , . . . , F 6 } ∈ H. It is enough to solve the spectral problem U − A1 U = F for some U ∈ D (A). If we denote U = {ϕ , Φ, ψ, Ψ, w , W }, the above equation becomes equivalent to ϕ − Φ = F 1 ∈ H 01 (0, L ), (2.9) 2 ρ1 Φ − κ (ϕx + ψ + lw )x − κ0l[ w x − lϕ ] = ρ1 F 2 ∈ L (0, L ), (2.10) ψ − Ψ = F 3 ∈ H 01 (0, L ), (2.11) 2 ρ2 Ψ − bψxx + κ (ϕx + ψ + lw ) = ρ2 F 4 ∈ L (0, L ), (2.12) w − W = F 5 ∈ H 01 (0, L ), (2.13) 2 ρ1 W − κ0 [ w x − lϕ ]x + κ l(ϕx + ψ + lw ) = ρ1 F 6 ∈ L (0, L ). (2.14) Isolating Φ , Ψ , W in (2.9), (2.11) and (2.13) and replacing into (2.10), (2.12) and (2.14) respectively, it remains to solve the following problems ρ1 ϕ − κ (ϕx + ψ + lw )x − κ0l[ w x − lϕ ] = f 1 , ρ2 ψ − bψxx + κ (ϕx + ψ + lw ) = f 2 , ρ1 w − k0 [ w x − lϕ ]x + κ l(ϕx + ψ + lw ) = f 3 where f 1 = ρ1 ( F 1 + F 2 ), f 2 = ρ2 ( F 3 + F 4 ), f 3 = ρ1 ( F 5 + F 6 ). In this way we define a bilinear form  2 a : H 01 (0, L ) × H 01 (0, L ) × H 01 (0, L ) →R given by   L a {ϕ , ψ, w }, {u , v , z} = ρ1 ϕ u + ρ2 ψ v + ρ1 w z + κ (ϕx + ψ + lw )(u x + v + lz) 0 + κ0 [ w x − lϕ ][zx − lu ] dx dt for {ϕ , ψ, w }, {u , v , z} ∈ H 01 (0, L ) × H 01 (0, L ) × H 01 (0, L ). It is not hard to prove that a is continuous and coercive. So, the conclusion follows from the Lax–Milgram theorem. Proof of (ii). From Assumptions 2.1 and 2.2, the operator − A 2 satisfies (− A 2 U , U )H  0 which proves the monotonicity of − A 2 . Now, let U i = {ϕi , Φi , ψi , Ψi , w i , Wi } ∈ H, i = 1, 2. We consider the following expression 2272 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290  − A 2 (U 1 + tU 2 ), U  H   = α1 (x) g 1 (Φ1 + t Φ2 ), Φ L 2   + α2 (x)(Ψ1 + t Ψ2 ), Ψ L 2   + α3 (x) g 3 ( W 1 + t W 2 ), W L 2 , for t > 0. Since, the desire is to show that     ⎧ lim α1 (x) g 1 (Φ1 + t Φ2 ), Φ L 2 = α (x)1 g 1 (Φ1 ), Φ L 2 , ⎪ ⎪ ⎪ ⎨ t →0    lim α2 (x) g 2 (Ψ1 + t Ψ2 ), Ψ L 2 = α (x)2 g 2 (Ψ1 ), Ψ L 2 , t →0 ⎪ ⎪ ⎪ ⎩ lim α3 (x) g 3 ( W 1 + t W 2 ), Φ  = α3 (x) g 3 ( W 1 ), W  , L2 L2 (2.15) t →0 it is enough to prove the first limit of (2.15), because the others are analogous. For this, consider a function f ∈ L 1 (0, L ) given by   f (x) = α (x) g 1 Φ1 (x) Φ(x) and define the sequence ( f n ) ⊂ L 1 (0, L ), given by  f n (x) = α (x) g 1 Φ1 (x) + 1 n  Φ2 (x) Φ(x), then we have lim f n (x) = f (x), n→∞ a.e. in (0, L ). Defining the set  Σn = x ∈ [0, L ];      Φ1 (x) + 1 Φ2 (x) < 1 ,   n it is not hard to prove that | f n (x)|  c 1 |Φ(x)| for almost every x ∈ Σn , where c 1 is a positive constant. On the other hand, by using (2.1) and (2.2), we also conclude that | f n (x)|  c 2 (|Φ1 (x)| + |Φ2 (x)|)|Φ(x)| for almost every in [0, L ] \ Σn . Both cases allow us to conclude that ( f n ) is limited by an integrable function on [0, L ]. Then, by Lebesgue’s Dominated Convergence Theorem we conclude the desired limit. So − A 2 is hemicontinuous. Finally, after some computations, by use of Assumptions 2.1 and 2.2 again, we conclude that the operator − A 2 maps every bounded subset into a bounded subset. Statements (i) and (ii) imply that the operator A is maximal monotone, so the conclusion of the proof follows from Theorem 3.1 in Brézis [3]. 2 3. Observability inequality In this section, we consider (1.1) without external forces: ⎧ ⎨ ρ1 ϕtt − κ (ϕx + ψ + lw )x − κ0l[ w x − lϕ ] = 0, ρ ψ − bψxx + κ (ϕx + ψ + lw ) = 0, ⎩ 2 tt ρ1 w tt − κ0 [ w x − lϕ ]x + κ l(ϕx + ψ + lw ) = 0 (3.16) W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 2273 Fig. 1. The circular arch. in (0, L ) × R+ , with Dirichlet boundary conditions (1.2). In the literature, system (3.16) consists of the equations for the theory of circular arch (see Fig. 1). Under these circumstances, we observe that the energy of conservative system (3.16) is the same energy for the damped system (1.1). Initially consider the result: Theorem 3.1. Let Γ := (a1 , a2 ) be an open interval contained in (0, L ). For T > 0 large enough, there exists a positive constant C 0 such that, any solution {ϕ , ψ, w } of (3.16) satisfies T  E0  C0  2  2  2 ρ1 ϕt (x, t ) + ρ2 ψt (x, t ) + ρ1  w t (x, t ) dx dt , (3.17) 0 Γ where E 0 := E (0) is the initial energy related to the solution {ϕ , ψ, w }. Proof. We highlight that using density arguments, it is enough to prove this result for strong solutions. Take |Γ | := a2 − a1 . Now consider ε0 , small enough, such that 0 < ε0 < |Γ | 2 . Let {ϕ , ψ, w } be a strong solution of (3.16) and define the following function continuous and piecewise C 1 : ⎧ ⎨ (λ − 1)x, g λ (x) = λ(x − a1 − ε0 ) + ⎩ (λ − 1)(x − L ), L −(a −a −2ε ) a1 −a2 +2ε0 (a1 L if x ∈ [0, a1 + ε0 ), + ε0 ), if x ∈ [a1 + ε0 , a2 − ε0 ], if x ∈ (a2 − ε0 , L ] 2 1 0 with λ := ∈ [0, 1[ and 0  a1 < a2  L, introduced in [9]. Multiplying the first, second L and third equations of the system (3.16) by the multipliers ϕx g λ , ψx g λ , w x g λ respectively, and performing integration by parts we obtain: 2274 0= W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 1 T  L  2 0  ρ1 |ϕt |2 + ρ2 |ψt |2 + ρ1 | w t |2 + b|ψx |2 + κ0 | w x − lϕ |2 + κ |ϕx + ψ + lw |2 gλ dx dt 0  L + T (ρ1 ϕt ϕx + ρ2 ψt ψx + ρ1 w t w x ) g λ dx T  L − κ0l[ w x − lϕ ]ϕ gλ dx dt 0 0 T L κ (ϕx + ψ + lw )ψ gλ dx dt − 0 + 0 0 T L 0 κ l(ϕx + ψ + lw ) w gλ dx dt . 0 0 As  g λ (x) = if x ∈ [a1 + ε0 , a2 − ε0 ], λ, (λ − 1), if x ∈∈ [0, a1 + ε0 ) ∪ (a2 − ε0 , L ], we have from equality above that  L T (1 − λ) E (t ) dt = − 0 T (ρ1 ϕt ϕx + ρ2 ψt ψx + ρ1 w t w x ) g λ dx 0 0 T L + + L κ0l[ w x − lϕ ]ϕ gλ dx dt − 0 0 T L 0 0 − + T κ (ϕx + ψ + lw )ψ gλ dx dt 0 0 κ l(ϕx + ψ + lw ) w gλ dx dt 1 2 1 2 T a2 −ε0 ρ1 |ϕt |2 + ρ2 |ψt |2 + ρ1 | w t |2 dx dt 0 a 1 + ε0 T a2 −ε0   b|ψx |2 + κ0 | w x − lϕ |2 + κ |ϕx + ψ + lw |2 dx dt . (3.18) 0 a 1 + ε0 Let us estimate the right side of (3.18). Using the equivalence between the norm of the energy and the usual norm in H, we obtain first that  L − T (ρ1 ϕt ϕx + ρ2 ψt ψx + ρ1 w t w x ) g λ dx  C E 0. (3.19) 0 0 After, using Young’s inequality with the fact that | g λ |  1, we obtain T L − T L |ϕ |2 dx dt , κ0l[ w x − lϕ ]ϕ gλ dx dt  ε T E 0 + C ε 0 0 0 0 (3.20) W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 T L T L κ (ϕx + ψ + lw )ψ gλ dx dt  ε T E 0 + C ε 0 0 0 T L |ψ|2 dx dt , (3.21) | w |2 dx dt . (3.22) 0 T L κ l(ϕx + ψ + lw ) w gλ dx dt  ε T E 0 + C ε 0 2275 0 0 0 Finally, we will estimate the last tranche of the right side of (3.18). For this purpose, we consider the following cutoff function η ∈ C 0∞ (0, L ) defined by:  0  η(x)  1, ∀x ∈ (0, L ),   η(x) = 0, in (0, a1 ) ∪ (a2 , L ),   η(x) = 1, in (a1 + ε0 , a2 − ε0 ). Multiplying the first, second and third equations of (3.16) by grating by parts on (0, T ) × (0, L ) we obtain: T  L ηϕ , ηψ , η w, respectively and inte-  b|ψx |2 + κ0 | w x − lϕ |2 + κ |ϕx + ψ + lw |2 0  η dx dt 0  L =− T (ρ1 ϕt ϕ + ρ2 ψt ψ + ρ1 w t w )η dx 0 0 T L + 0 +   ρ1 |ϕt |2 + ρ2 |ψt |2 + ρ1 | w t |2 η dx dt 0 1 T L 2 0 T   κ |ϕ |2 + b|ψ|2 + κ0 | w |2 ηxx dx dt 0 L − (κϕ ψ + κ lw ϕ − κ0lϕ w )ηx dx dt . 0 0 Consequently, by analogous calculations that we have made before, we infer T L  b|ψx |2 + κ0 | w x − lϕ |2 + κ |ϕx + ψ + lw |2 0  η dx dt 0  C E0 + C T a2  0 a1 2 1 t 2 2 ψt ρ ϕ +ρ 2 1 wt +ρ  dx dt + C T a2   ϕ 2 + ψ 2 + w 2 dx dt , 0 a1 where the positive constant C does not depend on the solution of (3.16). (3.23) 2276 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 From (3.18)–(3.23) we obtain: T a2  (1 − λ − 3ε ) T E 0  C E 0 + C  ρ1 ϕt2 + ρ2 ψt2 + ρ1 w t2 dx dt 0 a1 + Cε T a2   ϕ 2 + ψ 2 + w 2 dx dt . 0 a1 Thus, for ε< 1−λ , 3 we have T E0  C E0 + C T a2   ρ1 ϕt2 + ρ2 ψt2 + ρ1 w t2 dx dt 0 a1 + Cε T a2   ϕ 2 + ψ 2 + w 2 dx dt , 0 a1 and, finally, taking T > C , we can conclude that E0  C T a2  T L  ρ1 ϕt2 + ρ2 ψt2 + ρ1 w t2 dx dt + C 0 a1 0   ϕ 2 + ψ 2 + w 2 dx dt , (3.24) 0 where C is a positive constant which does not depend on the solution of (3.16). Our desire here is to estimate the second integral on the right side of (3.24) in terms of ϕt , ψt , and w t . For this purpose, it suffices to prove that there exists a positive constant C , which does not depend on the solutions of (3.16), such that the following inequality holds for every solution of (3.16): T L T a2 ϕ 2 + ψ 2 + w 2 dx dt  C 0 ρ1 ϕt2 + ρ2 ψt2 + ρ1 w t2 dx dt . (3.25) 0 a1 0 To prove this statement, we will argue by contradiction. Indeed, let us suppose that (3.25) is not true. So, we can find a sequence of non-null solutions of (3.16), namely {ϕν , ψν , w ν }ν ∈N , satisfying T L 0   T a2 ϕν2 + ψν2 + w 2ν dx dt > ν ρ1 ϕν t2 + ρ2 ϕν t2 + ρ1 w ν t2 dx dt , ν ∈ N, 0 a1 0 which implies that  T  a2 0 a1 ρ1 ϕν t2 + ρ2 ϕν t2 + ρ1 w ν t2 dx dt T L 0 0 (ϕν2 + ψν2 + w 2ν ) dx dt → 0, when ν → ∞. W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 2277 Denoting ϕν ϕ̃ν :=   T L , (ϕν2 + ψν2 + w 2ν ) dx dt 0 0 ψ̃ν :=   T L 0 0 w̃ ν :=   T L 0 0 ψν , (ϕν2 + ψν2 + w 2ν ) dx dt wν (ϕν2 + ψν2 + w 2ν ) dx dt we can easily see that 1 ν T a2 > 2 ρ1 ϕ˜ν t2 + ρ2 ψ˜ν t + ρ1 w˜ν t2 dx dt → 0, when ν → ∞, 0 a1 and also T L 0   ϕ̃ν2 + ψ̃ν2 + w̃ 2ν dx dt = 1, for all ν ∈ N. 0 From this result we have    {ϕ˜ν t }, {ψ˜ν t }, { w˜ν t } are bounded in L 2 0, T ; L 2 (Γ ) ,   {ϕ˜ν }, {ψ˜ν }, { w˜ν } are bounded in L 2 0, T ; L 2 (0, L ) . (3.26) Employing the already proved inequality (3.24) to the solutions {ϕ̃ν , ψ̃ν , w̃ ν } of (3.16) we obtain T a2 2 1 ˜ν t E ν (t ) = E ν (0)  C ρ ϕ ˜ 2 2 ψν t +ρ 2 1 w˜ν t dx dt +ρ T L +C 0 a1 0   ϕ̃ν2 + ψ̃ν2 + w̃ 2ν dx dt 0 and by using the boundedness in (3.26) and Poincaré’s inequality we can conclude the following ⎧   ⎪ {ϕ˜ }, {ψ˜ν t }, { w˜ν t } are bounded in L 2 0, T ; L 2 (0, L ) , ⎪ ⎨ νt   {ψ˜ν } are bounded in L 2 0, T ; H 01 (0, L ) , ⎪ ⎪ ⎩ {ϕ˜ + ψ˜ + l w˜ }, { w˜ − lϕ˜ } are bounded in L 2 0, T ; L 2 (0, L ). νx ν ν νx (3.27) ν Since the initial data are limited, rewrite t ϕ̃ν (x, t ) = ϕ̃ν (x, s) ds + ϕ̃ν (0) 0 and so {ϕ˜ν } are bounded in L 2 (0, T ; L 2 (0, L )). Then, from (3.27) it follows that { w̃ ν ,x } is bounded in L 2 (0, T ; L 2 (0, L )) as well. By Poincaré’s inequality, we conclude that 2278 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290   { w˜ν } are bounded in L 2 0, T ; H 01 (0, L ) . (3.28) Therefore, from (3.27) and (3.28), we obtain that are bounded in L 2 (0, T ; H 01 (0, L )). Employing the Aubin–Lions theorem, we also deduce that ⎧   ⎪ ϕ̃ → ϕ̃ strong in L 2 0, T ; L 2 (0, L ) , ν ⎪ ⎨   ψ̃ν → ψ̃ strong in L 2 0, T ; L 2 (0, L ) , ⎪ ⎪ ⎩ w̃ → w̃ strong in L 2 0, T ; L 2 (0, L ). ν (3.29) The strong convergences in (3.29) imply that T  L 1 = lim ν →∞ 0  T L  ϕ̃ν2 + ψ̃ν2 + w̃ 2ν dx dt = 0 0   ϕ̃ 2 + ψ̃ 2 + w̃ 2 dx dt . (3.30) 0 On the other hand, from the weak convergences for the derivatives in (3.27) it follows that T a2 T a2 ρ1 ϕ̃t2 + ρ2 ψ̃t2 + ρ1 w̃ t2 dx dt  lim inf ν →∞ 0 a1 2 ρ1 ϕ˜ν t2 + ρ2 ψ˜ν t + ρ1 w˜ν t2 dx dt = 0, 0 a1 which implies that ϕ̃t = ψ̃t = w̃ t = 0, in (a1 , a2 ) × (0, T ). However, since {ϕ̃ , ψ̃, w̃ } is a solution of ⎧ ⎪ ⎪ ρ1 ϕ̃tt − κ (ϕ̃x + ψ̃ + l w̃ )x − κ0l[ w̃ x − lϕ̃ ] = 0, ⎪ ⎨ ρ2 ψ̃tt − bψ̃xx + κ (ϕ̃x + ψ̃ + l w̃ ) = 0, ⎪ ρ1 w̃ tt − κ0 [ w̃ x − lϕ̃ ]x + κ l(ϕ̃x + ψ̃ + l w̃ )x = 0, ⎪ ⎪ ⎩ ϕ̃t = ψ̃t = w̃ t = 0, in Q , in Q , in Q , (3.31) in (a1 , a2 ) × (0, T ) where Q := (0, T ) × (0, L ). Taking the derivative of (3.31) with respect to the variable t in distributional sense and denoting z = ϕ̃t , u = ψ̃t and w̃ t = v we infer that { z, u , v } is a solution of ⎧ ρ1 ztt − κ (zx + u + lv )x − κ0l[ v x − lz] = 0, ⎪ ⎪ ⎪ ⎨ ρ2 utt − bu xx + κ (zx + u + lv ) = 0, ⎪ ρ1 v tt − κ0 [ v x − lz] + κ l(zx + u + lv ) = 0, ⎪ ⎪ ⎩ z = u = v = 0, in Q , in Q , in Q , in (a1 , a2 ) × (0, T ) and employing Holmgren’s Uniqueness Theorem we deduce that z = u = v = 0, or equivalently, ϕ̃t = ψ̃t = w̃ t = 0 a.e. in Q . Now, returning to (3.31) we obtain (3.32) W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 ⎧ ⎪ ⎨ −κ (ϕ̃x + ψ̃ + l w̃ )x − κ0l[ w̃ x − lϕ̃ ] = 0, −bψ̃xx + κ (ϕ̃x + ψ̃ + l w̃ ) = 0, ⎪ ⎩ −κ0 [ w̃ x − lϕ̃ ]x + κ l(ϕ̃x + ψ̃ + l w̃ ) = 0 in Q . Multiplying the first, second, and third equations of (3.33) by integrating by parts on Q we have 2279 (3.33) ϕ̃ , ψ̃ , and w̃ respectively, and T L κ (ϕ̃x + ψ̃ + l w̃ )2 + bψ̃x2 + κ0 [ w̃ x − lϕ̃ ]2 dx dt 0= 0 0 which implies by applying Poincaré’s inequality that ψ̃ = 0 on Q . As we have in (3.32), we obtain that the ODE system ⎧ ⎪ ⎨ ϕ̃x + l w̃ = 0, w̃ x − lϕ̃ = 0, ⎪ ⎩ ϕ̃ , w̃ ∈ H 01 (0, L ) provides us ϕ̃ = w̃ = 0, contradicting (3.30). Therefore, we must conclude that the statement done in (3.25) is absolutely true. Combining (3.25) in (3.24), we deduce, for T enough large, that T a2 ρ1 ϕt2 + ρ2 ψt2 + ρ1 w t2 dx dt . E0  C 2 0 a1 Remark 3.1. It is important to emphasize that the positive constant C does not depend on the solutions of (3.16). This is our major result of observability. This observability result will be related to the damped system (1.1) in order to allow us to take an arbitrarily small region for the damping mechanisms. 4. Main result The main purpose of the present section is to obtain some decay rates for the solutions of the damped Bresse system (1.1). Before stating our stability result, we will define some needed functions. In order to do that, we will follow the ideas first introduced in Lasiecka and Tataru [10] and adapted by Cavalcanti et al. in [6]. It will be presented here our adaptation of the last one. Let h be defined by h(x) = h1 (x) + h2 (x) + h3 (x) with h i (0) = 0, i = 1, 2, 3, where the h i are concave, strictly increasing functions such that   h i sg i (s)  s2 + g i2 (s), for |s|  1. (4.34) Note that such function can be straightforwardly constructed, given the hypotheses on the functions g i in Assumption 2.1. With those functions, we define  r (·) = h · |Q |  (4.35) 2280 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 where | Q | := meas( Q ). As r is monotone increasing, the function c I + r is invertible for all c  0. For M a positive constant, we then set p (x) = (c I + r )−1 ( Mx). (4.36) The function p is easily seen to be positive, continuous and strictly increasing with p (0) = 0. Finally, let q(x) = x − ( I + p )−1 (x). (4.37) The next lemma is due to Lasiecka and Tataru [10]. Lemma 4.1. Let the functions p, q be defined as above. Consider a sequence (sn ) of positive numbers which satisfies sm+1 + p (sm+1 )  sm . Then sn  S (m) where S (t ) is a solution of the differential equation ⎧ ⎨ d   S (t ) + q S (t ) = 0, ⎩ dt S (0) = s0 . Moreover, lim S (t ) = 0, t →∞ if p (x) > 0 for x > 0. Now, we are in a position to state our main result: Theorem 4.2. Assume Assumption 2.1 and Assumption 2.2. So, there will exist a positive constant T 0 > 0 such that if {ϕ , ψ, w } is a solution of problem (1.1)–(1.2) which initial energy satisfies E 0 < K then  E (t )  S t T0  −1 , ∀t > T 0 , (4.38) with limt →∞ S (t ) = 0, where S (t ) is the solution of the differential equation ⎧ ⎨ d   S (t ) + q S (t ) = 0, ⎩ dt S (0) = E 0 where q is as given in (4.37). Remark 4.1. If the feedbacks are linear, e.g., g i (s) = s, then, under the same assumptions as in the above theorem, we have that the energy of problem (1.1) decays exponentially with respect to the initial energy, e.g., there exist two positive constants C > 0 and γ > 0, which may depend on the initial data E 0 , such that E (t )  C e −γ t E 0 , for every solution satisfying E 0 < K . t > 0, W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 2281 In order to prove the above result, we will need the following essential lemma: Lemma 4.3. Let us assume Assumptions 2.1 and 2.2. So, for T > 0 large enough and K > 0 there exists a positive constant C , which depends on T and K , such that T L       α1 (x) ϕt2 + g1 (ϕt )2 + α2 (x) ψt2 + g2 (ψt )2 + α3 (x) w t2 + g3 ( w t )2 dx dt E (T )  C 0 0 (4.39) for any strong solution {ϕ , ψ, w } of (1.1)–(1.2) satisfying E 0  K . Proof of Theorem 4.2. Let the sets be as follows:   Πϕ = (x, t ) ∈ Q ; ϕt (x, t ) > 1   Πψ = (x, t ) ∈ Q ; ψt (x, t ) > 1   Π w = (x, t ) ∈ Q ;  w t (x, t ) > 1 and Ξϕ = Q \ Γϕ , and Ξψ = Q \ Γψ , and Ξw = Q \ Γw . By the essential lemma we have (4.39), then let us estimate each tranche of this inequality. Initially, note that     α1 (x) ϕt2 + g1 (ϕt )2 dx dt = Q    α1 (x) ϕt2 + g1 (ϕt )2 dx dt + Πϕ   α1 (x) ϕt2 + g1 (ϕt )2 dx dt . Ξϕ We analyze each integral on the right side of equality above. First, from Assumption 2.1 we see that     1 α1 (x) ϕt2 + g1 (ϕt )2 dx dt  k− 1 + K1 Πϕ   α1 (x) g1 (ϕt )ϕt dx dt . (4.40) Πϕ Now, by (4.34) we have  Ξϕ   α1 (x) ϕt2 + g1 (ϕt )2 dx dt     α1 (x)h1 g1 (ϕt )ϕt dx dt Ξϕ    1+ α1  Ξϕ   1 + α1 ∞  ∞ h1   h1  Ξϕ   1 + α1 ∞  | Q |h1 α1 1 + α1  ∞ g 1 (ϕt )ϕt dx dt  α1 (x) g1 (ϕt )ϕt dx dt  1 |Q |   α1 (x) g1 (ϕt )ϕt dx dt Q where the last inequality is obtained by Jensen’s inequality. Then, using (4.40) and (4.41), (4.41) 2282 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290     1 α1 (x) ϕt2 + g1 (ϕt )2 dx dt  k− 1 + K1  Q  α1 (x) g1 (ϕt )ϕt dx dt Q  + 1 + α1 ∞   | Q |h1   1 |Q | α1 (x) g1 (ϕt )ϕt dx dt . (4.42) Q Analogously, we can obtain     1 α2 (x) ψt2 + g1 (ψt )2 dx dt  k− 2 + K2  Q  α2 (x) g2 (ψt )ψt dx dt Q  + 1 + α2 ∞   | Q |h2   1 |Q | α2 (x) g2 (ψt )ψt dx dt (4.43) Q and     1 α3 (x) w t2 + g3 ( w t )2 dx dt  k− 3 + K3 Q   α3 (x) g3 ( w t ) w t dx dt Q  + 1 + α3 ∞   | Q |h3 1 |Q |   α3 (x) g3 ( w t ) w t dx dt . (4.44) Q Using (4.39), (4.42), (4.43) and (4.44), besides the fact that h i are strictly increasing functions, we gets E (T )  C 3   1 k− + Ki i  i =1  α1 (x) g1 (ϕt )ϕt + α2 (x) g2 (ψt )ψt + α3 (x) g3 ( w t ) w t dx dt Q + C|Q | 3   1+ αi   α1 (x) g1 (ϕt )ϕt + α2 (x) g2 (ψt )ψt + α3 (x) g3 ( w t ) w t dx dt ∞ r i =1  Q where r was defined in (4.35). Setting M= 1 3 i =1 (1 + C|Q | αi ∞) and c = |Q | 3 −1 + i =1 (k i 3 i =1 (1 + Ki) αi ∞) we arrived at  M E (T )  c α1 (x) g1 (ϕt )ϕt + α2 (x) g2 (ψt )ψt + α3 (x) g3 ( w t ) w t dx dt Q  +r  α1 (x) g1 (ϕt )ϕt + α2 (x) g2 (ψt )ψt + α3 (x) g3 ( w t ) w t dx dt Q   = (c I + r ) E 0 − E ( T ) W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 2283 using (2.7) in equality above. We can still rewrite the above inequality as   p E ( T )  E 0 − E ( T ). (4.45) To finish the proof, we replace T (respectively, 0) in (4.45) with (m + 1) T (respectively, mT ) to obtain     E (m + 1) T + p E (m + 1) T   E (mT ), for m = 0, 1, . . . . Using Lemma 4.1 with sm = E (mT ) results in E (mT )  S (m), Finally, from (2.7), we have for t = mT + τ , with  E (t )  E (mT )  S (m)  S m = 0, 1 , . . . . τ ∈ [0, T ], that t−τ T   S t T  −1 , for t > T , where we have used above the fact that S (·) is dissipative. The proof of the theorem is now completed. 2 Now we proceed to prove the essential lemma. Proof of Lemma 4.3. Let us put T > T 0 , with T 0 as in the previous section, and K > 0. Since E (t )  E 0 , ∀t > 0, it is enough for us to prove inequality (4.39) for E 0 instead of E ( T ). Let us suppose that for any C > 0 there exists a strong solution {ϕ , ψ, w }, which depends on C , satisfying E 0  K but not inequality (4.39), i.e. T L       α1 (x) ϕt2 + g1 (ϕt )2 + α2 (x) ψt2 + g2 (ψt )2 + α3 (x) w t2 + g3 ( w t )2 dx dt . E0 > C 0 0 For simplicity we shall denote u := ut . Choosing C = n with n ∈ N we will obtain, for each n ∈ N, a non-null strong solution {ϕn , ψn , w n } satisfying 0 < E n (0)  K and T L   α1 (x) ϕn2 + g1 ϕn E n (0) > n 0 2   2   2    + α2 (x) ψn 2 + g 2 ψn + α3 (x) w n2 + g 3 w n dx dt , 0 or equivalently T L 0 0 α1 (x)(ϕn2 + g1 (ϕn )2 ) + α2 (x)(ψn 2 + g2 (ψn )2 ) + α3 (x)( w n2 + g3 ( w n )2 ) dx dt E n (0) 1  . n (4.46) 2284 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 Since ( E n (0))n∈N is bounded, the inequality (4.46) yields ⎧ T L ⎪  2   ⎪ ⎪ ⎪ lim α1 (x) ϕn2 + g1 ϕn dx dt = 0, ⎪ ⎪ n→∞ ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ T L ⎪  ⎨  2   lim α2 (x) ψn 2 + g2 ψn dx dt = 0, n→∞ ⎪ ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ T L ⎪  ⎪ ⎪  2   ⎪ ⎪ ⎪ lim α3 (x) w n2 + g3 w n dx dt = 0 ⎪ ⎩ n→∞ 0 (4.47) 0 and here, using Assumption 2.2, we obtain T  T  2 0 = lim ϕn dx dt = lim n→∞ w n2 dx dt . ψn dx dt = lim n→∞ 0 I1 T  2 n→∞ 0 I2 (4.48) 0 I3 Also, the family of energy functionals ( E n )n∈N is uniformly bounded on (0, T ). So, the following convergences hold       ϕn  ϕ weak-star in L ∞ 0, T ; L 2 (0, L ) , ψn  ψ weak-star in L ∞ 0, T ; L 2 (0, L ) , wn  w weak-star in L ∞ 0, T ; L 2 (0, L ) ,   ψn,x  ψx weak-star in L ∞ 0, T ; L 2 (0, L ) ,   w n,x − lϕn  w x − lϕ weak-star in L ∞ 0, T ; L 2 (0, L ) ,   ϕn,x + ψn + lw n  ϕx + ψ + lw weak-star in L ∞ 0, T ; L 2 (0, L ) . Due to the previous section employing Poincaré’s inequality, compactness result and since the initial data are limited, we also deduce that ⎧   ⎪ ϕ → ϕ strongly in L 2 0, T ; L 2 (0, L ) , ⎪ ⎨ n   ψn → ψ strongly in L 2 0, T ; L 2 (0, L ) , ⎪ ⎪ ⎩ w → w strongly in L 2 0, T ; L 2 (0, L ). n At this moment we shall divide our proof into two cases:  0. (i) U = {ϕ , ϕ , ψ, ψ , w , w } = Consider the system   ⎧ ⎪ ⎨ ρ1 ϕn − κ (ϕn,x + ψn + lw n )x − κ0l[ w n,x − lϕn ] + α1(x) g 1 ϕn = 0, ρ2 ψn − bψn,xx + κ (ϕn,x + ψn + lw n ) + α2 (x) g2 ψn = 0, ⎪   ⎩ ρ1 w n − κ0 [ w n,x − lϕn ]x − κ l(ϕn,x + ψn + lw n ) + α3 (x) g3 w n = 0 (4.49) W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 2285 in Q . By convergences above, passing to the limit in (4.49), and using (4.47) and (4.48), we arrive at ⎧ ρ1 ϕ − κ (ϕx + ψ + lw )x − κ0l[ w x − lϕ ] = 0 in Q , ⎪ ⎪ ⎪ ⎨ ρ2 ψ − bψxx + κ (ϕx + ψ + lw ) = 0 in Q , ⎪ ρ w − κ [ w − l ϕ ] + κ l ( ϕ + ψ + lw ) = 0 in Q, ⎪ 1 0 x x x ⎪ ⎩ ϕt = 0 in Q 1 , ψt = 0 in Q 2 , w t = 0 in Q 3 (4.50) where Q i := I i × (0, T ). Taking the derivative of (4.50) on t in distributional sense and substituting ϕt = z, ψt = u, and w t = v, we infer ⎧ ρ1 z − κ (zx + u + lv )x − κ0l[ v x − lz] = 0 ⎪ ⎪ ⎪ ⎨ ρ u − bu + κ ( z + u + lv ) = 0 2 xx x ⎪ ρ v − κ [ v − lz ] + κ l( zx + u + lv ) = 0 ⎪ 1 0 x ⎪ ⎩ z=u=v =0 in Q , in Q , in Q , in Ĩ × (0, T ). Employing Holmgren’s uniqueness theorem we deduce that z = u = v = 0 in Q , and consequently, ϕt = ψt = w t = 0, in Q . (4.51) Returning to (4.50) we obtain in Q : ⎧ ⎨ −κ (ϕx + ψ + lw )x − κ0l[ w x − lϕ ] = 0, −bψxx + κ [ w x − lϕ ] = 0, ⎩ −κ0 [ w x − lϕ ]x + κ l[ w x − lϕ ] = 0. Multiplying the first equation of (4.52) by adding the obtained result, yields (4.52) ϕ , the second one by ψ , the third equation by w, and T L κ (ϕx + ψ + lw )2 + bψx2 + κ0 [ w x − lϕ ]2 dx dt = 0 0 0 which implies from Poincaré’s inequality, that ψ = 0. Resulting from differentiating (4.51) with respect to time, we obtain the ODE system ⎧ ⎪ ⎨ ϕx + lw = 0, w x − l ϕ = 0, ⎪ ⎩ ϕ , w ∈ H 01 (0, L ). It provides us ϕ = w = 0, which is a contradiction. (ii) U = {ϕ , ϕ , ψ, ψ , w , w } = 0. In this case, setting initially ! νn := E n (0), from (4.46) we obtain ϕ n := ϕn , νn ψ n := ψn νn , and w n := wn νn 2286 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290  0 = lim n→∞     g 1 (νn ϕ n )2 g 2 (νn ψ n )2 2 2 α1 (x) ϕ n + + α2 ψ n + 2 2 νn Q νn   g 3 (νn w n )2 dx dt . + α3 (x) w n2 + 2 (4.53) νn Furthermore, define for each n, the energy E n (t ) of the normalized problem as E n (t ) = 1 L ρ1 ϕ n2 + ρ2 ψ n2 + ρ1 w n2 + bψ n2,x + κ0 [ w n,x − lϕ ]2 + κ (ϕ n,x + ψ n + w n )2 dx, 2 0 then, E n (0) = E n (0) νn2 = 1, for all n ∈ N. This limitation implies the following convergences       ϕn  ϕ weak-star in L ∞ 0, T ; L 2 (0, L ) , ψn  ψ weak-star in L ∞ 0, T ; L 2 (0, L ) , wn  w weak-star in L ∞ 0, T ; L 2 (0, L ) ,   ψ n,x  ψ x weak-star in L ∞ 0, T ; L 2 (0, L ) ,   w n,x − lϕ n  w x − lϕ weak-star in L ∞ 0, T ; L 2 (0, L ) ,   ϕ n,x + ψ n + lw n  ϕ x + ψ + lw weak-star in L ∞ 0, T ; L 2 (0, L ) . Consequently, making use of the Poincaré inequality and the Aubin–Lions theorem, one has   ϕ n → ϕ strongly in L 2 0, T ; L 2 (0, L ) ,   ψ n → ψ strongly in L 2 0, T ; L 2 (0, L ) ,   w n → w strongly in L 2 0, T ; L 2 (0, L ) . Now considering the strong convergence above, and passing to the limit in the system ⎧ g 1 (νn ϕ n ) ⎪ ⎪ ρ1 ϕ n − κ (ϕ n,x + ψ n + lw n )x − κ0l[ w n,x − lϕ n ] + α1 (x) = 0, ⎪ ⎪ νn ⎪ ⎪ ⎪ ⎨ g 2 (νn ψ n ) ρ ψ − bψ n,xx + κ (ϕ n,x + ψ n + lw n ) + α2 (x) = 0, ⎪ 2 n νn ⎪ ⎪ ⎪ ⎪ ⎪ g (ν w ) ⎪ ⎩ ρ1 w n − κ0 [ w n,x − lϕ n ]x + κ l(ϕ n,x + ψ n + lw n ) + α3 (x) 3 n n = 0 νn in Q , we conclude W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 ⎧ ρ1 ϕ − κ (ϕ x + ψ + lw )x − κ0l[ w x − lϕ ] = 0, ⎪ ⎪ ⎪ ⎨ ρ2 ψ − bψ xx + κ (ϕ x + ψ + lw ) = 0, ⎪ ρ1 w − κ0 [ w x − lϕ ]x + κ l(ϕ x + ψ + lw ) = 0, ⎪ ⎪ ⎩ ϕ = ψ = w = 0, 2287 in Q , in Q , in Q , in Ĩ × (0, T ). Analogously to the case (i) we deduce that ϕ = ψ = w = 0. In order to reach a contradiction, consider { z, u , v } a solution of the following homogeneous Bresse system ⎧ ρ1 zn − κ (zn,x + un + lv n )x − κ0l[ v n,x − lzn ] = 0, ⎪ ⎪ ⎪ ⎪ ρ2 un − bun,xx + κ (zn,x + un + lv n ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ1 v n − κ0 [ v n,x − lzn ]x + κ l( zn,x + un + lv n ) = 0, zn ( L , t ) = zn (0, t ) = un ( L , t ) = un (0, t ) = v n ( L , t ) = v n (0, t ) = 0, ⎪ ⎪ ⎪ zn (x, 0) = ϕ n (x, 0), zn (x, 0) = ϕ n (x, 0), ⎪ ⎪ ⎪ ⎪ ⎪ u (x, 0) = ψ n (x, 0), un (x, 0) = ψ n (x, 0), ⎪ ⎪ ⎩ n v n (x, 0) = w n (x, 0), v n (x, 0) = w n (x, 0), in Q , in Q , in Q , in (0, T ), in (0, L ), in (0, L ), in (0, L ) with energy E (t ). Now define ⎧ ⎨ ϕ̃n = ϕ n − zn , ψ̃ = ψ n − un , ⎩ n w̃ n = w n − v n . In this case, it is not difficult to see that {ϕ̃n , ψ̃n , w̃ n } is a solution of the following inhomogeneous Bresse system ⎧ g 1 (νn ϕ n ) ⎪ ⎪ ρ1 ϕ̃n − κ (ϕ̃n,x + ψ̃n + l w̃ n )x − κ0l[ w̃ n,x − lϕ̃n ] = α1 (x) , ⎪ ⎪ νn ⎪ ⎪ ⎪ ⎪ ⎪ g 2 (νn ψ n ) ⎪ ⎪ , ⎪ ⎨ ρ2 ψ̃n − bψ̃n,xx + κ (ϕ̃n,x + ψ̃n + l w̃ n ) = α2 (x) ν in Q , in Q , n g 3 (νn w n ) ⎪ ⎪ ⎪ ρ1 w̃ n − κ0 [ w̃ n,x − lϕ̃n ]x + κ l(ϕ̃n,x + ψ̃n + l w̃ n ) = α3 (x) , in Q , ⎪ ⎪ νn ⎪ ⎪ ⎪ ⎪ ⎪ ϕ̃n ( L , t ) = ϕ̃n (0, t ) = ψ̃n ( L , t ) = ψ̃n (0, t ) = w̃ n ( L , t ) = w̃ n (0, t ) = 0, in (0, T ), ⎪ ⎪ ⎩ ϕ̃n (x, 0) = ϕ̃n (x, 0) = ψ̃n (x, 0) = ψ̃n (x, 0) = w̃ n (x, 0) = w̃ n (x, 0) = 0, in (0, L ) whose expression for energy Ẽ n (t ), related to the solution {ϕ̃n , ψ̃n , w̃ n }, is the same as in (2.6), then L ρ1 ϕ̃n2 + ρ2 ψ̃n 2 + ρ1 w̃ n2 dx 2 Ẽ n (t )  0  ρ1 ϕ n2 + ρ2 ψ n2 + ρ1 w n2 + ρ1 zn2 + ρ2 un2 + ρ1 v n2 dx  Ĩ 2288 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290  −2 ρ1 ϕ n zn + ρ2 ψ n un + ρ1 w n v n dx, (4.54) Ĩ for t ∈ [0, T ]. Estimating the second integral in (4.54), by using Cauchy–Schwarz’s inequality, we obtain  12      12 ρ1 ϕ n2 dx ρ1 ϕ n zn + ρ2 ψ n un + ρ1 w n v n dx  Ĩ Ĩ ρ1 zn2 dx Ĩ  12     12 2 2 ψ n dx + 2 2 un dx ρ ρ Ĩ Ĩ  12     12 2 2 w n dx + 2 2 v n dx ρ ρ Ĩ . Ĩ Although, since E (t ) = E (0) = 1, t ∈ [0, T ], (4.55) we obtain   12 ρ1 zn2 dx 1 2  2E (0) 1 = 22 , Ĩ as well as   12    12 1 2 , ρ1 v n dx  2 2 . 2 2 un dx ρ Ĩ Ĩ Returning to (4.54), having this three estimates in mind, we conclude that  ρ1 zn2 + ρ2 un2 + ρ1 v n2 dx 2 Ẽ (t )  Ĩ 3    12 ρ1 ϕ n2 dx − 22  12  ρ2 ψ n2 dx + Ĩ  12   ρ2 w n2 dx + Ĩ Ĩ and so,  ρ1 zn2 + ρ2 un2 + ρ1 v n2 dx Ĩ  2 sup Ẽ n (t ) + 2 3 2    12 ρ1 ϕ t ∈[0, T ] Ĩ 2 n dx  12  2 2 ψ n dx + ρ Ĩ  12   2 2 w n dx + ρ Ĩ . (4.56) W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 2289 Let ε > 0 small enough. Integrating both sides of (4.56) on (ε , T − ε ) and employing Young’s inequality we arrive at 3 2 2 ( T − 2ε ) 1 2 "  T −ε ρ1 ϕ ε  T −ε # 12 2 n dx dt + ρ ε Ĩ  T −ε # 12 2 2 ψ n dx dt # 12 $ 2 2 w n dx dt + ρ ε Ĩ Ĩ T −ε  ρ1 zn2 + ρ2 un2 + ρ1 v n2 dx dt . + 2( T − 2ε ) sup Ẽ n (t )  t ∈[0, T ] ε (4.57) Ĩ Using (4.53) we observe that each integral, the first tranche of this inequality converges to zero when n goes to infinity. On the other hand, the energy functional Ẽ n also satisfies  t L Ẽ n (t ) = − α1 (x) 0 g 1 (νn ϕ n (x, s))ϕ̃ + α2 (x) νn 0 g 2 (νn ψ n (x, s))ψ̃ νn + α3 (x) g 3 (νn w n (x, s)) w̃ νn dx ds, for t ∈ [0, T ], which yields T "  L sup Ẽ n (t )  M t ∈[0, T ] α1 (x) 0 g 12 (νn ϕ n (x, s)) 0 T " L +M α2 (x) 0 0 T " L +M α3 (x) 0 0 νn2 $ 12 dx g 22 (νn ψ n (x, s)) νn2 g 32 (νn w n (x, s)) νn2 ds $ 12 dx ds $ 12 dx ds where M > 0 is a constant which comes from boundedness of (ϕ̃n ), (ψ̃n ) and ( w̃ n ) in L ∞ (0, T ; L 2 (0, L )). Again, by (4.53) each term on the right hand side of this inequality converges to zero when goes to infinity. Therefore, T −ε  ρ1 zn2 + ρ2 un2 + ρ1 v n2 dx dt = 0. lim n→∞ ε (4.58) Ĩ By using Eq. (4.55) and the observability result, proved in the previous section, we have T −ε  ρ1 zn2 + ρ2 un2 + ρ1 v n2 dx dt , ∀n ∈ N, 1 = En (t ) = En (0)  C 1 ε Ĩ where C 1 = C 1 (ε , C 0 ). Passing to the limit when n goes to infinity in the previous inequality and having in mind (4.58) we must conclude a contradiction. This finishes the proof of the lemma. 2 2290 W. Charles et al. / J. Differential Equations 255 (2013) 2267–2290 Acknowledgments The authors would like to express their great gratitude to reviewer(s) for the useful discussions which improve the result. References [1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academici Române, Bucuresti, 1974. [2] F.A. Boussouira, J.E. Muñoz Rivera, D. 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