DECAY RATES FOR SOLUTIONS TO THERMOELASTIC BRESSE SYSTEMS OF TYPES I AND III

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1 Electronic Journal of Differential Equations, Vol , No. 73, pp. 6. ISSN: URL: or DECAY RATES FOR SOLUTIONS TO THERMOELASTIC BRESSE SYSTEMS OF TYPES I AND III FERNANDO A. GALLEGO, JAIME E. MUÑOZ RIVERA Communicate by Mokhtar Kirane Abstract. In this article, we stuy the energy ecay for the thermoelastic Bresse system in the whole line with two issipative mechanisms, given by heat conuction Types I an III. We prove that the ecay rate of the solutions are very slow. More precisely, we show that the solutions ecay with the rate of +t /8 in the L -norm, whenever the initial ata belongs to L R H s R for a suitable s. The wave spees of propagation have influence on the ecay rate with respect to the regularity of the initial ata. This phenomenon is known as regularity-loss. The main tool use to prove our results is the energy metho in the Fourier space.. Introuction In this article, we consier two Cauchy problems relate to the Bresse moel with two issipative mechanisms corresponing to the heat conuction couple to the system. The first of them is the Bresse system with thermoelasticity of Type I: ρ ϕ tt kϕ x ψ lω x k 0 lω x lϕ + lγθ = 0 ψ tt bψ xx kϕ x ψ lω + γθ x = 0 ρ ω tt k 0 ω x lϕ x klϕ x ψ lω + γθ x = 0 with the initial ata θ t k θ xx + ω x lϕ t = 0 θ t k θ xx + m ψ xt = 0 in R 0,, in R 0,, in R 0,, in R 0,, in R 0,, ϕ, ϕ t, ψ, ψ t, ω, ω t, θ, θ x, 0 = ϕ 0, ϕ, ψ 0, ψ, ω 0, ω, θ 0, θ 0 x. The secon one, is the Bresse system with thermoelasticity of Type III: ρ ϕ tt kϕ x ψ lω x k 0 lω x lϕ + lγθ t = 0 ψ tt bψ xx kϕ x ψ lω + γθ xt = 0 ρ ω tt k 0 ω x lϕ x klϕ x ψ lω + γθ xt = 0 θ tt k θ xx α θ xxt + ω x lϕ t = 0 θ tt k θ xx α θ xxt + m ψ xt = 0 in R 0,, in R 0,, in R 0,, in R 0,, in R 0,,.. 00 Mathematics Subject Classification. 35B35, 35L55, 93D0. Key wors an phrases. Decay rate; heat conuction; Bresse system; thermoelasticity. c 07 Texas State University. Submitte February 7, 06. Publishe March 5, 07.

2 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 with the initial ata ϕ, ϕ t, ψ, ψ t, ω, ω t, θ, θ, θ t, θ t x, 0 = ϕ 0, ϕ, ψ 0, ψ, ω 0, ω, θ 0, θ 0, θ, θ x, where α, α, ρ,, γ, b, k, k 0, k, k, l an m are positive constants. Terms k 0 ω x lϕ, kϕ ψ lω an bψ x enote the axial force, the shear force an the bening moment, where ω, ϕ an ψ are the longituinal, vertical an shear angle isplacements, respectively. Furthermore, ρ = ρa, = ρi, k 0 = EA, k = k GA, b = EI an l = R, where ρ enotes the ensity, E is the elastic moulus, G is the shear moulus, k is the shear factor, A is the crosssectional area, I is the secon moment of area of the cross-section an R is the raius of curvature of the beam. Here, we assume that all the above coefficients are positive constants. In what concerns of the Thermoelastic of type III, we refer the work of Green an Naghi [9, ]. They re-examine the classical moel of thermoelasticity an introuce the so-calle moel of thermoelasticity of type III, which the constitutive assumption on the heat flux vector is ifferent from Fourier s law. They evelope a moel of thermoelasticity that inclues temperature graient an thermal isplacement graient among the constitutive variables an propose a heat conuction law as qx, t = κθ x x, t + κ v x x, t,.3 where v t = θ an v is the thermal isplacement graient, κ an κ are constants. Combining.3 with the energy balance law lea to the equation ρθ t + ϱ iv q = 0,.4 ρθ tt ϱκθ xx ϱκ θ xx = 0, which permits propagation of thermal waves at finite spee. The common feature of these theories, is that all of them lea to hyperbolic ifferential equations an moel heat flow as thermal waves traveling at finite spee. More information about mathematical moeling can be foun in [3,, 5]. The main purpose of this article is to investigate the asymptotic behavior of the solutions to the Cauchy problems. an. pose on R. To the best of our knowlege, the stability of the Bresse moel oes not have any physical explanation when it is consiere in the real line. Be that as it may, from mathematical point of view, a consierable number of stability issues concerning the Bresse moel in a whole space, have receive consierable attention in the last years [0, 4, 5, 8, 3]. This has been ue to the regularity-loss phenomenon that usually appears in the pure Cauchy problems for instance, see [5, 6,, 3, 4, 3] an references therein. Roughly speaking, the ecay rate of the solution is of the regularity-loss type, when it is obtaine only by assuming some aitional orer regularity on the initial ate. Thus, base on this refinement of the initial ata, we investigate the relationship between amping terms, the wave spees of propagation an their influence on the ecay rate of the vector solutions V an V see below of the systems. an., respectively. Thus, our main result reas as follows. Theore.. Let s be a non-negative integer, suppose that Vj 0 Hs R L R for j =,. Then, the vector solutions V j of thermoelastic Bresse problems. an., respectively, satisfy the following ecay estimates:

3 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM 3 If ρ = k b an k = k 0, then k xv j t C + t 8 k 4 V 0 j + C + t l 4 k+l x V 0 j,.5 for j =, an t 0. If ρ k b or k k 0, then k xv j t C + t 8 k 4 V 0 j + C + t l 6 k+l x V 0 j,.6 for j =,, an t 0. where k + l s, C, C are two positive constants. Our proof is base on some estimates for the Fourier image of the solution as well as a suitable linear combination of series of energy estimates. The key iea is to construct functionals to capture the issipation of all the components of the vector solution. These functional allows to buil an appropriate Lyapunov functionals equivalent to the energy, which gives the issipation of all the components in the vector ˆV 0 ξ, t an ˆV 0 ξ, t See 3.3 below. Finally, we rely on the Plancherel theorem an some asymptotic inequalities to show the esire ecay estimates. The ecay rate +t /8 can be obtaine only uner the regularity V 0 H s R. This regularity loss comes to analyze the Fourier image of the solution. Inee, for ˆV ξ, t, we have see.3,.66 an 3.3 below that where sξ = ˆV ξ, t Ce βsξt ˆV ξ, 0,.7 ξ C 4 +ξ 8, ξ C 4 +ξ +ξ 8, ρ if = k b an k = k 0, ρ if k b or k k 0. As we will see, the ecay estimate.5-.6 epens in a critical way on the properties of the function sξ. Obviously, the function sξ behaves like ξ 4 in the low frequency region ξ an like ξ 4 near infinity whenever ρ = k b an k = k 0. Otherwise, if the wave spees of propagation are ifferent the function sξ behaves also like ξ 4 in the low frequency region an like ξ 6 near infinity, which means that the issipation in the high frequency region is very weak an prouces the regularity loss phenomenom. It has been known recently that this regularity loss leas to some ifficulties in the nonlinear problems, see [, 4] for more etails. There are many works on the global existence an asymptotic stability of solutions to the initial bounary value problem for the Bresse system with issipation. In this irection, we refer the IBVP associate to. consiere by Liu an Rao in [6]. They prove that the exponential ecay exists only when the velocities of the wave propagation are the same. If the wave spees are ifferent, they showe that the energy ecays polynomially to zero with the rate t an t 4, provie that the bounary conitions are Dirichlet-Neumann-Neumann ω x x, t = ϕx, t = ψ x x, t = θ x, t = θ x, t = 0, for x = 0, l, an Dirichlet-Dirichlet-Dirichlet type, ωx, t = ϕx, t = ψx, t = θ x, t = θ x, t = 0, for x = 0, l. An improvement of the above results was mae by Fatori an Muñoz Rivera in [8]. They showe that, in general, the Thermoelastic Bresse system of Type I is not

4 4 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 exponentially stable, but there exists polynomial stability with rates that epen on the wave propagation an the regularity of the initial ata. As far as we know, there exist just a few results relate to the stability of the pure Cauchy problem to the Bresse moel. The ecay rate of the solution of the IVP for Bresse system in the whole line has been first stuie by Sai-Houari an Soufyane in [3]. They consiere the system ϕ tt ϕ x ψ lω x k 0lω x lϕ = 0 ψ tt a ψ xx kϕ x ψ lωγ ψ t = 0 ω tt k 0ω x lϕ x lϕ x ψ lω + γ ω t = 0 in R 0,, in R 0,, in R 0,,.8 an investigate the relationship between the frictional amping terms, the wave spees of propagation an their influence on the ecay rate of the solution. In aition, they showe that the L -norm of the solution ecays with the rate + t /4. Later on, the same authors in [8], prove that the vector solution V of the Bresse system ampe by heat conuction: ϕ tt kϕ x ψ lω x k 0lω x lϕ = 0 ψ tt a ψ xx kϕ x ψ lω + mθ x = 0 ω tt k 0ω x lϕ x lϕ x ψ lω + γω t = 0 ecays with the rate k xv t L for a =, an k xv t L θ t k θ xx + mψ xt = 0 in R 0,, in R 0,, in R 0,, in R 0,,.9 C + t k 6 V0 L + C + t l k+l x V 0 L,.0 C + t k 6 V0 L + C + t l 4 k+l x V 0 L,. for a, k =,,..., s l. More recently, Sai-Houari an Hamaouche [4] stuie the ecay properties of the Bresse-Cattaneo system ϕ tt ϕ x ψ lω x k 0lω x lϕ = 0 ψ tt a ψ xx ϕ x ψ lω + mθ x = 0 ω tt k 0ω x lϕ x lϕ x ψ lω + γω t = 0 θ t + q x + mψ xt = 0 τ q q t + βq + θ x = 0 in R 0,, in R 0,, in R 0,, in R 0,, in R 0,,. obtaining the same ecay rate as the one of the solution for the Bresse-Fourier moel.9. This fact has been also seen in [6], where the authors investigate the Timoshenko-Cattaneo an Timoshenko-Fourier moels an showe the same behavior for the solutions of both systems. Finally, concerning to the Termoelasticity type III in one-imensional space, Sai-Houari an Hamaouche in [5] have been recently analyze the system ϕ tt ϕ x ψ lω x k 0lω x lϕ = 0 ψ tt a ψ xx ϕ x ψ lω + mθ tx = 0 ω tt k 0ω x lϕ x lϕ x ψ lω + γω t = 0 θ tt k θ xx + βψ tx k θ txx = 0 in R 0,, in R 0,, in R 0,, in R 0,.

5 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM 5 They prove that the solution ecay with the rate for a =, an k xv t L k xv t L C + t k 6 V0 L + C + t l k+l x V 0 L, C + t k 6 V0 L + C + t l 8 k+l x V 0 L, for a, k =,,..., s l. This paper is organize as follows: In Section, we analyze the ODE system generate by the Fourier transform applies to the Cauchy problem, obtaining appropriate ecay estimates for the Fourier image of the solution. Section 3 is eicate to proof our main result.. Energy metho in the Fourier space In this section, we establish ecay rates for the Fourier image of the solutions of thermoelastic Bresse systems. To obtain the estimates of the Fourier image is actually the harest an technical part. These estimates will play to a crucial role in proving the Theorems. an.3, below... Thermoelastic Bresse system of type I. Taking Fourier Transform in., we obtain the ODE system ρ ˆϕ tt ikξiξ ˆϕ ˆψ lˆω k 0 liξˆω l ˆϕ + lγ ˆθ = 0 in R 0,. ˆψtt + bξ ˆψ kiξ ˆϕ ˆψ lˆω + iγξ ˆθ = 0 in R 0,. ρ ˆω tt ik 0 ξiξˆω l ˆϕ kliξ ˆϕ ˆψ lˆω + iγξ ˆθ = 0 in R 0,.3 ˆθ t + k ξ ˆθ + iξˆω l ˆϕ t = 0 in R 0,.4 ˆθ t + k ξ ˆθ + im ξ ˆψ t = 0 in R 0,.5 The energy functional associate to the above system is efine by Êξ, t = ρ ˆϕ t + ˆψ t + ρ ˆω t + γ ˆθ + γ m ˆθ + b ξ ˆψ + k iξ ˆϕ ˆψ lˆω + k 0 iξˆω l ˆϕ..6 Lemma.. Consier the energy functional Ê associate to the system.-.5. Then têξ, t = γξ k ˆθ + k ˆθ..7 m Proof. Multiplying. by ˆϕ t,. by ˆψ t,.3 by ˆω t,.4 by γ ˆθ, an.5 by γ m ˆθ, aing an taking real part,.7 follows. We show that the ecay rate of the solution will epen on the wave spees of propagation. More precisely, we analyze two cases: First, we suppose that ρ = k b an k = k 0. Otherwise, we consier the case when the wave spees of propagation are ifferent ρ k b or k k 0. The proof of our main results in this section Theorems. an.3 below are base on the following lemmas:

6 6 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 Lemma.. The functional satisfies J ξ, t = Rei ξ ˆψ t ˆθ, t J ξ, t + m ξ ˆψ t b ξ 3 ˆψ ˆθ + k ξ ˆθ iξ ˆϕ ˆψ lˆω + C + ξ ξ ˆθ, where C is a positive constant. Proof. Multiplying.5 by i ξ ˆψ t an taking real part, we obtain t Re iξ ˆψ t ˆθ + Rei ξ ˆψ tt ˆθ Reik ξ 3 ˆψt ˆθ + m ξ ˆψ t = 0. By., we have Re t iξ ˆψ t ˆθ + m ξ ˆψ t k ξ 3 ˆψ t ˆθ + b ξ 3 ˆψ ˆθ + k ξ ˆθ iξ ˆϕ ˆψ lˆω + γξ ˆθ Applying Young s inequality, we obtain.8. Lemma.3. The functional satisfies T ξ, t = Re ρ ˆϕ t iξˆω l ˆϕ ρ ˆϕ t ˆθ t T ξ, t + k 0l iξˆω l ˆϕ ρ k ξ ˆϕ t ˆθ Reikξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ where C is a positive constant. + k ξ ˆθ iξ ˆϕ ˆψ lˆω + C ˆθ, Proof. Multiplying. by iξˆω l ˆϕ an taking real part, we have.8.9 t Re ρ ˆϕ t iξˆω l ˆϕ + Reρ ˆϕ t iξˆω l ˆϕ t + Re ikξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ + k 0 l iξˆω l ˆϕ Re lγ ˆθ iξˆω l ˆϕ = 0. Inequality.4 implies Re t ρ ˆϕ t iξˆω l ˆϕ ρ Re ˆϕ t ˆθt ρ k Reξ ˆϕ t ˆθ + Reikξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ + k 0 l iξˆω l ˆϕ Re lγ ˆθ iξˆω l ˆϕ = 0..0 On the other han, multiplying. by ˆθ an taking real part, it follows that Re t ρ ˆϕ t ˆθ + ρ ik Re ˆϕ t ˆθt + Re ξ m ˆθ iξ ˆϕ ˆψ lˆω k0 l + Re ˆθ iξˆω l ˆϕ lγ. ˆθ = 0.

7 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM 7 Aing.0 an., t T ξ, t + k 0 l iξˆω l ˆϕ ρ k ξ ˆϕ t ˆθ Reikξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ applying Young s inequality,.9 follows. Lemma.4. The functional satisfies + lγ ˆθ iξˆω l ˆϕ + k ξ ˆθ iξ ˆϕ ˆψ lˆω + k 0l ˆθ iξˆω l ˆϕ + lγ ˆθ, T ξ, t = Reiρ ξˆω t iξˆω l ˆϕ + i ρ ξˆω t ˆθ, t T ξ, t + k 0 ξ iξˆω l ˆϕ ρ k ξ 3 ˆω t ˆθ + Reiklξiξ ˆϕ m ˆψ lˆωiξˆω l ˆϕ + kl ξ ˆθ iξ ˆϕ m ˆψ lˆω + C 3 ξ ˆθ, where C 3 is a positive constant. Proof. Multiplying.3 by iξiξˆω l ˆϕ an taking real part, we obtain t Reiρ ξˆω t iξˆω l ˆϕ Reiρ ξˆω t iξˆω l ˆϕ t + k 0 ξ iξˆω l ˆϕ Reiklξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ Re γξ ˆθ iξˆω l ˆϕ = 0,..3 using.4, it follows that t Reiρ ξˆω t iξˆω l ˆϕ + ρ Reiξˆω t ˆθt + ρ k Reiξ 3 ˆω t ˆθ + k 0 ξ iξˆω l ˆϕ Reiklξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ Reγξ ˆθ iξˆω l ˆϕ = 0. On the other han, multiplying.3 by iξ ˆθ an taking real part, t Reiρ ξ ˆω t ˆθ ρ Reiξˆω t ˆθt + Re k 0ξ iξˆω l ˆϕˆθ Re iklξ iξ ˆϕ m ˆψ lˆωˆθ γ ξ ˆθ = 0. Aing.4 an.5, applying Young s inequality,. follows. Lemma.5. Consier the functional Then, there exists δ > 0 such that J ξ, t := lt ξ, t + T ξ, t. t J ξ, t + k 0 δ iξˆω l ˆϕ ρ lk ξ ˆϕ t ˆθ + ρ k ξ 3 ˆω t ˆθ.4.5

8 8 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 where C 4 is a positive constant. Proof. Lemmas.3 an.4 imply that t J ξ, t + k 0 l + ξ iξˆω l ˆϕ + kl ξ iξ ˆϕ ˆψ lˆω ˆθ + C 4 + ξ ˆθ, ρ lk ξ ˆϕ t ˆθ + kl ξ ˆθ iξ ˆϕ m ˆψ lˆω + ρ k ξ 3 ˆω t ˆθ + C 4 + ξ ˆθ. Note that there exists δ > 0 such that δ l +ξ +ξ. Thus, t J ξ, t + k 0 δ + ξ iξˆω l ˆϕ Lemma.6. Consier the functional If ρ = k b an k = k 0, then t J 3ξ, t + k iξ ˆϕ ˆψ lˆω ρ lk ξ ˆϕ t ˆθ + kl ξ ˆθ iξ ˆϕ m ˆψ lˆω + ρ k ξ 3 ˆω t ˆθ + C 4 + ξ ˆθ. J 3 ξ, t = Re ˆψt iξ ˆϕ ˆψ lˆω i ρ b k ξ ˆψ ˆϕ t. ˆψ t + l Re ˆψ t ˆω t bl Reiξ ˆψiξˆω l ˆϕ + blγ k ξ ˆψ ˆθ + C 5 ξ ˆθ. Moreover, if ρ k b or k k 0, then t J 3ξ, t + k iξ ˆϕ ˆψ lˆω ˆψ t + l Re ˆψ t ˆω t + bρ k Reiξ ˆψ t ˆϕ t k 0bl k Reiξ ˆψiξˆω l ˆϕ + blγ k ξ ˆψ ˆθ + C 5 ξ ˆθ, where C 5 is a positive constant Proof. Multiplying. by iξ ˆϕ ˆψ lˆω an taking real part, t Re ˆψt iξ ˆϕ ˆψ lˆω Rei ξ ˆψ t ˆϕ t ˆψ t Re l ˆψ t ˆω t Rebξ ˆψiξ ˆϕ ˆψ lˆω + k iξ ˆϕ ˆψ lˆω Re iγξ ˆθ iξ ˆϕ ˆψ lˆω = 0..9 First, suppose that ρ = k b an k = k 0..0

9 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM 9 By. an.0, Re bξ ˆψiξ ˆϕ ˆψ lˆω = b Re iξ ˆψiξiξ ˆϕ ˆψ lˆω = ρ b k Reiξ ˆψ ˆϕ tt bl Reiξ ˆψiξˆω l ˆϕ + blγ k Reiξ ˆψˆθ = t Reiξ ˆψ ˆϕ t Reiξ ˆψ t ˆϕ t bl Re iξ ˆψiξˆω l ˆϕ Substituting this in.9, we have t J 3ξ, t + k iξ ˆϕ ˆψ lˆω + blγ k Reiξ ˆψˆθ. ˆψ t + Re l ˆψ t ˆω t bl Reiξ ˆψiξˆω l ˆϕ + blγ k ξ ˆψ ˆθ + γ ξ ˆθ iξ ˆϕ ˆψ lˆω. Applying Young s inequality,.7 follows. Now, suppose that ρ k b or k k 0.. Proceeing as above,. implies Re bξ ˆψiξ ˆϕ ˆψ lˆω = ρ b k t Reiξ ˆψ ˆϕ t ρ b k Reiξ ˆψ t ˆϕ t k 0 k bl Reiξ ˆψiξˆω l ˆϕ + blγ k Reiξ ˆψˆθ. Substituting in.9 an applying Young s inequality, we obtain.8. Lemma.7. Let 0 < ε < ρl ρ Then an consier the functional J 4 ξ, t = Re ρ l ˆψt ˆψ ρ lˆω t ˆψ. ρ t J 4ξ, t + b l ε ρ ξ ˆψ ρ l ρ ˆψ t l Re ˆψ t ˆω t + k 0 l Reiξ ρ ˆψiξˆω l ˆϕ + Cε ˆθ + ˆθ, where Cε is a positive constant.. Proof. Multiplying. by ˆψ an taking real part, t Re ˆψ t ˆψ ρ ˆψ t + bξ ˆψ Rek ˆψiξ ˆϕ ˆψ lˆω + Reiγξ ˆψˆθ = 0..3 Then.3 implies t Reρ ˆψ t ˆψ t Reρ ˆψˆω t + bξ l ˆψ = ˆψ t ρ l Re ˆψ t ˆω t k 0 l Reiξ ˆψiξˆω l ˆϕ + γ l Reiξ ˆψˆθ Reiγξ ˆψˆθ.

10 0 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 Multiplying by ρl ρ, we have t J 4ξ, t + bl ρ ξ ˆψ = ρ l ρ ˆψ t l Re ˆψ t ˆω t k 0 l Reiξ ρ ˆψiξˆω l ˆϕ + γl Reiξ ρ ˆψˆθ γl Reiξ ρ ˆψˆθ. Applying Young s inequality an using the wave spees of propagation, we obtain.. Lemma.8. Let 0 < ε < ρl ρ an k = k 0, then an consier Kξ, t = J 3 ξ, t+j 4 ξ, t. If ρ = k b t Kξ, t + l ε bξ ρ ˆψ + k iξ ˆϕ ˆψ lˆω s ˆψ t + Cε ˆθ + Cε + ξ ˆθ. Moreover, if ρ k b or k k 0, then t Kξ, t + l ε bξ ρ ˆψ + k iξ ˆϕ ˆψ lˆω s ˆψ t + ρ b k k 0l Reiξ ˆψiξˆω l ˆϕ + bρ k Reiξ ˆψ t ˆϕ t + Cε ˆθ + Cε + ξ ˆθ, where s = ρl ρ +. The above lemma follows from Lemmas.6 an.7, applying Young s inequality. For the next lemma, it is necessary to observe that iξ ˆϕ ˆψ lˆω t iξ ˆϕ t + ˆψ t + lˆω t = 0 iξˆω l ˆϕ t iξˆω t + l ˆϕ t = 0 Lemma.9. Consier the functional Hξ, t = ρ Re iξ ˆϕ ˆψ lˆωˆω t + ρ Reiξˆω l ˆϕ ˆϕ t. If ρ = k b an k = k 0, then t Hξ, t + ρ l ˆϕ t + ρ l ˆω t k bl ˆψ t + 3kl iξ ˆϕ ˆψ lˆω + 3k 0l iξˆω l ˆϕ + C 6 + ξ ˆθ. Moreover, if ρ k b or k k 0, then t Hξ, t + ρ l ˆϕ t + ρ l ˆω t ρ l ˆψ t + C k, k 0 iξ ˆϕ ˆψ lˆω + C k, k 0 + ξ iξˆω l ˆϕ + C 6 + ξ ˆθ, where C k, k 0 an C 6 are positive constants..4.5

11 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM Proof. Multiplying. by iξˆω l ˆϕ,. by ρ ˆϕ t, aing these equalities an taking the real part, we obtain t Reρ iξˆω l ˆϕ ˆϕ t ρ Reiξˆω t ˆϕ t + ρ l ˆϕ t k Reiξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ k 0 l iξˆω l ˆϕ + lγ Reˆθ iξˆω l ˆϕ = 0..6 Multiplying.3 by iξ ˆϕ ˆψ lˆω,. by ρ ˆω t, aing, an taking the real part, t Re ρ iξ ˆϕ ˆψ lˆωˆω t ρ Reiξ ˆϕ t ˆω t + ρ Re ˆψ t ˆω t + ρ l ˆω t kl iξ ˆϕ ˆψ lˆω k 0 Re iξiξˆω l ˆϕiξ ˆϕ ˆψ lˆω.7 + γ Re iξ ˆθ iξ ˆϕ ˆψ lˆω = 0. Aing.6 an.7, t Hξ, t + ρ l ˆϕ t + ρ l ˆω t kl iξ ˆϕ ˆψ lˆω + k 0 l iξˆω l ˆϕ + γ ξ ˆθ iξ ˆϕ ˆψ lˆω + γl ˆθ iξˆω l ˆϕ + k k 0 ξ iξˆω l ˆϕ iξ ˆϕ ˆψ lˆω + ρ ˆψ t ˆω t. Applying Young s inequality an using the wave propagation properties,.4 an.5 follow. To prove our main result, we nee to establish a functional equivalent to the energy.6 in a polynomial sense. In particular, this kin of functional gives some issipative terms of the vector solution. First, let us efine J + ε ξ K + ξ J + ε 3 ξ H, if ρ = k b an k = k 0, L ξ, t = ξ ξ +ξ +ξ 4 λ ε 3 J + +ξ +ξ 4 ε 3λ K + J + ε 3 H, if ρ k b or k k 0. where λ, λ, ε, ε 3 are positive constants to be etermine later..8 Proposition.0. There exist constants M, M > 0 such that, if ρ = k b an k = k 0, then t L ξ, t + Mξ { bξ ˆψ + k iξ ˆϕ ˆψ lˆω + ˆψ t + k 0 iξˆω l ˆϕ + ρ ˆϕ t + ρ ˆω t } Cε, ε, ε 3 + ξ + ξ 4 + ξ 6 ξ ˆθ + Cε, ε + ξ + ξ 4 ˆθ. Moreover, if ρ k b or k k 0, we obtain t L ξ, t + M ξ 4 { bξ + ξ + ξ 4 ˆψ + k iξ ˆϕ ˆψ lˆω + ˆψ t + k 0 iξˆω l ˆϕ + ρ ˆϕ t + ρ ˆω t } Cε, ε 3, λ, λ + ξ ξ ˆθ + ˆθ.9.30

12 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 Proof. First, we suppose that ρ = k b an k = k 0. By using Lemmas. an.8, we have t {J ξ, t + ε ξ Kξ, t} + bε l ε ξ 4 ρ ˆψ + k ε ξ iξ ˆϕ ˆψ lˆω + m s ε ξ ˆψ t b ξ 3 ˆψ ˆθ + k ξ ˆθ iξ ˆϕ + ˆψ + lˆω + Cε, ε ξ ˆθ + Cε, ε + ξ ξ ˆθ. On the other han, By Lemmas.5 an.9, it follows that t {ξ J ξ, t + ε 3 ξ Hξ, t} + k 0 δξ iξˆω l ˆϕ + ρ ε 3 lξ ˆϕ t + ρ ε 3 l ξ ˆω t ρ lk ξ 4 ˆϕ t ˆθ + ρ k ξ 5 ˆθ ˆω t + kl ξ 3 ˆθ iξ ˆϕ ˆψ lˆω + C 4 + ξ ξ ˆθ + 3k 0ε 3 l ξ iξˆω l ˆϕ + 3klε 3 ξ iξ ˆϕ ˆψ lˆω + ε 3 k ξ bl ˆψ t + Cε 3 + ξ ξ ˆθ. Aing an using Young s inequality, we obtain t L ξ, t + bε l ε ξ 4 ρ ˆψ + k ε 4 lε 3ξ iξ ˆϕ ˆψ lˆω + m s ε k bl ε 3ξ ˆψ t + k 0 δ 3l ε 3ξ iξˆω l ˆϕ + ρ l ε 3 ξ ˆϕ t + ρ l 4 ε 3 ξ ˆω t Cε, ε, ε 3 + ξ + ξ 4 + ξ 6 ξ ˆθ + Cε, ε + ξ + ξ 4 ˆθ. We choose our constants as follows: ε < l, ε < m, ε 3 < min { δ ρ s 3l, ε 8l, bl k m s ε }. Consequently, we euce that there exists M > 0, such that t L ξ, t + Mξ { bξ ˆψ + k iξ ˆϕ ˆψ lˆω + ˆψ t + k 0 iξˆω l ˆϕ + ρ ˆϕ t + ρ ˆω t } Cε, ε, ε 3 + ξ + ξ 4 + ξ 6 ξ ˆθ + Cε, ε + ξ + ξ 4 ˆθ. At last, we assume that ρ k b or k k 0. Consier the functional P = ξ + ξ + ξ 4 λ ξ 4 ε 3 J ξ, t + + ξ + ξ 4 λ ε 3 Kξ, t. By Lemmas. an.8 an by using Young s inequality, it follows that t P ξ, t + l λ ε 3 bξ 6 3ε ρ + ξ + ξ 4 ˆψ ε 3 λ kξ ξ + ξ 4 iξ ˆϕ ˆψ lˆω + λ m Cε 4, λ ε 3 ξ 4 + ξ + ξ 4 ˆψ t λ ε 3 ε 4 ξ 4 ˆϕ t + ξ + ξ 4 + Cε + ξ ξ 4 iξˆω l ˆϕ, λ ε 3 + ξ + ξ 4

13 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM 3 + Cε, ε 3, λ, λ + ξ ξ ˆθ + ˆθ. In the above estimate we use the following inequalities: + ξ + ξ 4, + ξ 4 + ξ + ξ 4, ξ + ξ + ξ 4..3 On the other han, we consier the functional P = ξ 4 + ξ + ξ 4 J ξ, t + ε 3 Hξ, t. By.6 in Lemma.5, Lemma.9, Young s inequality an.3, we obtain t P + ξ ξ 4 + k 0 δ Ck, k 0 ε 3 + ξ + ξ 4 iξˆω l ˆϕ + ρ ε 3 lξ 4 + ξ + ξ 4 ˆϕ t ρ ε 3 lξ ξ + ξ 4 ˆω t ρ ε 3 ξ 4 l + ξ + ξ 4 ˆψ t ξ 4 + Ck, k 0 ε 3 + ξ + ξ 4 iξ ˆϕ ˆψ lˆω + Cε 3 + ξ ξ ˆθ. Thus, aing the above estimates of P an P, we obtain t L ξ, t + l λ ε 3 bξ 6 3ε ρ + ξ + ξ 4 ˆψ + λ 4 Ck, k ε 3 kξ ξ + ξ 4 iξ ˆϕ ˆψ lˆω λ m + Cε 4, λ ρ ε 3 ξ 4 l + ξ + ξ 4 ˆψ t ρ ε 3 lξ ξ + ξ 4 ˆω t + δ Ck, k 0, ε, λ ε 3 k 0 + ξ ξ 4 + ξ + ξ 4 iξˆω l ˆϕ + l ε 4λ ρ ε 3 ξ 4 ρ + ξ + ξ 4 ˆϕ t Cε, ε 3, λ, λ + ξ ξ ˆθ + ˆθ We choose our constants as follows: ε < l, λ > 4Ck, k 0, ε 4 < ρ l, 3ρ λ λ > ρ + l Cε 4, λ δ, ε 3 < m l Ck, k 0, ε, λ. Consequently, by using.3, we euce that there exists M > 0, such that t L ξ, t + M ξ 4 { bξ + ξ + ξ 4 ˆψ + k iξ ˆϕ ˆψ lˆω + ˆψ t + k 0 iξˆω l ˆϕ + ρ ˆϕ t + ρ ˆω t } Cε, ε 3, λ, λ + ξ ξ ˆθ + ˆθ

14 4 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 With the Proposition.0 in han an making some appropriate combinations, we buil a Lyapunov functional L, which plays a crucial role in the proof of our main result. Theorem.. For any t 0 an ξ R, we obtain the following ecay rates { C Êξ, 0e βsξt, if ρ ρ Êξ, t = k b an k = k 0, C Êξ, 0e β s ξt, if ρ k b or k k.3 0, where C, β, C, β are positive constants an s ξ = ξ 4 + ξ + ξ 4 + ξ 6 + ξ 8, s ξ 4 ξ = + ξ + ξ + ξ 4. Proof. Consier the Lyapunov functional { ξ L ξ, t + N + ξ + ξ 4 + ξ 6 8 ρ + ξ Êξ, t, if ρ Lξ, t = = k b an k = k 0, L ξ, t + N ρ + ξ Êξ, t, if k b or k k where N an N are positive constants to be fixe later. First, we suppose that ρ = k b an k = k 0. By Lemma. an Proposition.0, t Lξ, t Mξ4{ bξ ˆψ + k iξ ˆϕ ˆψ lˆω + ˆψ t + k 0 iξˆω l ˆϕ + ρ ˆϕ t + ρ ˆω t } where η = min{ k M > 0, such that It follows that γηn Cε, ε, ε 3 + ξ + ξ 4 + ξ 6 + ξ 8 ξ ˆθ + ˆθ,, k m }. On the other han, by efinition of L, there exists ξ L ξ, t M + ξ + ξ ξ Êξ, t. N M + ξ + ξ 4 + ξ ξ Êξ, t Lξ, t N + M + ξ + ξ 4 + ξ 6 + ξ 8 Êξ, t..34 Choosing N > maxm, Cε,ε,ε3 γη an by using there exists M > 0, such that Note that.34 implies where β = Now,.34 yiels + ξ + ξ 4 + ξ 6 + ξ 8 ξ, t Lξ, t M ξ 4 Êξ, t..35 t Lξ, t β ξ 4 + ξ + ξ 4 + ξ 6 + ξ 8 Lξ, t,.36 M N+M. By using Gronwall inequality, it follows that Lξ, t Lξ, 0e βsξt..37 Êξ, t CÊξ, 0e βsξt,

15 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM 5 where C = N+M N M > 0. At last, we assume that ρ k b or k k 0. By Lemma. an Proposition.0, t Lξ, t M ξ 4 { bξ + ξ + ξ 4 ˆψ + k iξ ˆϕ ˆψ lˆω + ˆψ t + k 0 iξˆω l ˆϕ + ρ ˆϕ t + ρ ˆω t } γηn Cε, ε 3, λ, λ + ξ ξ ˆθ + ˆθ On the other han, by efinition of L an by using Young s inequality, there exists M > 0 such that N M + ξ Êξ, t Lξ, t N + M + ξ Êξ, t..38 Choosing N > maxm, Cε,ε3,λ,λ γη an by using there exists M > 0, such that From.38, we obtain + ξ ξ + ξ + ξ 4, t Lξ, t M + ξ + ξ 4 Êξ, t..39 ξ 4 t Lξ, t ξ 4 β + ξ + ξ + ξ 4 Lξ, t,.40 where β = M N +M. By using Gronwall inequality, we conclue that Lξ, t Lξ, 0e β s ξt..4 The last inequality together with.38 leas to the secon inequality of the theorem, which completes the proof... Thermoelastic Bresse system of type III. In this subsection, we establish ecay rates for the Fourier image of the solutions of Thermoelastic Bresse system of Type III. Taking Fourier Transform in., we obtain the ODE system ρ ˆϕ tt ikξiξ ˆϕ ˆψ lˆω k 0 liξˆω l ˆϕ + lγ ˆθ t = 0 in R 0,,.4 ˆψtt + bξ ˆψ kiξ ˆϕ ˆψ lˆω + iγξ ˆθt = 0 in R 0,,.43 ρ ˆω tt ik 0 ξiξˆω l ˆϕ kliξ ˆϕ ˆψ lˆω + iγξ ˆθ t = 0 in R 0,,.44 ˆθ tt + k ξ ˆθ + α ξ ˆθt + iξˆω l ˆϕ t = 0 in R 0,,.45 ˆθ tt + k ξ ˆθ + α ξ ˆθt + im ξ ˆψ t = 0 in R 0,..46 The energy functional associate with the above system is Êξ, t = ρ ˆϕ t + ˆψ t + ρ ˆω t + γ ˆθ t + k γ ξ ˆθ + γ m ˆθ t + k γ m ξ ˆθ + b ξ ˆψ + k iξ ˆϕ ˆψ lˆω + k 0 iξˆω l ˆϕ.47

16 6 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 Lemma.. Let Then Ê the energy functional associate with system têξ, t = γξ α ˆθ t + α m ˆθ t..48 Proof. Multiplying.4 by ˆϕ t,.43 by ˆψ t,.44 by ˆω t,.45 by γ ˆθt,.46 by γ m ˆθt, aing these equalities an taking the real part,.48 follows. To establish the main result of this subsection an base in the approach one in the previous subsection, we establish the following lemmas: Lemma.3. The functional J ξ, t = Rei ξ ˆψ t ˆθt + Reik ξ 3 ˆψˆθ, satisfies t J ξ, t + m ξ ˆψ t k ξ 3 ˆψ ˆθ t + b ξ 3 ˆψ ˆθ t + k ξ ˆθ t iξ ˆϕ ˆψ lˆω + C + ξ ξ ˆθ t, where C is a positive constant..49 Proof. Multiplying.46 by i ξ ˆψ t an taking real part, we obtain t Re iξ ˆψ t ˆθt + Rei ξ ˆψ tt ˆθt t Reik ξ 3 ˆψˆθ By.43, it follows that + Reik ξ 3 ˆψˆθt Reiα ξ 3 ˆψt ˆθt + m ξ ˆψ t = 0. t J ξ, t + m ξ ˆψ t k ξ 3 ˆψ ˆθ t + α ξ 3 ˆψ t ˆθ t + b ξ 3 ˆψ ˆθ t + k ξ ˆθ t iξ ˆϕ ˆψ lˆω + γξ ˆθ t. Applying Young s inequality,.49 follows. Lemma.4. The functional satisfies T ξ, t = Re ρ ˆϕ t iξˆω l ˆϕ ρ ˆϕ t ˆθt, t T ξ, t + k 0l iξˆω l ˆϕ α ρ ξ ˆϕ t ˆθ t Reikξiξ ˆϕ m ˆψ lˆωiξˆω l ˆϕ k Reiξ ˆθ t iξ ˆϕ ˆψ lˆω + ρ k Reξ ˆϕ t ˆθ + C ˆθ t, where C is a positive constant..50

17 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM 7 Proof. Multiplying.4 by iξˆω l ˆϕ an taking real part, we have t Re ρ ˆϕ t iξˆω l ˆϕ + Reρ ˆϕ t iξˆω l ˆϕ t + Reikξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ + k 0 l iξˆω l ˆϕ Relγ ˆθ t iξˆω l ˆϕ = 0, by using.45, we have Re t ρ ˆϕ t iξˆω l ˆϕ ρ t Re ˆϕ t ˆθ t + ρ Re ˆϕ tt ˆθt ρ k Reξ ˆϕ t ˆθ α ρ Reξ ˆϕ t ˆθt + Reikξiξ ˆϕ m ˆψ lˆωiξˆω l ˆϕ + k 0 l iξˆω l ˆϕ Relγ ˆθ t iξˆω l ˆϕ = 0. Note that.4 implies t T ξ, t + k Re iξ m ˆθ t iξ ˆϕ ˆψ lˆω + k 0 l iξˆω l ˆϕ ρ k Reξ ˆϕ t ˆθ α ρ Reξ ˆϕ t ˆθt lγ ˆθ t + lγ + k 0l ˆθ t iξˆω l ˆϕ. Applying Young s inequality, we obtain Re ikξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ Lemma.5. The functional satisfies T ξ, t = Re t T ξ, t + k 0 ξ iξˆω l ˆϕ α ρ ξ 3 ˆω t ˆθ t + Re iρ ξˆω t iξˆω l ˆϕ + i ρ ξˆω t ˆθt, iklξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ + kl Reiξ ˆθ t iξ ˆϕ ˆψ lˆω k ρ Reiξ 3 ˆω t ˆθ + C 3 ξ ˆθ t, where C 3 is a positive constant..5 Proof. Multiplying.44 by iξiξˆω l ˆϕ an taking real part, t Re iρ ξˆω t iξˆω l ˆϕ Re iρ ξˆω t iξˆω l ˆϕ t + k 0 ξ iξˆω l ˆϕ Re iklξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ Re γξ ˆθt iξˆω l ˆϕ = 0, by using.45, we have t Re iρ ξˆω t iξˆω l ˆϕ + ρ t Reiξˆω t ˆθ t ρ Reiξˆω tt ˆθt + ρ k Reiξ 3 ˆω t ˆθ + α ρ Reiξ 3 ˆω t ˆθt + k 0 ξ iξˆω l ˆϕ m Re iklξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ Re γξ ˆθt iξˆω l ˆϕ = 0.

18 8 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 Note that,.44 implies that t T ξ, t kl Reiξ m ˆθ t iξ ˆϕ ˆψ lˆω Re iklξiξ ˆϕ ˆψ lˆωiξˆω l ˆϕ + k 0 ξ iξˆω l ˆϕ + k ρ Reiξ 3 ˆω t ˆθ + α ρ Reiξ 3 ˆω t ˆθt γξ ˆθ t + γ + k 0 ξ ˆθ t iξˆω l ˆϕ. Applying Young s inequality, we obtain.5. Lemma.6. Consier the functional J ξ, t := lt ξ, t + T ξ, t + ρ k Then, there exists δ > 0 such that t J ξ, t + k 0 δ iξˆω l ˆϕ Re ξ ˆθ iξˆω l ˆϕ. α ρ l ξ ˆϕ t ˆθ t + α ρ ξ 3 ˆω t ˆθ t + C 4 + ξ ˆθ t.5 where C 3 is a positive constant. Proof. By Lemma.4 an Lemma.5, t J ξ, t + k 0 l + ξ iξˆω l ˆϕ ρ lα ξ ˆϕ t ˆθ t + ρ k ξ ˆθ t iξˆω l ˆϕ + α ρ ξ 3 ˆω t ˆθ t + C 4 + ξ ˆθ t Note that there exists δ > 0 such that 4δ l +ξ +ξ. Thus, t J ξ, t + k 0 δ + ξ iξˆω l ˆϕ ρ lα ξ ˆϕ t ˆθ t + ρ k + ξ ˆθ t iξˆω l ˆϕ + α ρ ξ 3 ˆω t ˆθ t + C 4 + ξ ˆθ t. Applying Young s inequality,.5 follows..53 Lemma.7. Consier the functional J 3 ξ, t = Re ˆψt iξ ˆϕ ˆψ lˆω i ρ b k ξ ˆψ ˆϕ t. If ρ = k b an k = k 0, then t J 3ξ, t + k iξ ˆϕ ˆψ lˆω ˆψ t + l Re ˆψ t ˆω t bl Re + blγ k ξ ˆψ ˆθ t + C 5 ξ ˆθ t iξ ˆψiξˆω l ˆϕ.54

19 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM 9 Moreover, if ρ k b or k k 0, then t J 3ξ, t + k iξ ˆϕ ˆψ lˆω ˆψ t + l Re ˆψ t ˆω t + bρ k Reiξ ˆψ t ˆϕ t k 0bl k Re iξ ˆψiξˆω l ˆϕ + blγ k ξ ˆψ ˆθ t + C 5 ξ ˆθ t, where C 5 is a positive constant. Proof. Proceeing as proof of Lemma.6, we obtain.54 an Lemma.8. Let 0 < ε < ρl ρ If ρ = k b an k = k 0, then J 4 ξ, t t + b l ρ ε ξ ˆψ an consier the functional J 4 ξ, t = Re ρ l ˆψt ˆψ ρ lˆω t ˆψ. ρ l ρ ˆψ t l Re ˆψ t ˆω t + bl Reiξ ˆψiξˆω l ˆϕ + Cε ˆθ t + ˆθ t Moreover, If ρ k b or k k 0, then ρ t J 4ξ, t + b l ε ρ ξ ˆψ ρ l ρ ˆψ t l Re ˆψ t ˆω t + k 0 l Re iξ ρ ˆψiξˆω l ˆϕ where Cε is a positive constant. + Cε ˆθ t + ˆθ t, Proof. Proceeing as proof of Lemma.7, we obtain.56 an.57. Lemma.9. Let 0 < ε < ρl ρ an k = k 0, then an consier Kξ, t = J 3 ξ, t+j 4 ξ, t, If ρ = k b t Kξ, t + bl ε ξ ρ ˆψ + k iξ ˆϕ ˆψ lˆω s ˆψ t + Cε ˆθ t + Cε + ξ ˆθ t Moreover, if ρ k b or k k 0, then t Kξ, t + bl ε ξ ρ ˆψ + k iξ ˆϕ ˆψ lˆω s ˆψ t + k 0 ρ b k l Reiξ ˆψiξˆω l ˆϕ + bρ k Reiξ ˆψ t ˆϕ t + Cε ˆθ t + Cε + ξ ˆθ t,

20 0 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 where s = ρl ρ +. The above lemma follows from Lemmas.7 an.8, applying Young s inequality. Lemma.0. Consier the functional Hξ, t = ρ Re iξ ˆϕ ˆψ lˆωˆω t + ρ Reiξˆω l ˆϕ ˆϕ t. If ρ = k b an k = k 0, then t Hξ, t + ρ l ˆϕ t + ρ l ˆω t k bl ˆψ t + 3kl iξ ˆϕ ˆψ lˆω + 3k 0l iξˆω l ˆϕ + C 6 + ξ ˆθ t Moreover, if ρ k b or k k 0, then t Hξ, t + ρ l ˆϕ t + ρ l ˆω t ρ l ˆψ t + C k, k 0 iξ ˆϕ ˆψ lˆω + C k, k 0 + ξ iξˆω l ˆϕ + C 6 + ξ ˆθ t, where C k, k 0, C 6 are positive constants. Proof. Proceeing as proof of Lemma.9, we obtain.58 an.59. Lemma.. The functional Sξ, t = γ Reξ ˆθt ˆθ + γ m Reξ ˆθt ˆθ + γ ξ4 α ˆθ t + α m ˆθ t + γ Reiξ 3 ˆψˆθ + ξ ˆθ iξˆω l ˆϕ satisfies t Sξ, t + k γ ξ 4 ˆθ + k γ m ξ 4 ˆθ γξ ˆθ t iξˆω l ˆϕ + γ ξ 3 ˆψ ˆθ t + γ ξ ˆθ t + γ m ξ ˆθ t.60 Proof. Multiplying.45 by γ ξ ˆθ an taking real part, we obtain { γ Reξ ˆθt ˆθ + α γ ξ 4 ˆθ t + γ Reξ ˆθ iξˆω l ˆϕ } t γ + k ξ 4 ˆθ γ ξ ˆθ t + γξ ˆθ t iξˆω l ˆϕ. Moreover, multiplying.46 by γ m ξ ˆθ an taking real part, { γ Reξ ˆθt ˆθ + α γ ξ 4 ˆθ t + γ Reiξ 3 ˆψˆθ } γ + k ξ 4 ˆθ t m m m γ m ξ ˆθ t + γ ξ 3 ˆψ ˆθ t. Aing.6 an.6, we obtain

21 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM Now, Consier the functional J ξ, t + ε ξ Kξ, t + ξ J ξ, t + ε 3 ξ Hξ, t + Sξ, t, if ρ = k b L ξ, t = an k = k 0, ξ +ξ +ξ 4 λ ε 3 J + +ξ +ξ 4 ε 3λ ξ K + ξ J + ε 3 ξ H + S, if ρ k b or k k where λ, λ, ε, ε 3 are positive constants to be etermine later. Proposition.. There exist constants M, M k = k 0, then > 0 such that if ρ = k b an t L ξ, t + Mξ { bξ ˆψ + k iξ ˆϕ ˆψ lˆω + ˆψ t + k 0 iξˆω l ˆϕ + ρ ˆϕ t + ρ ˆω t + k γ ξ ˆθ + k γ m ξ ˆθ } Cε, ε, ε 3 + ξ + ξ 4 + ξ 6 ξ ˆθ t + Cε, ε + ξ + ξ 4 ˆθ t Moreover, if ρ k b or k k 0, we obtain.64 t L ξ, t + M ξ 4 { bξ + ξ + ξ 4 ˆψ + k iξ ˆϕ ˆψ lˆω + ˆψ t + k 0 iξˆω l ˆϕ + ρ ˆϕ t + ρ ˆω t + k γ ξ ˆθ + k γ m ξ ˆθ } Cε, ε 3, λ, λ + ξ ξ ˆθ + ˆθ..65 Proof. We can prove.64 an.65 following the ieas use on the proof of Proposition.0, thus we omit some etails. First, we suppose that ρ = k b an k = k 0. By Lemmas.3,.9,.6 an.0, it follows that t {L ξ, t Sξ, t} + l ε ε bξ 4 ρ ˆψ + ε 4 3l ε 3kξ iξ ˆϕ ˆψ lˆω + m s ε k bl ε 3 ξ ˆψ t + δ 3l ε 3k 0 ξ iξˆω l ˆϕ + ρ l ε 3 ξ ˆϕ t + ρ l 4 ε 3 ξ ˆω t Cε, ε, ε 3 + ξ + ξ 4 + ξ 6 ξ ˆθ t + Cε, ε + ξ + ξ 4 ˆθ t. Aing Sξ, t in the above inequality, applying Lemma. an Young s inequality, we obtain t L ξ, t + l 3ε ε bξ 4 ρ ˆψ + ε 4 3l ε 3kξ iξ ˆϕ ˆψ lˆω + m s ε k bl ε 3 ξ ˆψ t + δ lε 3 k 0 ξ iξˆω l ˆϕ + ρ l ε 3 ξ ˆϕ t + ρ l 4 ε 3 ξ ˆω t + k γ ξ 4 ˆθ + k γ m ξ 4 ˆθ Cε, ε, ε 3 + ξ + ξ 4 + ξ 6 ξ ˆθ t + Cε, ε + ξ + ξ 4 ˆθ t.

22 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 We choose our constants as follows: ε < l 3ρ, ε < m s, ε 3 < min{ δ l, ε 6l, bl k m s ε } Consequently, we euce that there exists M > 0, such that.64 hols. Secon, we assume that ρ k b an k k 0. By Lemmas.3,.9, the estimate.53 in Lemma.6, Lemma.0, aing these inequalities an by using Young s inequality, we obtain { ξ Sξ, t } l λ ε 3 bξ 6 L ξ, t + t + ξ + ξ 4 3ε ρ + ξ + ξ 4 ˆψ ρ ε 3 lξ ξ + ξ 4 ˆω t + λ m Cε 4, λ ρ ε 3 ξ 4 + ξ + ξ 4 ˆψ t + λ 4 Ck, k ε 3 kξ ξ + ξ 4 iξ ˆϕ ˆψ lˆω + δ Ck, k 0, ε, λ ε 3 k 0 + ξ ξ 4 + ξ + ξ 4 iξˆω l ˆϕ + l ε 4λ ρ ε 3 ξ 4 ρ + ξ + ξ 4 ˆϕ t Cε, ε 3, λ, λ + ξ ξ ˆθ t + ˆθ t. ξ In the last estimate, we use also the inequalities.3. Aing +ξ +ξ 4 Sξ, t in the above inequality, applying Lemma. an Young s inequality, it follows that t L ξ, t + l λ ε 3 bξ 6 4ε ρ + ξ + ξ 4 ˆψ + λ 4 Ck, k ε 3 kξ ξ + ξ 4 iξ ˆϕ ˆψ lˆω + λ m Cε 4, λ ρ ε 3 ξ 4 + ξ + ξ 4 ˆψ t + ρ lε 3 ξ ξ + ξ 4 ˆω t + δ Ck, k 0, ε, λ ε 3 k 0 + ξ ξ 4 + ξ + ξ 4 iξˆω l ˆϕ + l ε 4λ ρ ε ξ 4 ρ + ξ + ξ 4 ˆϕ t + k γ ξ 6 + ξ + ξ 4 ˆθ + k γ m ξ 6 + ξ + ξ 4 ˆθ Cε, ε 3, λ, λ + ξ ξ ˆθ t + ˆθ t. We choose our constants as follows: ε < l 4ρ, λ > 4Ck, k 0, ε 4 < ρ l λ, λ > ρ + Cε 4, λ δ, ε 3 < m Ck, k 0, ε, λ. Consequently, by using.3, we euce that there exists M > 0, such that.65 hols. Theorem.3. For any t 0 an ξ R, we obtain the following ecay rates, Êξ, t { C Êξ, 0e βsξt, if ρ = k b an k = k 0, C Êξ, 0e β s ξt, if ρ k b or k k 0,.66

23 EJDE-07/73 THERMOELASTIC BRESSE SYSTEM 3 where C, β, C, β are positive constants an s ξ = ξ 4 + ξ + ξ 4 + ξ 6 + ξ 4, s ξ 4 ξ = + ξ + ξ + ξ 4. Proof. We prove.66, by Proposition. an using the same approach one in the proof of Theorem., Thus, we omit the etails. 3. Main result In this section, we establish ecay estimates of the solutions to systems. an.. For Bresse system., thermoelasticity of Type I, we consier the vector solution V := ρ / ϕ t, ρ / ψ t, ρ / ω t, γ / θ, γ / θ, b / ψ x, m k / ϕ x ψ x lω x, k / 0 ω x lϕ 3. an for Bresse system., thermoelasticity of type III, V := ρ / ϕ t, ρ / ψ t, ρ / ω t, γ / θ t, k γ / θ x, γ / θ t, m k 3. γ / θ x, b / ψ x, k / ϕ x ψ x lω x, k / 0 ω x lϕ m Note that Êξ, t = ˆV ξ, t, Êξ, t = ˆV ξ, t, 3.3 where Ê an Ê are efine in.6 an.47, respectively. position to prove our main result. We are now in a Proof of Theore.. Applying the Plancherel ientity an 3.3, we have xv k t L R = iξk ˆV t L R = ξ k Êξ, t ξ, R xv k t L R = iξk ˆV t L R = ξ k Êξ, t ξ. By Theorems. an.3, it follows that xv k j t C ξ k e βsξt ˆVj 0, ξ ξ R C ξ k e βsiξt ˆVj 0, ξξ + C ξ = I + I, i, j =,. R ξ ξ k e βsiξt ˆVj 0, ξξ It is not ifficult to see that if ρ = k b an k = k 0, then the function s ξ satisfies s ξ 5 ξ4 if ξ, s ξ 5 ξ 4 if ξ. 3.4

24 4 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 Thus, we estimate I as follows, I C V ˆ j 0 L ξ ξ k e β 5 ξ4t ξ C ˆ V 0 j L + t 4 +k C + t 4 +k Vj 0 L, j =,. On the other han, using the secon inequality in 3.4, we obtain I C ξ k e β 5 ξ 4 t V ˆ0 j ξξ ξ C sup { ξ l e β 5 ξ 4t } ξ k+l V ˆ0 j ξξ ξ C + t l k+l x V 0 j, j =,. R 3.5 Combining the estimates of I an I, we obtain.5. On the other han, if ρ k b or k k 0, the function s ξ satisfies s ξ 8 ξ4 if ξ, s ξ 8 ξ 6 if ξ 3.6 Thus, we estimate I as follows, I C ˆ V 0 j 0 L ξ ξ k e β 8 ξ4t ξ C ˆ V 0 j L + t 4 +k C + t 4 +k V 0 j L Moreover, by using the secon inequality in 3.6, it follows that I C ξ k e β 8 ξ 6 t V ˆ0 j ξξ ξ C sup { ξ l e β 8 ξ 6t } ξ k+l V ˆ0 j ξξ ξ C + t l 3 k+l x V 0 j Combining the estimates of I an I, we obtain.6. R References [] M. Alves, L. Fatori, J. Silva, R. Monteiro; Stability an optimality of ecay rate for a weakly issipative bresse system. Mathematical Methos in the Applie Sciences, 38, 5 05, [] F. A. Boussouira, J. E. M. Rivera, D.. S. A. Júnior; Stability to weak issipative bresse system. Journal of Mathematical Analysis an applications, 374, 0, [3] D. Chanrasekharaiah; Hyperbolic thermoelasticity: a review of recent literature. Applie Mechanics Reviews, 5, 998, [4] L. H. Fatori, R. N. Monteiro. The optimal ecay rate for a weak issipative bresse system. Applie Mathematics Letters, 53, 0, [5] L. Djouamai, B. Sai-Houari; Decay property of regularity-loss type for solutions in elastic solis with vois. J. Math. Anal. Appl., 409, 04,

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26 6 F. A. GALLEGO, J. E. MUÑOZ RIVERA EJDE-07/73 [3] Y. Uea, S. Kawashima; Decay property of regularity-loss type for the Euler-Maxwell system. Methos Appl. Anal., 83, 0, [33] X. Zhang, E. Zuazua; Decay of solutions of the system of thermoelasticity of type iii. Communications in Contemporary Mathematics, 50, 003, Fernano A. Gallego Centre e Robotique CAOR, MINES Paristech, PSL Research University, 60 boulevar Saint-Michel, 757 Paris Ceex 06, France aress: ferangares@gmail.com Jaime E. Muñoz Rivera Laboratório e Computução Científica, LNCC, Petrópolis, , RJ, Brazil. Instituto e Matemática, Universiae Feeral o Rio e Janeiro, UFRJ, P.O. Box 68530, , Rio e Janeiro, RJ, Brazil aress: rivera@lncc.br, rivera@im.ufrj.br