On the regularity and stability of three-phase-lag thermoelastic plate

UPCommons Portal del coneixement obert de la UPC http://upcommons.upc.edu/e-prints This is an Accepted Manuscript of an article published by Taylor & Francis in Applicable analysis on 25/02/2021, available online: https://www.tandfonline.com/doi/full/10.1080/00036811.2021.18920 79. Published paper : Liu, Z.; Quintanilla, R.; Wang, Y. On the regularity and stability of three-phase-lag thermoelastic plate. "Applicable analysis: an international journal", 25 Febrer 2021. doi: 10.1080/00036811.2021.1892079 URL d'aquest document a UPCommons E-prints: https://upcommons.upc.edu/handle/2117/341856 On the Regularity and Stability of Three-Phase-Lag Thermoelastic Plate Zhuangyi Liu Department of Mathematics and Statistics University of Minnesota Duluth, MN 55812-2496, USA School of Mathematics Beijing Institute of Technology Beijing 10081, China E-mail:[email protected] Ramon Quintanilla Department of Mathematics UPC Terrassa Colom 11, 08222 Terrassa, Spain E-mail:[email protected] Yang Wang School of Mathematics, Physics and Statistics Shanghai University of Engineering Science Shanghai, 201600, P. R. China E-mail: [email protected] Abstract In this paper we consider the following three-phase-lag thermoelastic plate ρ¨u=−∆(µ∆u−mθ), c(˙ θ+τq¨ θ) = m∆( ˙u+τq¨u) + ∆(k∗α+τ∗ νθ+kτθ˙ θ). We obtain analyticity and exponential stability for the associated C0−semigroup when the coefficients satisfy τν≥k∗τq. The analyticity of the semigroup still holds when τν< k∗τq. However, instability of solution was proved by the Routh-Hurwitz rule. Key words: three-phase-lag thermoelastic plate, analyticity, exponential stability. MSC 2000 35Q35,35Q30,35L65,76N10 1 Introduction Causality principle is violated by the theory based on the Fourier law when we consider the heat equation c˙ θ=qi,i, where cis the thermal capacity and qiis the heat flux vector and θis the temperature. To overcome this drawback many scientists have been interested in proposing alternative heat conduction theories. Cattaneo and Maxwell proposition is the first of these alternatives and it leads to a damped hyperbolic equation[1]. It has been extended by Lord and Shulman to a thermoelastic theory[2]. Some other theories have been proposed (see[3–6]). In the decade of the 1990’s several theories were proposed. On one side, Green and Naghdi proposed three thermoelastic theories[7–9] which were based on some axioms of thermoelasticity. On the other side, Tzou[10] proposed an alternative possibility based on an intuitive point of view which generalizes the Cattaneo and Maxwell theory in the case that we consider the Taylor approximations. More recently, Choudhuri[11] considered an extension of these two last theories. This is called three-phase-lag theory and recovers the Green and Naghdi theory when we consider Taylor approximations. This theory has deserved much attention in the last years[13, 14]. Quintanilla and Racke[16] gave the suitable conditions to clarify the stability of the heat equation. Other studies concerning the stability have been considered[17]. It 1 is also worth recalling that Borgmeyer[18] studied the thermoelastic problem and she obtained several stability results. In this note we focus on the problem determined for a thermoelastic plate[19] in the case of a threephase-lag theory. It is worth noting that Choudhuri proposed two particular heat conduction equations, and we here consider the case corresponding to a parabolic heat conducting equation. We will prove that in this case the solutions of the problem are determined by an analytic semigroup. Similar results have been obtained for other heat conduction theories[20, 21]. In addition, we also investigate the stability of this system. In section 2 we will prove the analyticity and exponential stability of the semigroup in the case that certain inequalities for the relaxation parameters hold. Section 3 is devoted to the case that one of the inequalities does not hold. We will prove that the semigroup remains to be analytic, but the stability is lost, which is proved by the Routh-Hurwitz rule. System of three-phase-lag thermoelastic plate are given by ρ¨u=−∆(µ∆u+mθ),(1.1) c(˙ θ+τq¨ θ) = m∆( ˙u+τq¨u) + ∆(k∗α+τ∗ νθ+kτθ˙ θ).(1.2) Here ˙α=θ, and the system coefficients ρ > 0, µ > 0, c > 0, k∗>0, τθ>0, τ∗ ν=k+k∗τν, τν>0, m 6= 0. After the change of variable ˆu=u+τq˙u, we can get (ρ¨u=−∆(µ∆u+m(θ+τq˙ θ)),(1.3) c(˙ θ+τq¨ θ) = m∆ ˙u+ ∆(k∗α+τ∗ νθ+kτθ˙ θ).(1.4) We still use notation uinstead of ˆuhereafter. Initial condition and boundary condition are u(x, 0) = u0(x),˙u(x, 0) = v0(x),(1.5) α(x, 0) = α0(x),˙α(x, 0) = θ0(x),¨α(x, 0) = η0(x),(1.6) u= ∆u=α= 0, on ∂Ω,(1.7) where Ω is a bounded domain in Rnwith smooth boundary ∂Ω. 2 Analyticity and Exponential Stability for the case τ∗ ν≥k∗τq It is important to identify a proper state space so that the “energy” of the system (1.3)-(1.7) is dissipative. For this purpose, we take the inner product of ˙uwith (1.3) and θ+τq˙ θwith (1.4) in L2(Ω) to get 1 2 d dtρk˙uk2+1 2 d dtµk∆uk2=mh∇(θ+τq˙ θ),∇˙ui(2.1) and 1 2 d dtckθ+τq˙ θk2+1 2 d dtk∗k∇αk2+1 2 d dt2k∗τqh∇α, ∇θi+1 2 d dt(τ∗ ντq+kτθ)k∇θk2 =−mh∇ ˙u, ∇(θ+τq˙ θ)i − (τ∗ ν−k∗τq)k∇θk2−kτθτqk∇ηk2(2.2) with η=˙ θ. It follows from (2.1) and (2.2) that 1 2 dE(t) dt =−(τ∗ ν−k∗τq)k∇θk2−kτθτqk∇ηk2,(2.3) where the “energy” of the system (1.3)-(1.7) is E(t) = ρk˙uk2+µk∆uk2+ckθ+τq˙ θk2+k∗k∇αk2+ 2k∗τqh∇α, ∇θi+ (τ∗ ντq+kτθ)k∇θk2 =ρkvk2+µk∆uk2+ckθ+τqηk2+k∗k∇α+τq∇θk2+ (τ∗ ντq+kτθ−k∗τ2 q)k∇θk2. 2 where v= ˙u. Let H:= H1 0∩H2×L2×H1 0×H1 0×L2. Denoting U= (u, v, α, θ, η)Tand U∗= (u∗, v∗, α∗, θ∗, η∗)T, we can define the inner product hU, U∗iH=µh∆u, ∆u∗i+ρhv, v∗i+chθ+τqη, θ∗+τqη∗i+k∗h∇(α+τqθ),∇(α∗+τqθ∗)i +(τ∗ ντq+kτθ−k∗τ2 q)h∇θ, ∇θ∗i, i.e., kUk2 H=µk∆uk2+ρkvk2+ckθ+τqηk2+k∗k∇α+τq∇θk2+ (τ∗ ντq+kτθ−k∗τ2 q)k∇θk2 =E(t). We can convert the system into a first-order evolution equation on Hilbert space H,    dU dt =AU, (2.4) U(0) = (u0, v0, α0, θ0, η0)T,(2.5) where the operator Ais given by AU=             v −1 ρ∆(µ∆u+m(θ+τqη)) θ η m cτq ∆v+1 cτq ∆(k∗α+τ∗ νθ+kτθη)−1 τq η             (2.6) and D(A) = {U= (u, v, α, θ, η)T∈ H|∆u= 0, on ∂Ω, v ∈H1 0∩H2, ∆(k∗α+τ∗ νθ+kτθη)∈L2, µ∆u+m(θ+τqη)∈L2}.(2.7) Theorem 2.1. Ais the infinitesimal generator of a C0−semigroup of contractions on the Hilbert space H. Proof. By(2.3) and τ∗ ν≥k∗τq, RehAU, UiH=1 2 d dtkUk2 H=−(τ∗ ν−k∗τq)k∇θk2−kτθτqk∇ηk2≤0.(2.8) Thus, Ais dissipative. It is clear that D(A) is dense in H. Let F= (F1, F2, F3, F4, F5)T∈ H. To solve AU=F, we first have v=F1∈H1 0∩H2,θ=F3∈H1 0, η=F4∈H1 0. Then we have ∆(µ∆u+m(θ+τqη)) = −ρF2∈L2,(2.9) −∆(k∗α+τ∗ νθ+kτθη) = −cF4−cτqF5+m∆F1∈L2.(2.10) Define a bilinear form B: ((H1 0∩H2)×H1 0)×((H1 0∩H2)×H1 0)→C ((u, α),(˜u, ˜α)) →µh∆u, ∆˜ui+k∗h∇α, ∇˜αi and a continuous linear function f: (H1 0∩H2)×H1 0→C, 3 (˜u, ˜α)→f(˜u, ˜α) := −ρhF2,˜ui+mh∇(F3+τqF4),∇˜ui−hcF4+cτqF5+m∆F1,˜αi−τ∗ νh∇F3+kτθ∇F4,∇˜αi. Then Bis a strong coercive sesquilinear form. By the Lax-Milgram Theorem, there exists a unique (u, α)∈(H1 0∩H2)×H1 0satisfying B((u, α),(˜u, ˜α)) = f(˜u, ˜α), for ∀(˜u, ˜α)∈(H1 0∩H2)×H1 0,k(u, α)kH2×H1≤CkFkH, where Cis a positive constant. This solves (2.9)-(2.10), i.e., we obtain U= (u, v, α, θ, η)T∈D(A) with AU =F and kUkH≤CkFkH which implies 0 ∈ρ(A). By the modified Lumer-Philips Theorem[22], we conclude that Agenerates a C0−semigroup of contractions on H. Theorem 2.2. The semigroup eAtis analytic and exponentially stable. Remark 2.1. It is worth noting that the analyticity of solutions implies the exponential stability of solutions and the impossibility of localization of solutions. That is the only solution which can be identically null after a finite time is the null solution. We will use the following theorem to prove Theorem 2.2. Theorem 2.3. [22] Let S(t) = eAtbe a C0−semigroup of contractions in a Hilbert space H. Suppose that iR⊂ρ(A).(2.11) Then, S(t)is analytic if and only if lim |β|→∞kβ(iβI −A)−1kH<∞(2.12) holds. Proof. 1. We first check the condition (2.11). Note that, from well-posedness, we have proved that 0∈ρ(A). Then, defining β∗:= sup β > 0 : (−iβ, iβ)⊂ρ(A), Suppose 0 < β∗<∞, then there exists a unit sequence Un= (un, vn, αn, θn, ηn)T∈D(A), such that k(iβnI− A)UnkH→0 when βn%β∗(w.l.o.g.),(2.13) For convenience, we omit the subscript n. We can rewrite (2.13) as                iβu −v=o(1),in H2,(2.14) iβρv + ∆(µ∆u+m(θ+τqη)) = o(1),in L2,(2.15) iβα −θ=o(1),in H1,(2.16) iβθ −η=o(1),in H1,(2.17) iβτqcη −m∆v−∆(k∗α+τ∗ νθ+kτθη) + τqη=o(1),in L2.(2.18) Thus by dissipation, RehAU, UiH=−(τ∗ ν−k∗τq)k∇θk2−kτθτqk∇ηk2=o(1).(2.19) Since τ∗ ν≥k∗τq, from(2.19), we only have k∇ηk2=o(1). By (2.16)-(2.17), we also get k∇θk2=o(1) and k∇αk2=o(1). Therefore, kαk,kθk,kηk=o(1). Taking the inner product of (2.18) with β−1∆uin L2, hiτqcη, ∆ui − mhβ−1∆v, ∆ui − β−1h∆(k∗α+τ∗ νθ+kτθη),∆ui+β−1τqhη, ∆ui=o(1).(2.20) 4 The first and last inner product term in (2.20) is of o(1) due to the boundedness of k∆uk. By (2.14) and (2.15), −mhβ−1∆v, ∆ui − β−1h∆(k∗α+τ∗ νθ+kτθη),∆ui =−imk∆uk2+o(1) −β−1h(k∗α+τ∗ νθ+kτθη),∆2ui =−imk∆uk2−β−1µ−1h(k∗α+τ∗ νθ+kτθη),−iρβv −m(∆θ+τq∆η)i+o(1) =−imk∆uk2−β−1µ−1h(k∗∇α+τ∗ ν∇θ+kτθ∇η),−m(∇θ+τq∇η)i+o(1) =−imk∆uk2+o(1).(2.21) (2.20) now is simplified into k∆uk2=o(1).(2.22) Taking the inner product of (2.15) with vin L2, iβρkvk2+µh∆u, ∆vi − mh∇(θ+τqη),∇vi=o(1).(2.23) By (2.14), (2.22) and dissipation, we can derive kvk2=o(1).(2.24) We can conclude that kUk2 H=o(1), which is a contradiction to the assumption kUk2 H= 1. Therefore, iR⊆ρ(A). 2.We now check the condition (2.12). Assume that (2.12) is false. Then by the uniform boundedness theorem, there exist a sequence β→ ∞ and a unit sequence U= (u, v, α, θ, η)T∈D(A) such that k(iI −1 βA)UkH→0.(2.25) We can write (2.25) as                                  iu −1 βv=o(1),in H2,(2.26) iρv −1 β∆µ∆u+m(θ+τqη)=o(1),in L2,(2.27) iα −1 βθ=o(1),in H1.(2.28) iθ −1 βη=o(1),in H1.(2.29) icτqη−m β∆v−1 β∆(k∗α+τ∗ µθ+kτθη)−τq βη=o(1),in L2.(2.30) From (2.25) and (2.19), Reh(iI −1 βA)U, UiH=−1 βRehAU, UiH=−1 β(τ∗ ν−k∗τq)k∇θk2−1 βkτθτqk∇ηk2=o(1).(2.31) Hence, kβ−1 2∇ηk=o(1), which further leads to k∇αk2=o(1) and k∇θk2=o(1) (2.32) due to (2.28) and (2.29). Taking inner product of (2.30) with ηin L2, we can obtain icτqkηk2+mhβ−1 2∇v, β−1 2∇ηi+hβ−1 2∇(k∗α+τ∗ µθ+kτθη), β−1 2∇ηi − 1 βτqkηk2=o(1).(2.33) 5 By Interpolation Inequality, kβ−1 2∇vk2≤Cβ−1k∆vkL2kvkL2 for some constant C. By (2.26), we have β−1k∆vkis bounded. Thus kβ−1 2∇vkis also bounded. Now (2.33) implies that kηk2=o(1).(2.34) Taking inner product of (2.30) with −∆uin L2, 1 βmh∆v, ∆ui+1 βh∆(k∗α+τ∗ µθ+kτθη),∆ui=o(1).(2.35) Applying (2.26) again, we convert (2.35) into imk∆uk2− hβ−1 2∇(k∗α+τ∗ µθ+kτθη), β−1 2∇(∆u)i=o(1).(2.36) Taking inner product of (2.27) with ∆uin L2, iρhv, ∆ui+µkβ−1 2∇(∆u)k2+mhβ−1 2∇(θ+τqη), β−1 2∇(∆u)i=o(1).(2.37) So we can get kβ−1 2∇(∆u)kis bounded. Then (2.36) implies k∆uk2=o(1).(2.38) Taking inner product of (2.27) with vin L2, iρkvk2−1 βµh∆u, ∆vi+m1 βh∇(θ+τqη),∇vi=o(1).(2.39) By (2.31) and (2.38), we have kvk2=o(1).(2.40) Therefore, we obtain kUk2 H=o(1) again. This is a contradiction with the assumption that kUk2 H= 1. 3 The case of τ∗ ν< k∗τq In the previous section, we have proved the analyticity and the exponential decay of the semigroup associated with the system (2.4)-(2.5). The operator Ais dissipative as the coefficients satisfy τ∗ ν≥k∗τq. In this section, we will consider the case when τ∗ ν< k∗τq. The first thing we need to do is to define a new inner product. We consider hU, U∗i=µh∆u, ∆u∗i+ρhv, v∗i+chθ+τqη, θ∗+τqη∗i+k∗h∇(α+τqθ),∇(α∗+τqθ∗)i +(τ∗ ντq+kτθ−k∗τ2 q)h∇θ, ∇θ∗i+εh∇θ, ∇θ∗i, where εis a sufficiently large positive number to guarantee that this inner product which is equivalent to the usual one in our Hilbert space H. We can even define the inner product in the case that τ∗ ντq+kτθ−k∗τ2 q<0.(3.1) Our first aim in this section is to prove the analyticity of the semigroup even in the case that (3.1) holds. Theorem 3.1. There exists a constant λlarge enough to guarantee that the semigroup generated by Aλ=A − λI is analytic. 6 Proof. We note that RehAλU, Ui=−λk∆uk2−λρkvk2−(τ∗ ν−k∗τq)k∇θk2−kτθτqk∇ηk2+εk∇θk2 −λckθ+τqηk2−λk∗k∇α+τq∇θk2−λ(τ∗ ντq+kτθ−k∗τ2 q)k∇θk2. It is clear that RehAλU, Ui≤−λk∆uk2−λρkvk2−λ(ε+τ∗ ντq+kτθ−k∗τ2 q)−(τ∗ ν−k∗τq)−εε1 2k∇θk2 −λk∗k∇α+τq∇θk2−(kτθτq−ε 2ε1 )k∇θk2−λckθ+τqηk2, where ε1is an arbitrary positive constant. We choose ε1large enough to guarantee that kτθτq−ε 2ε1 >0 and λlarge enough to guarantee that λ(ε+τ∗ ντq+kτθ−k∗τ2 q)−(τ∗ ν−k∗τq)−εε1 2>0. We obtain that RehAλU, Ui ≤ −K(k∆uk2+kvk2+k∇αk2+k∇θk2+k∇ηk2), where Kis a positive constant. Therefore a similar argument to the one used in the proof of Theorem 2.1 allow us to show the analyticity of Aλ. Theorem 3.2. The operator Agenerates a quasi-contractive and analytic semigroup. Proof. As Aλis analytic and λI is a bounded perturbation we obtain the analyticity of A. On the other side, Agenerates a quasi-contractive semigroup since Aλgenerates a semigroup of contraction. In general, we cannot expect the exponential stability of the solutions for every domain, but we can conclude the impossibility of localization. In the remain of this section we want to give examples of domain Ω such that the solutions of the problem (2.4)-(2.5) are unstable. That is, they can be unbounded when the time is increasing. We look for solutions to the system (1.3)-(1.4) of the form u=Aexp(wt)φn(x), α=Bexp(wt)φn(x), where φn(x) satisfies that ∆φn+λnφn= 0, ∆2φn=λ2 nφand φ= ∆φ= 0 on ∂Ω. We have ((ρw2+µλ2 n)A−mw(1 + τqw)λnB= 0, mwλnA+cw2(1 + τqw) + k∗λn+τ∗ νwλn+kτθw2λnB= 0. If we want solutions for A, B are different from zero, we need that (ρw2+µλ2 n)cw2(1 + τqw) + k∗λn+τ∗ νwλn+kτθw2λn+m2w2(1 + τqw)λ2 n= 0. Therefore, we need that wmust be a solution of the equation x5+q1x4+q2x3+q3x2+q4x+q5= 0,(3.2) 7 where                                      q1=cρ +ρkτθλn ρcτq , q2=cµτqλ2 n+m2τqλ2 n+ρτ∗ νλn ρcτq , q3=µkτθλ3 n+cµλ2 n+m2λ2 n+ρk∗λn ρcτq , q4=µτ∗ νλ3 n ρcτq q5=µk∗λ3 n ρcτq . We want to see that there are solutions to (3.2) such that the real parts are positive. It follows from the Routh-Hurwitz rule that the solutions have negative real part if and only if the main minors of the matrix     q1100 q3q2q11 q5q4q3q2 0 0 q5q4     are positive. Let us consider q1q2−q3. We have (1 τq +kτθλn cτq )(cµτqλ2 n+ρτ∗ νλn+m2τqλ2 n ρcτq )−cµλ2 n+ρk∗λn+µkτθλ3 n+m2λ2 n ρcτq =1 cτ2 q (τ∗ ν−k∗τq)λn+kτθτ∗ ν c2τ2 q λ2 n+kτθm2 ρc2τq λ3 n.(3.3) It is clear that there exist domain Ω such that the eigenvalues λ1, λ2,· · · are so small to guarantee that (3.3) is negative since τ∗ ν−k∗τq<0. Instability of solutions follows from this. A similar result could be obtained with the other minors. Results of this kind have been proved previously in the context of the dual-phase-lag thermoelasticity[12] and the Moore-Gibson-Thompson thermoelasticity[15]. Acknowledgements. The work of R. Quintanilla was supported by projects “An´alisis Matem´atico de Problemas de la T ermomec´anica” (MTM2016-74934-P) (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness, and “An´alisis Matem´atico Aplicado a la T ermomec´anica” (Ref. PID2019- 105118GB-I00), (AEI/FEDER, UE) of the Spanish Ministry of Science, Innovation and Universities. References [1] C. Cattaneo, On a form of heat equation which eliminates the paradox of instantaneous propagation, C. R. Acad. Sci. Paris, 247 (1958), 431–433. [2] H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15 (1967), 299–309. [3] R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-470. [4] R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, International J. Solids Structures, 37 (2000), 215-224. 8