Stability of Timoshenko systems with past history

Transcription

1 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P. - J. Math. Anal. Appl. Stability of Timosheno systems with past history Jaime E. Muñoz Rivera a,b, Hugo D. Fernánez Sare b, a Department of Research an Development, National Laboratory for Scientific Computation, Rua Getulio Vargas 333, Quitaninha CEP 565-7, Petrópolis, Rio e Janeiro, Brazil b Mathematics Institute, Feeral University of Rio e Janeiro, Rio e Janeiro, Brazil Receive 9 October 6 Submitte by M. Naao Abstract We consier vibrating systems of Timosheno type with past history acting only in one equation. We show that the issipation given by the history term is strong enough to prouce exponential stability if an only if the equations have the same wave spees. Otherwise the corresponing system oes not ecay exponentially as time goes to infinity. In the case that the wave spees of the equations are ifferent, which is more realistic from the physical point of view, we show that the solution ecays polynomially to zero, with rates that can be improve epening on the regularity of the initial ata. 7 Elsevier Inc. All rights reserve. Keywors: Timosheno system; Exponential stability; Polynomial rate of ecay. Introuction We consier the following linear Timosheno system with past history: ρ ϕ tt ϕ x + ψ x = in,l,,. ρ ψ tt bψ xx + an initial conitions gsψ xx x, t ss + ϕ x + ψ= in,l,,. ϕ, = ϕ, ϕ t, = ϕ, ψ, = ψ, ψ t, = ψ in,l.3 with positive constants ρ,ρ,,b. Here we are intereste in the asymptotic behavior of the solutions. Our main tools are Prüss results on the exponential stability of semigroups, see [7,8]. In orer to use these results, it is necessary to embe the problem into Supporte by a CNPq-DLR grant an FAPERJ. * Corresponing author. aresses: rivera@lncc.br J.E. Muñoz Rivera, hugo@pg.im.ufrj.br H.D. Fernánez Sare. -47X/$ see front matter 7 Elsevier Inc. All rights reserve. oi:.6/j.jmaa.7.7. Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7.

2 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P. - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. the context of semigroup, so some moifications in our original system.. shoul be mae. We introuce the notation η t x, s := ψx,t ψx,t s, then system.. is rewritten as.4 ρ ϕ tt ϕ x + ψ x =, ρ ψ tt b gss ψ xx η t + η s ψ t =, Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa gsη t xx x, s s + ϕ x + ψ=,.6 where the thir equation is obtaine ifferentiating.4 with respect to s. The initial conitions are given by ϕ, = ϕ, ϕ t, = ϕ, ψ, = ψ, ψ t, = ψ in,l,.8 η,s= ψ, ψ, s in,l,,.9 which means that the history is consiere as an initial value. We consier Dirichlet bounary conitions, but our arguments can be use to prove similar results for other bounary conitions. Concerning the ernel g we consier the following hypotheses: gt >,,, > : gt g t gt, g t gt, t,. b := b gss >..7. Let us mention some energy ecay results for issipative Timosheno systems. In [], Kim an Renary consiere a conservative Timosheno system with two bounary feebacs an they prove exponential stability for the energy associate to the system. If the history term in. is replace by the control function bxψ t, b >, then Soufyane [9] prove exponential stability for the linearize system if an only if the waves spee of Eqs..,. are equal, that is, ρ = ρ b.. Similar results are obtaine by Rivera an Race [5], where semigroups techniques are use. In [4] the same authors consier a issipative Timosheno system with a issipation through a coupling to a heat equation, an they show exponential stability if an only if. hols. In [], Ammar Khoja et al. consier also a Timosheno system with memory effect but consiering null history, in that case the system is calle a Volterra integro-ifferential system. For the Volterra problem they prove the exponential stability provie the wave spees are equal. When the wave spees are ifferent, the authors consier a class of ernels for which there is no exponential stability. No information is given concerning the ecay in this case. Introucing non-zero history on ψ maes the problem ifferent from that consiere in [], so ifferent techniques have to be use. The main result of this paper is to show that the system is exponentially stable if an only if the wave spees of Eqs..,. are equal, that is, if an only if. hols. Moreover, the class of ernel that we consier here to prove the lac of exponential stability is larger than that consiere in []. In particular our result implies the non-exponential stability for singular ernels. When the ientity. oes not hol, which is more interesting from the physical point of view, we show that the first-orer energy ecays polynomially with rates that epen on the regularity of the initial ata. The paper is organize as follows. In Section we establish the existence an uniqueness results to system.5.7. The exponential stability of the semigroup associate to this system is stuie in Section 3. In Section 4 we show the non-exponential stability of the semigroup. Finally, in Section 5 we show the polynomial ecay when the wave spees are ifferent.

3 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.3 - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. 3. Existence an uniqueness To facilitate our analysis we consier the following bounary conitions: ϕ,t= ϕl,t = ψ,t= ψl,t= η t,s= η t L, s =, s,t.. In view of., let L g R+,H be the Hilbert space of H -value functions on R+, enowe with the inner prouct L ϕ,ψ L g R +,H = gsϕ x sψ x s s x. Now, we use the Lumer Phillips theorem see [6] to obtain existence an uniqueness results. Therefore, we formulate our problem as an abstract Cauchy problem. We efine U := ϕ, ϕ t,ψ,ψ t,η t, so the system.5.7 is equivalent to U t = AU, U = U where U := ϕ,ϕ,ψ,ψ,η an A is given by I ρ x ρ x A := I.. ρ x b ρ x ρ I ρ gs x,ss I s Let H := H,L L,L H,L L,L L g R +,H. It is not ifficult to prove that H together with the norm U H = u,u,u 3,u 4,η H.3 = ρ u L + ρ u 4 L + b u 3 x L + u x + u 3 L + η L g R+,H.4 is a Hilbert space. The operator A has the following omain: { DA = U H: u,u 3 H,L; u,u 4 H,L, } gsηxx t x, s s L,L, η s L g R +,H,η =. Then the operator A, formally given by., is the infinitesimal generator of a contraction semigroup. In fact, note that A is issipative, because for any U DA we have Re AU,U H = g s η x sx gs η x sx. We also have ImI A = H. Therefore, by the Lumer Phillips theorem, it follows that A is the infinitesimal generator of a contraction semigroup. Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7.

4 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.4-4 J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. Theorem.. Assume that g satisfies.. an that U DA, then there exists a unique solution U = ϕ, ϕ t,ψ,ψ t,ηto system.5.7 with bounary conitions. satisfying U C R + ; DA C R + ; H. Moreover, if U DA n, then U C n R + ; D A, =,,...,n. Remars. For another bounary conitions, we enote A := x. We consier the following cases: DA = H,L H,L, { } DA = v H,L: vx=, DA = { v H,L: v = v x L = }, DA = { v H,L: v x = vl = }. We efine H = L,Lor L,L, with L {v L },L:= L,L: vx=. Then the semigroups formulation is mae in the Hilbert spaces of type H := D A / H D A / H L g R +,D A / Exponential stability First we consier the system.5.7 with bounary conitions. an the hypotheses over the ernel g.. hol. We shall emonstrate that the energy Et = [ ρ ϕt + ρ ψt + bψx + ϕ x + ψ + Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. gs η t x s ] x 3. ecays to zero exponentially as time goes to infinity provie conition. hols. We shall use Prüss results [3], which states that a semigroup e At is exponentially stable if an only if the following conitions hol: ir ϱa resolvent set 3. an C >, U DA, λ R: iλi A H C. 3.3 In fact, note that the resolvent equation iλi AU = F is given by iλu u = f, 3.4 iλρ u u x + u3 x = ρ f, 3.5 iλu 3 u 4 = f 3, 3.6 iλρ u 4 bu 3 xx gsηxx t x, s s + u x + u3 = ρ f 4, 3.7 iλη + η s u 4 = f 5, 3.8

5 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.5 - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. 5 where b := gss, b := b b >. To prove conition 3.3 we will use a series of lemmas. 3.9 Lemma 3.. Let us suppose that conitions. an. on g hol. Then there exists a positive constant C>, being inepenent of F H, such that gs η x sx C U H F H. Proof. Multiplying 3.5 by u in L,L we get iλρ u x + an, using Eq. 3.4, iλρ u x iλ u x + u 3 u x x = ρ u x + u 3 u x x = ρ f u x f u x + On the other han, multiplying Eq. 3.7 by u 4 in L,L we get iλρ u 4 x + b L u 3 x u4 x x + gsη x u 4 x sx + } {{ } :=I Substituting u 4 given by 3.8, 3.6, into I an I, respectively, we get iλρ = ρ u 4 x iλ b f 4 u 4 x + b L u 3 x x iλ Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. gs η x sx iλ L fx 3 u3 x x + u x + u 3 f 3 x + u x + u 3 fx x. 3. u x + u 3 u 4 x = ρ } {{ } :=I u x + u 3 u 3 x + Aing 3. an 3., using. an taing the real part our conclusion follows. Lemma 3.. With the same hypotheses as in Lemma 3. there exists C> such that ρ u 4 x C U H F H + C U / H F / H Proof. Multiplying 3.7 by gs ηs in L,Lwe get u 3 x L + u x + u3 L. f 4 u 4 x. gsη x η xs s gsη x fx 5 sx. 3.

6 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.6-6 J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. iλρ gs ηu 4 sx+ b } {{ } :=I 3 = ρ gs ηf 4 sx. From Lemma 3. we obtain gsη x s x gss Substituting η given by 3.8 into I 3,using Re { L } gsη s u 4 sx ρ L gsη x u 3 x sx + gsη x s x + gs η x sx C U H F H. u 4 x + C g s ηx sx an., our conclusion now immeiately follows from Lemma 3.. To estimate u 3 we introuce the multiplier w given by the solution of the Dirichlet problem gs u x + u3 ηsx w xx = u 3 x, w = wl =. Note that w can be written as wx = x u 3 y y + x L Uner the above conitions we have u 3 y x G u 3 x. 3. Lemma 3.3. With the same hypotheses as in Lemma 3., for any ε > there exists C ε > such that b u 3 x x C ε U H F H + C ε U / H Proof. Multiplying 3.7 by u 3 yiels iλρ u 4 u 3 x+ b } {{ } :=I 4 u 3 x x + Substituting u 3 given by 3.6 into I 4 we get b L u 3 x x + u x u3 x + = ρ u 4 x F / H u x + u3 L + ε ρ u L. L gsη x u 3 x sx + u x u3 x + u 3 x gsη x u 3 x sx + ρ Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. f 4 u 3 x + ρ u 3 x = ρ L f 4 u 3 x. u 4 f 3 x. 3.3

7 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.7 - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. 7 On the other han, multiplying 3.5 by w we have Since u x w x x w x x = ρ u [ G u 4 + G f 3] x + ρ f wx. 3.4 u x w x x = u x u3 x we conclue from that b u 3 x x L w x x u 3 x = ρ f 4 u 3 x + ρ u 4 f 3 x + ρ u G f 3 x + ρ f wx+ ρ gsη x u 3 x sx + ρ u G u 4 x. Note that, for any ε > there exists C ε > such that Re { ρ Finally, since } u G u 4 x ε ρ u L + C ε ρ u 4 L. u 4 x w x x u 3 x, taing real part an using Lemmas our conclusion follows. Our next step is to estimate the term u x + u3 L. Here we shall use conition.. Lemma 3.4. With the same hypotheses as in Lemma 3. together with conition., for any ε > there exists C ε > such that [ u x + u 3 ] x C ε U H F H + Re bu 3x + gsη x s where ε is given by the Lemma 3.3. u x x=l x= + ε + ε u L, Proof. Multiplying 3.7 by u x + u3 we have iλρ [ u 4 ] u x + u3 x bu 3x + gsη x s u x Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. x=l x= + u x + u3 x

8 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.8-8 J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. [ u + bu 3x + gsη x s] x + u3 x x = ρ } {{ } :=I 5 Substituting u x + u3 x given by 3.5 into I 5 we get iλρ u 4 u x x + iλρ } {{ } :=I 6 } {{ } :=I 7 iλ ρ gsη x u sx } {{ } :=I 8 bρ u 3 x f x + [ ] u 4 u 3 x bu 3x + gsη x s u x + u3 x = ρ gsη x f sx L f 4 u x + u3 x. Substituting u given by 3.4 an u 4 given by 3.6 into I 6 we obtain I 6 = iλρ Using 3.6 we get I 7 = ρ u 3 u x x ρ u 4 x ρ L u 4 f x x + ρ Finally, a substitution of η given by 3.8 yiels I 8 = ρ From 3.6 we can rewrite I 8 as I 8 = ρ gsη xs u sx ρ b Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. u x x=l x= iλ bρ u 3 x u x f 4 u x + u3 x. 3.5 f 3 u x x. 3.6 u 4 f 3 x. 3.7 g sη x u sx iλ ρ b Using in 3.5 we obtain ρ iλb ρ b }{{} = L [ ] = bu 3x + gsη x s + ρ u 4 x u x ρ L u 3 u x x + u x + u3 x u x x=l x= gsη x f sx + ρ + ρ u 3 x u x + ρ b u 4 x + ρ f 4 u x + u3 x gsf 5 x u sx. f 3 x u x ρ g sη x u sx + ρ b gsf 5 x u sx. 3.8 u 3 x f x

9 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.9 - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. 9 + ρ u 4 f 3 x + ρ u 4 fx x + ρ ρ L b fx 3 u x + ρ Now, using. an the previous lemmas, our claim follows. gsf 5 x u sx. Noting that, when the bounary conitions are of mixe type, the bounary term in Lemma 3.4 is equal to zero. In the case., this bounary term is not equal to zero. In the next lemma we mae an estimation of the bounary term. Lemma 3.5. Uner the above notations, let us tae q C [,L] such that q = ql =, then there exist C,C q > such that i ii an qx bu 3 x + gsη x s x=l x= C U H F H + C U / H + ε Cρ u L + C q u 3 x F / H qx x=l u x C U H F H + C q u x + u3 L + ρ u L. x= L u x + u3 L u x + u3 L, Proof. To prove i, multiplying 3.7 by qx bu 3x + gsη x s in L,Lwe have iλρ u 4 qx bu 3x + gsη x s x } {{ } :=I 9 + qx u x + u3 bu 3 x + ReI 9 = ρ b gsη x s From 3.6 an 3.8 we obtain q x { u 4 x + Re bρ ρ Note that qx ReI = bu 3 x + bu 3xx + gsη xx s qx bu 3x + gsη x s x Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. } {{ } :=I x = ρ f 4 qx bu 3x + gsη x s x. 3.9 qxu 4 f 3 x x ρ g sqxu 4 η x sx gsqxu 4 f 5 x sx }. 3. gsη x s x=l x= + q x bu 3 x + gsη x s x. 3.

10 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P. - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. From , an using the previous lemmas we obtain i. To get ii, multiply 3.5 by qxu x, iλρ u qxu x x } {{ } :=I L u xx qxu x x u 3 x qxu x x = ρ f qxu x x. Substituting u given by 3.4 into I, using Lemma 3.3 an taing the real part our conclusion follows. Lemma 3.6. There exists C> such that ρ u x C U H F H + 4 u x + u 3 L. Proof. Multiplying Eq. 3.5 by u we get iλρ u u x+ } {{ } :=I u x + u 3 u x x = ρ f u x. Substituting u given by 3.4 into I an taing real parts we get ρ u x C U H F H + u x + u 3 L + C u 3 x Using Lemma 3.3 for ε sufficiently small, our conclusion follows. Now we are in the position to prove the main result of this section. Theorem 3.7. Let us assume hypotheses. an. on g, suppose that initial ata satisfies ϕ,ψ H,L, η L g R +,H an ϕ,ψ L,L an suppose that conition. hols. Then the energy Et ecays exponentially to zero as time tens to infinity, that is, there exist positive constants C an α, being inepenent of the initial ata, such that Et CEe αt, t. Proof. We shall prove conitions 3. an 3.3 see [7]. Let U = u,u,u 3,u 4,η an F = f,f,f 3,f 4,f 5 satisfy , then, from Lemma 3., we get L. η L g C F H U H. 3. From Lemma 3., for ε >, there exists C := C ε > such that ρ u 4 L C F H U H + b u 3 x L + ε u x + u3 L. 3.3 Also, from Lemma 3.3, we obtain b u 3 x L C ε F H U H + ε ρ u L + ε u x + u3 L. 3.4 Then, aing 3.3 an 3.4 we get Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7.

11 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P. - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. ρ u 4 L + b u 3 x L C F H U H + ε ρ u L + ε u x + u 3 L. 3.5 On the other han, from Lemmas 3.5 an 3.3, we obtain for each Ñ>an δ>, qx Ñ bu x=l 3 x + gsη x s CÑ F H U H + u x 4 + u3 L + ε CÑ ρ u L 3.6 x= an qx x=l δ u x C δ F H U H + u x= 4 x + u 3 L + δc q ρ u L. 3.7 Aing 3.6, 3.7, using Lemma 3.4 an using Rez z σ z + C σ z, z,z C, σ >, we obtain, for any <τ<, τ := τδ,ε,ε >, that there exists C τ > such that u x + u3 L C 4 F H U H + τρ u L. 3.8 Finally, from Lemma 3.6, we have τρ u L τc F H U H + 8τ u x + u 3 L. 3.9 Aing 3.8 an 3.9 we conclue 8τ u x + u3 L + τρ u L C 5 F H U H. 3.3 From 3., 3.5 an 3.3, we obtain for ε,ε sufficiently small, that there exists C> inepenent of λ an F, U such that U H C F H, This completes the proof. U DA. Remar. For another bounary conitions, the elliptic problem 3. must change. For example, for ϕ,t= ϕl,t = ψ x,t= ψ x L, t = ηx t,s= ηt x L, s =, w is given by the solution of the problem w xx = u 3 x, w x = w x L =. 4. Non-exponential ecay Now we shall prove that conition. is also necessary for exponential stability in the case where the bounary conitions are of mixe type. We use the following lemma. Lemma 4.. Let us suppose that g satisfies the conitions. an let us assume that sgs =. lim s Then there exists C>such that λ gse iλs s C, uniformly in λ R. Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7.

12 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P. - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. Proof. Note that Then Setting gse iλs s = π/λ π/λ gse iλs s gse iλs s π/λ gss = π/λ νλ = sup sgs asλ, s, π λ the above integral is less than or equal to νλ π/λ s = πνλ. s λ π/λ sgs s e iλs gs + π/λs s. Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. π/λ [ s+π/λ ] e iλs g y y s. The estimation of the secon integral is similar. Concerning the last term, changing the orer of integration, an maing use of., we get [ s+π/λ ] [ s ] e iλs g y y s g y y s = π λ gπ/λ, π/λ s π/λ s+π/λ which, multiplie by λ, tens to zero as λ. Theorem 4.. Let us suppose that. oes not hol. Then the semigroup associate to system.5.7, with bounary conitions ϕ x,t= ϕ x L, t = ψ,t= ψl,t= η t,s= η t L, s =, s,t, 4. is not exponentially stable. Proof. From.5 let us consier the Hilbert space H := H,L L,L H,L L,L L g R +,H,L. Here the omain of the operator A is efine by { DA = U H : u H,L, u x H,L, u H,L, u3 H,L, } u 4 H,L, gsηxx t x, s s L,L, η s L g R +,H,η =. Now, from the previous analysis, we have that U = ϕ, ϕ t,ψ,ψ t,η satisfies t Ut= AUt, U = U. To show the lac of ecay it is enough to show that the solution of iλ n I AU n = F n s

13 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.3 - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. 3 satisfies lim U n H =, n where λ λ n := nπ δl As F we choose n N, δ := F F n :=,f,,f 4,, where ρ. f x := cosδλx, f 4 x := sinδλx. The solution U = v,v,v 3,v 4,η to iλ AU = F, shoul satisfy iλv v =, iλv ρ v xx ρ v 3 x = f, iλv 3 v 4 =, iλv 4 b ρ v 3 xx + b ρ v 3 xx ρ iλη + η s v 4 =, where b := gss. Eliminating v,v 4 we get 4.4 gsη t xx x, s s + ρ v x + ρ v 3 = f 4, λ v ρ v xx ρ v 3 x = f, 4.7 λ v 3 b ρ v 3 xx + b ρ v 3 xx ρ iλη + η s iλv 3 =. This can be solve by gsη t xx x, s s + ρ v x + ρ v 3 = f 4, 4.8 v x = A cosδλx, v 3 x = B sinδλx, ηx, s = ϕssinδλx, where A, B, ϕs epen on λ an will be etermine explicitly in the sequel. Note that this choose is just compatible with the bounary conitions. The system is equivalent to 4.9 λ A + ρ δ λ A ρ δλb =, 4. λ B + b ρ δ λ B b ρ δ λ B + δ λ ρ iλϕs + ϕ s iλb =. Solving 4. we get gsϕss ρ δλa + ρ B =, ϕs = Ce iλs + B. Since η =, then C = B, an 4.3 becomes ϕs = B Be iλs Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7.

14 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.4-4 J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. Then, from 4.4 we have gsϕss = gs [ B Be iλs] s = Bb B gse iλs s. 4.5 Using 4.5, we have from that A an B satisfy δ λ A δλb =, 4.6 ρ ρ b δ λ B δ λ gse iλs s B δλa + B =. 4.7 ρ ρ ρ ρ Since ρ δ =, we conclue from 4.6 that ρ B = λ. Substituting 4.8 into 4.7 we get A = λ ρ ρ λ + ρ Recalling that v = iλv = iλacosδλx we get v x = i λ iρ ρ + iρ λ Note that v L L = v x gse iλs s + b gse iλs s + ib ρ b ρ. ρ b ρ λ cosδλx. 4.8 = L λ ρ ρ + ρ λ gse iλs s + b ρ b ρ λ L λ ρ ρ + ρ λ gse iλs s }{{} boune as λ using Lemma 4.. Therefore we have lim U n λ H lim v λ L = which completes the proof. Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa L Remar. The result also hols for the following bounary conitions: b ρ b ρ λ ϕ,t= ϕl,t = ψ x,t= ψ x L, t = ηx t,s= ηt x L, s =, ϕ x,t= ϕl,t = ψ,t= ψ x L, t = η t,s= ηx t L, s =, ϕ,t= ϕ x L, t = ψ x,t= ψl,t= ηx t,s= ηt L, s =.

15 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.5 - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl Polynomial ecay In this section we shall show the polynomial ecay of the solutions of the system.5.7 with bounary conitions., when. oes not hol. Let us introuce the secon-orer energy E t := Eϕ t,ψ t,η t. Then from 3. an., results t Et t E t We efine w as the solution of the Dirichlet problem w xx = ψ x, w = wl =, an we introuce the functional F t := [ρ ψ t ψ + ρ ϕ t w] x. Then the following lemma hols. gs η t x s x, 5. gs ηxt t s x. 5. Lemma 5.. For any ε > there exists a positive constant C ε > such that t F t b ψ x x + ε Proof. Multiplying Eq..6 by ψ we get t ρ ψ t ψx= ρ ψ t x b Multiplying Eq..5 by w we obtain t ρ ϕ t wx= Equations 5.4 an 5.5 lea to t F t = ρ + ρ ψ t x b ϕ t x + C ε ϕ x ψx ϕψ x x + ϕ t w t x L ψx x L ψx x ψ x + ρ w x x + ρ ψ t x + C Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. ψ x + gsη t x s ψ x x. gsη t x s ψ x x gs ηx t s x. 5.3 ψ t x. 5.4 ϕ t w t x. 5.5 w x x

16 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.6-6 J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. Therefore, using our conclusion follows. gsη t x s ψ x x δ ψ x x + C δ Let us enote by K the functional Kt := ρ ψ t gsη t x, s s x. Let b := gss. Using Eqs..6.7 we get Since an t Kt = b + ψ x gsηx t t, s s x + ϕ x + ψ ψ t gsη s x, s s x = gsη t x x, s s x b Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. gsη t x, s s x + ρ gs ηx t s x gsη t x x, s s x ρ b ψ t g sη s x, s s x ψ t gsη s x, s s x. ψ t x gs ηx t s x, 5.6 using Poincare s inequality we conclue that, for any ε > there exists C ε > such that t Kt ρ b ψ t x + ε ψ x x + ε ϕ x x + C ε gs ηx t s x. 5.7 Now, we efine the functional Et as Et := N Et + E t + F t + N Kt, 5.8 where N := Nε,ε >. Then, from an 5.7 we get L t Et NK gs L ηxt t N s x + C + N C ε gs ηx t s x b N ε L Also, we efine the functional F t as F t := ρ ψ t ϕ x + ψx + ρ b ψ x x N ρ b ψ x ϕ t x + ρ L C ε ψ t x + ε ϕ t x + N ε gsη t x x, s s ϕ t x. ϕx x. 5.9

17 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.7 - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. 7 Uner the above conitions we have Lemma 5.. For any ε 3 >, there exists a constant C ε3 > such that [ ] x=l t F t bψ x + gsηx t x, s s ϕ x ϕ x + ψ x + ε 3 + ρ ψ t x + C ε3 x= gs ηx t s x + C ε3 Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. Proof. Multiplying Eq..6 by ϕ x + ψ an using.5 we get [ ] x=l t F t = bψ x + gsηx t x, s s ϕ x ϕ x + ψ x L ρ b + ρ On the other han, from.7 we have ψ xt ϕ t x + ψ xt = ηxt t + ηt xs. Recalling that b = gss, we euce b Therefore ψ xt ϕ t x = = = = ψ xt ϕ t x = b gsψ xt s ϕ t x x= gs [ ηxt t + ] ηt xs s ϕ t x gsη t xt s ϕ t x + gsη t xt s ϕ t x gsηxt t s ϕ t x b ϕ t x gs ηxt t s x. gsη t x x, s s ϕ t x. 5. gsη t xs s ϕ t x g sη t x s ϕ t x. Substituting 5. into 5. results [ ] x=l t F t = bψ x + gsηx t x, s s ϕ x + b ρ b ρ L x= g sη t x s ϕ t x. 5. ϕ x + ψ x gsηxt t s ϕ t x ρ b b ρ L g sηx ϕ t s t x

18 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.8-8 J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. + gsη t x x, s s ϕ t x. 5. Finally, using the hypothesis on g in 5., our conclusion follows. Remar. When the bounary conitions are of mixe type, the bounary term in the last lemma is equal to zero. For bounary conitions of Dirichlet type, we nee to estimate this term. In orer to eal with the bounary term we shall prove the following lemma. Lemma 5.3. Let q C [,L] satisfy q = ql =, an let us introuce the following functionals: J t := ρ J t := ρ ψ t qx bψ x + ϕ t qxϕ x x. gsη t x x, s s x, Then there exist C > an, for any ε>, a positive constant C ε > such that i [ t J t bψ x L, t + + C ψ t x + ε gsηx t L, s s + bψ x,t+ ϕx x + C ε ψ x x + C ε gsη t x,ss ] gs η t x s x, ii t J t [ ϕ x L, t + ϕ x,t] + C Proof. Using Eqs..6.7 we get [ t J t = ] x=l qx bψ x + gsηx t s + ρ qxϕ x + ψ bψ x + ψ t qx [ = bψ x L, t + x= [ ϕ t + ϕx + ] ψ x x. Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. q x bψ x + gsηx t x, s s x ρ b gs [ η xs + ψ xt ] s x gsηx t L, s s + bψ x,t+ gsη t x x, s s x q xψ t x gsη t x,ss ]

19 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P.9 - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. 9 b ρ q x bψ x + qxϕ x ψ x x b qxψ gsη t x x, s s x q xψ x Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7. gsηx t x, s s x ρ b + qxψ t gsηs t x, s s x. qxϕ x gsηx t x, s s x q xψ t x Then, using 5.6 an the hypotheses. on g, conclusion i follows. To prove ii we use.5, that is t J t = qxϕ xx ϕ x x + qxψ x ϕ x x + ρ = L [ ] qxϕ x=l x x= + qxψ x ϕ x x ρ We efine, for δ> an Ñ>, the following functional: F 3 t := F t + ÑJ t + δj t. From Lemmas 5. an 5.3 we conclue, observing ϕ x + ψ x 4 L ϕx x + C ψ x x C >, qxϕ t ϕ xt x q xϕ t x. that, for sufficiently small ε,ε 3,δ>, large Ñ an for any <τ<, there exist C τ > an C > such that t F 3t + C τ Finally, we efine the functional F 4 t := Using.5 an.6 we get t F 4t ρ ϕ x + ψ x + C τ [ ψx + ψ t + [ρ ϕ t ϕ + ρ ψ t ψ] x. ϕ t x ρ Choosing τ small enough, we have ϕ t x 5.3 gs η t x s + gs ] η t xt s x. 5.4 ψ t x + ϕ x + ψ x + C 5.5 [ψ x + gs ] ηx t s x. 5.6

20 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P. - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. { F 3 t + C } τ F 4 t t ρ 4 + C τ ϕ x + ψ x C τ [ gs ηx t s + Now we are in the position to prove the polynomial rate of ecay. ϕ t x + C τ [ ψ t + ψx ] x gs ] ηxt t s x. 5.7 Theorem 5.4. Suppose that. hols an initial ata verifies ϕ,ψ H H,L, η L g R +,H H an ϕ,ψ H,L. Then the first-orer energy Et ecays polynomially to zero, that is, there exists a positive constant C, being inepenent of the initial ata, such that Et C E + E. t Moreover, if U := ϕ,ϕ,ψ,ψ,η DA, then TtUH C A t U H. 5.8 Proof. We efine Lt as { Lt := Et + μ F 3 t + C } τ F 4 t. ρ Choosing μ, ε,ε small, N,N large, an using the inequalities 5.9 an 5.7, we get Lt αet, t for some α>. Therefore t α Ess L Lt, t. 5.9 On the other han, it is not ifficult to prove that there exists a constant β> such that L Lt β E + E, t. 5. From we obtain t Ess β α E + E. Finally, since { } tet = Et + t Et Et, t t from 5. we get 5. Et C E + E, t where C := β α >. Finally, if U DA, we use Prüss results [8] to obtain 5.8, which completes the proof. Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7.

21 YJMAA:69 JID:YJMAA AID:69 /FLA [m3sc+; v.73; Prn:9/8/7; :7] P. - J.E. Muñoz Rivera, H.D. Fernánez Sare / J. Math. Anal. Appl. References [] F. Ammar Khoja, A. Benaballah, J.E. Muñoz Rivera, R. Race, Energy ecay for Timosheno systems of memory type, J. Differential Equations [] J.U. Kim, Y. Renary, Bounary control of the Timosheno beam, SIAM J. Control Optim [3] Z. Liu, S. Zheng, Semigroups Associate with Dissipative Systems, Res. Notes Math., vol. 398, Chapman & Hall/CRC, Boca Raton, 999. [4] J.E. Muñoz Rivera, R. Race, Milly issipative nonlinear Timosheno systems Global existence an exponential stability, J. Math. Anal. Appl [5] J.E. Muñoz Rivera, R. Race, Global stability for ampe Timosheno systems, Discrete Contin. Dyn. Syst [6] A. Pazy, Semigroups of Linear Operators an Applications to Partial Differential Equations, Springer-Verlag, New Yor, 983. [7] J. Prüss, On the spectrum of C -semigroups, Trans. Amer. Math. Soc [8] J. Prüss, A. Bátai, K. Engel, R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr [9] A. Soufyane, Stabilisation e la poutre e Timosheno, C. R. Aca. Sci. Paris Sér. I Please cite this article in press as: J.E. Muñoz Rivera, H.D. Fernánez Sare, Stability of Timosheno systems with past history, J. Math. Anal. Appl. 7, oi:.6/j.jmaa.7.7.