On the Long-Time Behavior of the Thermoelastic Plates with Second Sound

The relation between the thermoelastic Cattaneo model and the thermoelastic Gurtin-Pipkin model is established. The existence of the compact global attractor of the Cattaneo-Mindlin plate model is proved and its properties are studied. Установлена зависимость между моделями термоупругости Каттанео и...

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Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2013
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Zitieren:On the Long-Time Behavior of the Thermoelastic Plates with Second Sound / T.B. Fastovska // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 191-206. — Бібліогр.: 14 назв. — англ.

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citation_txt On the Long-Time Behavior of the Thermoelastic Plates with Second Sound / T.B. Fastovska // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 191-206. — Бібліогр.: 14 назв. — англ.
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description The relation between the thermoelastic Cattaneo model and the thermoelastic Gurtin-Pipkin model is established. The existence of the compact global attractor of the Cattaneo-Mindlin plate model is proved and its properties are studied. Установлена зависимость между моделями термоупругости Каттанео и Гертина-Пипкина. Доказано существование компактного глобального аттрактора модели термоупругости Каттанео-Миндлина и изучены его свойства.
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fulltext Journal of Mathematical Physics, Analysis, Geometry 2013, vol. 9, No. 2, pp. 191–206 On the Long-Time Behavior of the Thermoelastic Plates with Second Sound T.B. Fastovska Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61077, Ukraine E-mail: Tamara.B.Fastovska@univer.kharkov.ua Received December 27, 2011, revised March 27, 2012 The relation between the thermoelastic Cattaneo model and the ther- moelastic Gurtin–Pipkin model is established. The existence of the com- pact global attractor of the Cattaneo–Mindlin plate model is proved and its properties are studied. Key words: second sound, asymptotic behavior, global attractor. Mathematics Subject Classification 2010: 35B40, 45K05, 74D05. 1. Introduction and Functional Settings In this paper we consider the nonlinear Mindlin–Timoshenko model of a ther- moelastic plate with heat conduction of Cattaneo type (for review see [1]). The Mindlin–Timoshenko model describes dynamics of a plate in view of transverse shear effects (see, e.g., [2, 3] and references therein). Unlike the classical Fourier constitutive law, the Cattaneo model describes heat conduction processes under the assumption of finite speed propagation of disturbances. We assume that the plate has a uniform thickness h and, when in equilib- rium, its middle surface lies in the bounded domain Ω ⊂ (x1, x2, 0) with the sufficiently smooth boundary ∂Ω. The integro-differential equations for the vec- tor of angles of deflection of the filament v(x, t) = (v1(x, t), v2(x, t)) ∈ R2, the transverse displacement of the middle surface w(x, t) ∈ R, the temperature varia- tion θ(x, t) and the heat flax q(x, t) averaged with respect to the thickness, where x = (x1, x2) ∈ Ω and t ≥ 0, are the following: α0vtt + β0vt −Av + µ(v +∇w) + β∇θ +∇vΦ(v) = 0, α1wtt + β1wt − µdiv(v +∇w) + g(w) = 0, γθt + κdivq + βdivvt = 0, ωqt + q +∇θ = 0. (1) c© T.B. Fastovska, 2013 T.B. Fastovska Here the vector function ∇vΦ(v) = (∂v1Φ(v1, v2), ∂v2Φ(v1, v2)) and the scalar function g(w) are the feedback forcing terms. The parameters α0, α1, β0, β1, β, γ, κ, µ, ω are positive constants. The operator A has the structure A =   ∂2 x1 + 1−ν 2 ∂2 x2 1+ν 2 ∂x1x2 1+ν 2 ∂x1x2 1−ν 2 ∂2 x1 + ∂2 x2   = ∇div− 1− ν 2 rotrot, where 0 < ν < 1 is the viscoelastic Poisson’s ratio. The Timoshenko systems have been treated by many authors. There are several works investigating the presence or the lack of the exponential stability of linear and nonlinear problems with various types of damping and boundary conditions for the Timoshenko ( see, e.g., [4, 5]) and Timoshenko–Cattaneo prob- lems [6, 7]. The existence and the properties of attractors for the related systems were established in [8–10]. In [8], the existence of a compact global attractor for the Mindlin–Timoshenko elasticity and its upper semicontinuity, as the shear modulus tends to infinity, are shown. Paper [10] is devoted to the existence of a compact global attractor and its properties of the Mindlin–Timoshenko vis- coelastic system of memory type coupled with Gurtin–Pipkin heat conduction equations (see [11] for the model description). The long-time behavior of the Mindlin–Timoshenko problem α0vtt + β0vt −Av + µ(v +∇w) + β∇θ +∇vΦ(v) = 0, α1wtt + β1wt − µdiv(v +∇w) + g(w) = 0, γθt − 1 ω2 ∞∫ 0 η( s ω )∆τ(s)ds + βdivvt = 0, θ = τt + τs, s ≥ 0 (2) for the model with Gurtin–Pipkin heat conduction with Dirichlet boundary con- ditions was studied in [9]. The main goal of the paper is to study the long-time behavior of the semilin- ear thermoelastic Mindlin–Timoshenko–Cattaneo system with locally Lipschitz nonlinearities of any polynomial growth (of odd degrees) and to establish the closeness of the family of attractors to the attractor of the Fourier thermoelastic model in a suitable sense in limit case ω → 0. In the present paper, we establish the relation between the dynamics of problems (1) and (2). It is shown that (1) can be decomposed into two systems. The energy of the first system decays exponentially to zero. The dynamics of the second one is connected with the dynamics of system (2) according to the law given in Lemma 2. Additionally, we describe the relation between the structures of the attractors of systems (1) and (2). We also establish the upper-semicontinuity of the family of attractors of (1) with respect to the relaxation time. 192 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 On the Long-Time Behavior of the Thermoelastic Plates with Second Sound Rewrite the thermoelastic model with second sound (1) in the following way: Putt + Mut + Au + Rθ = F (u), γθt + κdivq + βdivvt = 0, ωqt + q +∇θ = 0, (3) and subject it to the initial and boundary conditions u(x, 0) = u0(x) ∈ [H1 0 (Ω)]3, ut(x, 0) = u1(x) ∈ [L2(Ω)]3, x ∈ Ω, θ(x, 0) = θ0(x) ∈ L2(Ω), q(x, 0) = q0(x) ∈ [L2(Ω)]2, x ∈ Ω, θ(x, t) = 0, u(x, t) = 0, x ∈ ∂Ω, t ≥ 0. (4) The operator A with the domain D(A) = {u = (v1, v2, w) ∈ [(H2 ∩H1 0 )(Ω)]3} has the structure A = ( −A + µI µ∇ −µdiv −µ∆ ) . Obviously, A is a positive self-adjoint operator with the square root possessing the domain D(A1/2) = [H1 0 (Ω)]3. It is easy to see that the operators {R : H1 0 (Ω) → [L2(Ω)]3, Rθ = β(∂1θ, ∂2θ, 0)} and {Q : [H1 0 (Ω)]3 → L2(Ω), Qu = −β(∂1u1 + ∂2u2), u = (u1, u2, u3)} possess the property (Rθ, u) = (θ, Qu) for any θ ∈ H1 0 (Ω) and u ∈ [H1 0 (Ω)]3. The bounded in [L2(Ω)]3 operators P and M are defined by the formulas P = ( α0I 0 0 α1I ) , M = ( β0I 0 0 β1I ) . The nonlinear term has the structure F (u) =   −∂v1Φ(v1, v2) −∂v2Φ(v1, v2) −g(w)   , u = (v1, v2, w). (5) We assume that the memory kernel η(s) possesses the properties η ∈ C1(R+) ∩ L1(R+), (6) moreover, η(s) ≥ 0, (7) Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 193 T.B. Fastovska and for any s ∈ R+ there exists l > 0 such that for any s ∈ R+ η′(s) + lη(s) ≤ 0. (8) We will use the notation ηω(s) = 1 ω2 η( s ω ). (9) Introduce a Hilbert space L2 ω(R+,H l(Ω)) of H l-valued functions on R+ (l ∈ R) such that ‖τ‖2 L2 ω(R+,Hl(Ω)) ≡ ω ∞∫ 0 ηω(s)‖τ(s)‖2 Hl(Ω)ds < ∞, which denotes a norm in this space. We endow L2 ω(R+,H l(Ω)) with the inner product (φ1, φ2)L2 ω(R+,Hl(Ω)) = ω ∞∫ 0 ηω(s)(φ1(s), φ2(s))Hl(Ω)ds. We will also need the space H1 ω(R+,H1 0 (Ω)) = {φ : φ(s), φs(s) ∈ L2 ω(R+, H1 0 (Ω))}. Define the operator Tω : D(Tω) → L2(Ω) with the domain D(Tω) = {τ ∈ L2 ω(R+, H1 0 (Ω)) : ∞∫ 0 ηω(s)∆τds ∈ L2(Ω), τ |s=0 = 0} by the formula Tωτ = − ∞∫ 0 ηω(s)∆τds. We assume that the nonlinearities of the problem satisfy the conditions g ∈ C1(R), Φ ∈ C2(R2), (10) and there exist q > 0 and C̃ > 0 such that |g′(z)| ≤ C̃(1 + |z|q), |∂2 1Φ(z)|+ |∂2 2Φ(z)|+ |∂1∂2Φ(z)| ≤ C̃(1 + |z|q). (11) Moreover, there exist bi ∈ R, i = 1, 2 such that Φ(z1, z2) ≥ −b1, G(z) ≡ z∫ 0 g(ζ)dζ ≥ −b2. (12) 194 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 On the Long-Time Behavior of the Thermoelastic Plates with Second Sound We also assume that there exist ai > 0, i = 1, 4 such that −a1Φ(z) +∇zΦ(z) z ≥ −a2, −a3G(z) + g(z) z ≥ −a4. (13) In the paper we will establish the relation between problem (3)–(4) and the Gurtin–Pipkin thermoelastic model considered in [9]: Putt + Mut + Au + Rθ = F (u), (14) γθt + Tωτ −Qut = 0, (15) θ = τt + τs, s ≥ 0 (16) u|t=0 = u0, ut|t=0 = u1, θ|t=0 = θ0, τ(t, s)|t=0 = τ0(s), s ≥ 0. (17) Introduce the space Xω = D(A1/2)× [L2(Ω)]3 × L2(Ω)× [L2(Ω)]2 with the inner product     u1 v1 θ1 q1  ,   u2 v2 θ2 q2     Xω =(A1/2u1, A 1/2u2)+(P 1/2v1, P 1/2v2)+γ(θ1, θ2)+ωκ(q1,q2) and the space D̃ = {φ ∈ [L2(Ω)]2 : divφ ∈ L2(Ω)} with the inner product (q1,q2)D̃ = (divq1, divq2) + (q1,q2). We define the operator Bω : Xω ⊃ D(Bω) → Xω with the domain D(Bω) = {(u, u, θ,q) : u ∈ D(A), u ∈ D(A1/2), θ ∈ H1 0 (Ω),q ∈ D̃} by the formula Bω =   0 I 0 0 −P−1A −P−1M −P−1R 0 0 1 γ Q 0 −κ γ div 0 0 − 1 ω∇ − 1 ω   . We will use the notations Z(t) = (u(t), u(t), θ(t),q(t)) and Z0 = (u0, u1, θ0,q0) ∈ Xω, where u(t) = ut(t). Then the problem (3)–(4) can be rewritten as follows: d dtZ(t)− BωZ(t) = f(Z(t)), Z(0) = Z0. (18) Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 195 T.B. Fastovska Introduce the space Hω = D(A1/2)× [L2(Ω)]3 × L2(Ω)× L2 ω(R+,H1 0 (Ω)) with the inner product     u1 v1 θ1 τ1   ,   u2 v2 θ2 τ2     Hω = (A1/2u1, A 1/2u2) + (v1, v2) + γ(θ1, θ2) + 1 ω 〈∇τ1,∇τ2〉. Denote Aη ω : D(Aη ω) → Hω to be Aη ω =   0 I 0 0 −A −D −R 0 0 1 γ Q 0 1 γ Tω 0 0 I −∂s   with the domain D(Aη ω) = D(A)×D(A1/2)×H1 0 (Ω)× [D(Tω) ∩H1 ω(R+,H1 0 (Ω))]. For problem (14)–(17), we use the result obtained in [9]: Proposition 1. The operator Aη ω is the generator of the C0-semigroup Uη ω(t) on the space Hω. Relying on this result we give the definition of the mild solution to (14)–(17). Definition 1. The function Z = (u, ut, θ, τ) ∈ C(0, T ; Hω) is a mild solution to (14)–(17) on the interval [0, T ] subjected to the initial conditions Z(0) = Z0 = (u0, u1, θ0, τ0) if the relation Sω(t)Z0 = Z(t) = Uη ω(t)Z0 + t∫ 0 Uη ω(t− s)f(Z(s))ds holds true for any t ∈ [0, T ]. The long-time behavior of the dynamical system generated by problem (14)– (17), i.e., the existence and the properties of the compact global attractor, was studied in paper [9]. By definition (see, e.g., [12, 13]), a global attractor is a bounded closed set Aω ⊂ Hω such that Sω(t)Aω = Aω for all t ≥ 0, and lim t→+∞ sup y∈B dist(S(t)y, Aω) = 0 for any bounded set B ⊂ Hω . 196 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 On the Long-Time Behavior of the Thermoelastic Plates with Second Sound We recall here the theorem proven in [9]. Theorem 1. Let the assumptions (6)–(8) and (10)–(13) hold. Then for any ω > 0 the dynamical system (Sω(t),Hω) generated by (14)–(17) possesses a com- pact global attractor Aω whose fractal dimension is finite. The family of attractors {Aω} is upper semicontinuous at zero, i.e., sup y∈Aω distHω(y, A0) → 0, ω → 0, where A0 =    y =   u0 u1 θ0 0   :   u0 u1 θ0   ∈ A    , where A is the compact global attractor of the dynamical system generated by the problem Putt + Mut + Au + Rθ = F (u), γθt − κ∆θ −Qut = 0, u|t=0 = u0, ut|t=0 = u1, θ|t=0 = θ0 (19) in the space [H1 0 (Ω)]3 × [L2(Ω)]2 × L2(Ω). 2. Long-Time Behavior To study the dynamics of Mindlin–Timoshenko–Cattaneo system (3)–(4), we consider the problems Putt + Mut + Au + Rθ = F (u), γθt + κ∆q + βdivvt = 0, x ∈ Ω, t > 0, ωqt + q + θ = 0, u(x, 0) = u0(x) ∈ [H1 0 (Ω)]3, ut(x, 0) = u1(x) ∈ [L2(Ω)]3, θ(x, 0) = θ0(x) ∈ L2(Ω), q(x, 0) = q0(x) ∈ H1 0 (Ω) (20) and ωpt + p = 0, p(0) = p0 ∈ [L2(Ω)]2. (21) Define the space Vω = D(A1/2)× [L2(Ω)]3 × L2(Ω)×H1 0 (Ω) with the norm ‖(u, v, θ, q)‖2 Vω = ‖A1/2u‖2 + ‖P 1/2v‖2 + γ‖θ‖2 + ωκ‖∇q‖2. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 197 T.B. Fastovska Obviously problem (21) generates an exponentially stable dynamical system in the space [L2(Ω)]2, moreover, p = p0e − t ω tends to zero as ω → 0. Problem (20) can be rewritten as (14)–(17) with the kernel η(s) = κe−s if we prolong θ in (20) backward in time, for instance, in the following way: θ(x,−t) = −q0(x), t > 0. It follows from the last equation in (20) that q(x, t) = e− t−ξ ω q0(x)− 1 ω t∫ ξ e− (t−s) ω θ(x, s)ds, ξ ≤ t, t ≥ 0. Consequently, letting ξ tend to −∞ and changing variables, we get q(x, t) = − 1 ω2 ∞∫ 0 e− s ω τ(x, t, s)ds. (22) Substituting (22) into (20) and defining τ0 = −sq0(x) ∈ L2 ω(R+,H1 0 (Ω)), we arrive at problem (14)–(17). It is easy to see that the kernel η(s) = κe−s satisfies conditions (6)–(8), and η0 = η1 = κ. Consequently, Theorem 1 is valid for the obtained problem of the kind (14)–(17). To establish the results analogous to those stated in Theorem 1 for problem (3)–(4), we have to study the relation between problems (14)–(17) and (20). At first we will show that the existence of the dynamical system (Sω(t),Hω) generated by problem (14)–(17) with the kernel η(s) = κe−s leads to the existence of the C0-semigroup Σω(t) on the space Vω generated by the mild solutions of problem (20). To give the definition of the solutions, we introduce the operatorsK : H1 0 (Ω) → L2 ω(R+,H1 0 (Ω)) and N : L2 ω(R+,H1 0 (Ω)) → H1 0 (Ω) acting by the formulas Kq = −sq and N τ = − 1 ω2 ∞∫ 0 e− s ω τds. The operators K : Vω → Hω and N : Hω → Vω are defined by K =   I 0 0 0 0 I 0 0 0 0 I 0 0 0 0 K   , N =   I 0 0 0 0 I 0 0 0 0 I 0 0 0 0 N   . 198 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 On the Long-Time Behavior of the Thermoelastic Plates with Second Sound Let the operator Dω : Vω ⊃ D(Dω) → Vω with the domain D(Dω) = {(u, u, θ, q) : u ∈ D(A), u ∈ D(A1/2), θ ∈ H1 0 (Ω),q ∈ H2 ∩H1 0 (Ω)} have the structure Dω =   0 I 0 0 −P−1A −P−1M −P−1R 0 0 1 γ Q 0 −κ γ ∆ 0 0 − κ ω − 1 ω   . The operators N and K possess the following properties. Proposition 2. For any Z ∈ D(Dω), NAωKZ = DωZ, (23) where Aω is the generator of the semigroup generated by problem (14)–(17) with the kernel η(s) = κe−s. Moreover, NK = I ∈ L(Vω). (24) P r o o f. It is obvious that if Z ∈ D(Dω), then KZ ∈ D(Aω). Therefore the statement of the lemma can be easily proved by the straightforward calculations. Let Z(t) = (u(t), u(t), θ(t), q(t)) and Z0 = (u0, u1, θ0, q0) ∈ Vω, where u(t) = ut(t). Then problem (20) can be rewritten as follows: d dtZ(t)− DωZ(t) = f(Z(t)), Z(0) = Z0. (25) Lemma 1. The operator Dω is the generator of the C0-semigroup Wω(t) = NUω(t)K on the space Vω, where Uω(t) is the exponentially stable C0-semigroup on the space Hω generated by the operator Aω. P r o o f. Consider the operator Wω(t) = NUω(t)K. To prove that Wω(t) is a C0-semigroup on the space Vω, we have to check the semigroup properties. It follows from (24) that Wω(0) = NUω(0)K = NK = I. Now we prove that Wω(r)Wω(t) = NUω(r)KNUω(t)K = NUω(t + r)K = Wω(t + r). (26) Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 199 T.B. Fastovska Define the operators Ui(t), i = 1, 4 as follows: U1(t)Z0 = u(t), U2(t)Z0 = ut(t), U3(t)Z0 = θ(t), U4(t)Z0 = τ(t), where (u(t), ut(t), θ(t), τ(t)) is a mild solution to problem (14)–(17) with initial conditions Z0. Note that Uω(t) = (U1(t), U2(t), U3(t), U4(t)). Point out that Wω(r)Wω(t)Z0 = NUω(r)(U1(t)Z0, U2(t)Z0, U3(t)Z0,KNU4(t)KZ0) = (U1(t + r)Z0, U2(t + r)Z0, U3(t + r)Z0,N r∫ r−s θ(t + ξ)dξ). In order to derive (26), we must consider more precisely the fourth component N r∫ r−s θ(t + ξ)dξ = N r∫ t+r−s θ(ξ)dξ =    N [ r+t∫ 0 U3(ξ)KZ0dξ + (t + r − s)Kq0], t + r − s < 0 N r+t∫ 0 U3(ξ)Z0dξ, t + r − s ≥ 0 = NU4(t + r)KZ0. Consequently, (26) holds true. It is easy to see that N ∈ L(Hω, Vω) and K ∈ L(Vω,Hω). This entails that for any Z ∈ Hω, lim t→0 ‖Wω(t)Z − Z‖Vω = lim t→0 ‖NUω(t)KZ −NKZ‖Vω ≤ lim t→0 ‖N‖L(Hω ,Vω)‖Uω(t)KZ − KZ‖Hω = 0. Thus, the semigroup Wω(t) is strongly continuous. Now we will describe the generator of this semigroup. It follows from property (23) that for any Z ∈ D(Dω), lim t→0 Wω(t)Z − Z t = N lim t→0 Uω(t)KZ − KZ t = NAωKZ = DωZ. The lemma is proved. Now we establish the well-posedness of problem (25). Lemma 2. Assume that conditions (10)–(12) hold true. Then problem (25) generates the dynamical system (Σω(t), Vω). Its evolution operator has the form Σω(t) = NSω(t)K, where Sω(t) is the evolution operator of problem (14)–(17) with η(s) = κe−s. 200 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 On the Long-Time Behavior of the Thermoelastic Plates with Second Sound P r o o f. For any Z0 = (u0, u1, θ0, q0) ∈ Vω, Z(t) = Σω(t)Z0 = NSω(t)KZ0 = NUω(t)KZ0 + t∫ 0 NUω(t− ξ)Kf(Z(ξ))dξ = Wω(t)Z0 + t∫ 0 Wω(t− ξ)f(Z(ξ))dξ, where t ∈ [0, T ] for any 0 < T < ∞. Thus, the function Z(t) = Σω(t)Z0 ∈ C(0, T ; Vω) is a mild solution to (25). The lemma is proved. Now we are ready to state the existence result for the attractor of system (20). Lemma 3. Let conditions (10)–(13) hold true. Then for any ω > 0 the dy- namical system (Σω(t), Vω), generated by (25), possesses a compact global attractor Qω. P r o o f. It is easy to see that ‖Σω(t)(u0, u1, θ0, q0)‖Vω ≤ ‖N‖L(Hω ,Vω)‖Sω(t)(u0, u1, θ0, τ0)‖Hω . Consequently, the dynamical system (Σω(t),Vω) possesses an absorbing ball. Consider the set K = { ⋃ Y0∈Aω ⋃ t∈R NSω(t)Y0} = N{ ⋃ Y0∈Aω ⋃ t∈R Sω(t)Y0}. (27) Since the operator N is bounded and the set { ⋃ Y0∈Aω ⋃ t∈R Sω(t)Y0} is compact in Hω, set (27) is compact in Vω. Let B ∈ Vω be a positively invariant set, i.e., Σω(t)B ⊂ B. Then lim t→+∞ sup z∈B distVω(Σω(t)z,K) ≤ lim t→+∞ sup y∈G distVω [NSω(t)y, Nỹ], where G = {y = Kz : z ∈ B} ⊂ Hω is a bounded set in Hω, ỹ is the element on which the minimal distance in Hω from the point Sω(t)y to the set Aω is reached. Then, obviously, lim t→+∞ sup y∈B distVω(Σω(t)y, K) ≤ C lim t→+∞ sup z∈G distHω(Sω(t)y, Aω) = 0. The above estimate implies that the dynamical system (Σω(t), Vω) is asympto- tically compact. This property together with the existence of an absorbing ball is a necessary condition for the existence of a compact global attractor Qω (see, e.g., [12, 13]). Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 201 T.B. Fastovska Now we study the relation between problems (20), (21) and initial statement (3). Let us introduce the operator F : Vω → Xω F =   I 0 0 0 0 I 0 0 0 0 I 0 0 0 0 ∇   . It is well known (see [14]) that [L2(Ω)]2 = E1 ⊕ E2, (28) where E1 = {∇φ : φ ∈ H1 0 (Ω)} and E2 = {ψ ∈ [L2(Ω)]2 : (divψ, φ) = 0, φ ∈ H1 0 (Ω)}. By virtue of (28), the initial condition, the problem (3) is subjected to, can be split as q0 = ∇q0 + p0, where q0 ∈ H1 0 (Ω) and p0 ∈ E2. Therefore, we can define the operator P1 : [L2(Ω)]2 → H1 0 (Ω) by the formula P1q0 = −(−∆)−1divq0 = q0, where −∆ is the Laplace operator with the Dirichlet boundary conditions and the projector P2 : [L2(Ω)] → E2 as P2q0 = p0. Introduce the operators P1 : Xω → Vω and P2 : Xω → Xω defined by the formulas P1 =   I 0 0 0 0 I 0 0 0 0 I 0 0 0 0 P1   , P2 =   I 0 0 0 0 I 0 0 0 0 I 0 0 0 0 P2   . The following properties of these operators can be easily checked by the straight- forward calculations. Proposition 3. For any Z ∈ D(Bω), [FNAωKP1 + 1 ω P2]Z = BωZ. (29) Moreover, FP1 + P2 = I ∈ L(Xω), (30) and for any Z ∈ Vω, P1FZ = Z. (31) Define the operator Sω(t)(u0, u1, θ0,q0) = (0, 0, 0, e− t ω p0) + FΣω(t)(u0, u1, θ0, q0) = [e− t ω P2 + FΣω(t)P1](u0, u1, θ0,q0) (32) in the space Xω. We are in position to show that Sω(t) is the evolution operator of problem (3)–(4). 202 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 On the Long-Time Behavior of the Thermoelastic Plates with Second Sound Lemma 4. The operator Bω is the generator of the C0-semigroup Ũω(t) = FNUω(t)KP1 + e− t ω P2 on the phase space Xω. P r o o f. First, we prove that Ũω(t) is the generator of the C0-semigroup on the space Vω. To this end, we check the semigroup properties. We obtain by (30) and (31) Ũω(0) = FWω(0)P1 + P2 = FP1 + P2 = I and Ũω(r)Ũω(t) = FWω(r)P1FWω(t)P1 + e− r+t ω P2 = FWω(r + t)P1 + e− r+t ω P2 = Ũω(r + t). It is easy to see that F ∈ L(Xω, Vω). Consequently, for any Z ∈ Xω, lim t→0 ‖Ũω(t)Z − Z‖Xω = lim t→0 [‖F‖L(Xω ,Vω)‖Wω(t)P1Z −P1Z‖Vω + ‖e− t ω P2Z −P2Z‖Xω ] = 0. Therefore the semigroup Ũω(t) is strongly continuous. Now we describe the generator of the semigroup. It follows from (29) that for any Z ∈ D(Bω), lim t→0 Ũω(t)Z − Z t = F lim t→0 Wω(t)− I t P1Z + lim t→0 e− t ω − I t P2Z = [FNAωKP1 + 1 ω P2]Z = BωZ. The lemma is proved. Now we are in position to prove the well-posedness result for (18). Lemma 5. Assume that conditions (10)–(12) hold true. Then (18) generates the nonlinear dynamical system (Sω(t), Xω) with the evolution operator defined in (32). P r o o f. For any Z0 = (u0, u1, θ0,q0) ∈ Xω, Z(t) = Sω(t)Z0 = FΣω(t)P1Z0 + e− t ω P2Z0 = Ũω(t)Z0 + t∫ 0 Ũω(t− ξ)f(Z(ξ))dξ, where t ∈ [0, T ] for any 0 < T < ∞. Therefore the function Z(t) = Sω(t)Z0 ∈ C(0, T ; Xω) is a mild solution to problem (18). The lemma is proved. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 203 T.B. Fastovska It follows from the above arguments that (Sω(t), Xω) is a dynamical system possessing the compact global attractor Uω = {FQω}. By the definition, the fractal dimension of a set is the value dimfA = lim ε→0 ln n(A, ε) ln 1/ε , where n(A, ε) is the minimal number of closed balls with radius ε covering the compact set A. Obviously, dimf Qω = dimf Uω. (33) Thus, to show the finite dimensionality of the attractor of the dynamical system (Sω(t), Xω) it remains to prove the lemma bellow. Lemma 6. Let (10)–(12) hold. Then the attractor Qω of the dynamical system (Σω(t),Vω) generated by problem (25) has the finite fractal dimension. P r o o f. Consider the restriction of N onto the attractor Aω, P = N|Aω . Let us show that PAω ⊂ Qω. If Y ∈ Aω, then there exists a whole trajectory Y (t) = (u(t), ut(t), θ(t), τ(t)) ∈ Aω passing through this point. Then PY (t) is a bounded whole trajectory passing through the point X = PY , i.e., X belongs to the attractor Qω. On the contrary, let Z = (u, v, θ, q) ∈ Qω, then Y = KZ ∈ Aω PZ = Y , i.e., Qω ⊂ PAω. Therefore, PAω = Qω, and the mapping P is continuous. Consequently, dimf Qω ≤ dimf Aω < ∞, and the lemma is proved. Now we will establish the upper semicontinuity of the family of attractors {Uω} with respect to the parameter ω and show that problem (3)–(4) is a singular perturbation of the classical thermoelastic Mindlin–Timoshenko problem (19). Theorem 2. Let the assumptions (6)–(8), (10)–(13) hold. Then problem (3)– (4) generates the dynamical system (Sω(t), Xω), where the operator Sω(t) is de- fined by formula (32). For any ω > 0 the dynamical system (Sω(t), Xω) possesses a compact global finite dimensional attractor Uω. The family of attractors {Uω} is upper semicontinuous at zero , i.e., sup y∈Uω distXω(y, B0) → 0, ω → 0, 204 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 On the Long-Time Behavior of the Thermoelastic Plates with Second Sound where B0 =    y =   u0 u1 θ0 −∇θ0   :   u0 u1 θ0   ∈ A    . Here A is a compact global attractor of the dynamical system generated by problem (19) on the space [H1 0 (Ω)]3 × [L2(Ω)]2 × L2(Ω). P r o o f. The existence of the dynamical system and the compact global attractor was proved in Lemmas 1–5. The finite dimensionality of the attractor follows from Lemma 6 and (33). Since ‖ 1 ω ∇ ∞∫ 0 e− s ω (τ − θ)ds‖ ≤ ω ∣∣∇τt ∣∣ → 0 ω → 0, it follows from Theorem 1 that lim ω→0 sup y∈Uω distXω(y, B0) = lim ω→0 sup z∈Aω distVω(Nz,NKP1B0) = 0. The theorem is proved. References [1] D.S. Chandrasekharaiah, Hyperbolic Thermoelasticity: A Review of Recent Litera- ture. — Appl. Mech. Rev. 51 (1998), 705–729. [2] J. Lagnese, Boundary Stabilization of Thin Plates. Philadelphia: SIAM, 1989. [3] P. Schiavone and R.J. Tait, Thermal Effects in Mindlin-Type Plates. — Q. Jl. Mech. Appl. Math. 46 (1993), 27–39. [4] J.E. Muñoz Rivera and R. Racke, Global Stability for Damped Timoshenko Systems. — Disc. Cont. Dyn. Sys. 9 (2003), 1625–1639. [5] J.E. Muñoz Rivera and R. Racke, Mildly Dissipative Nonlinear Timoshenko Systems — Global Existence and Exponential Stability. — J. Math. Anal. Appl. 276 (2002), No. 1, 248–278. [6] H.D. Fernández Sare and R. Racke, On the Stability of Damped Timoshenko Sys- tems: Cattaneo Versus Fourier Law. — Arch. Rational Mech. Anal. 194 (2009), 221–251. [7] S.A. Messaoudi, M. Pokojovy, and B. Said-Houari, Nonlinear Damped Timoshenko Systems with Second Sound — Global Existence and Exponential Stability. — Math. Meth. Appl. Sci. 32 (2009), No. 5, 505–534. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 205 T.B. Fastovska [8] I. Chueshov and I. Lasiecka, Global Attractors for Mindlin–Timoshenko Plates and for Their Kirchhoff Limits. — Milan J. Math. 74 (2006), 117–138. [9] T. Fastovska, Upper Semicontinuous Attractor for a 2D Mindlin–Timoshenko Ther- moelastic Model with Memory. — Commun. Pure Appl. Anal. 6 (2007), No. 1, 83–101. [10] T. Fastovska, Upper Semicontinuous Attractor for a 2D Mindlin–Timoshenko Ther- mo-Viscoelastic Model with Memory. — Nonlinear Analysis TMA 71 (2009), No. 10, 4833–4851. [11] M.E. Gurtin and A.C. Pipkin, A General Theory of Heat Conduction with Finite Wave Speeds. — Arch. Rational Mech. Anal. 31 (1968), 113–126. [12] I.D. Chueshov, Introduction to the theory of infinite-dimensional dissipative sys- tems. Acta, Kharkov, 1999. (Russian). (Engl. transl.: Acta, Kharkov, 2002). [13] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988). [14] S. Jiang and R. Racke, Evolution Equations in Thermoelasticity. π Monographs Surveys Pure Appl. Math. 112, Chapman&Hall/CRC, Boca Raton, 2000. 206 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
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institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1812-9471
language English
last_indexed 2025-12-07T17:17:57Z
publishDate 2013
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
record_format dspace
spelling Fastovska, T.B.
2016-10-04T17:16:44Z
2016-10-04T17:16:44Z
2013
On the Long-Time Behavior of the Thermoelastic Plates with Second Sound / T.B. Fastovska // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 191-206. — Бібліогр.: 14 назв. — англ.
1812-9471
https://nasplib.isofts.kiev.ua/handle/123456789/106745
The relation between the thermoelastic Cattaneo model and the thermoelastic Gurtin-Pipkin model is established. The existence of the compact global attractor of the Cattaneo-Mindlin plate model is proved and its properties are studied.
Установлена зависимость между моделями термоупругости Каттанео и Гертина-Пипкина. Доказано существование компактного глобального аттрактора модели термоупругости Каттанео-Миндлина и изучены его свойства.
en
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
Журнал математической физики, анализа, геометрии
On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
Article
published earlier
spellingShingle On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
Fastovska, T.B.
title On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
title_full On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
title_fullStr On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
title_full_unstemmed On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
title_short On the Long-Time Behavior of the Thermoelastic Plates with Second Sound
title_sort on the long-time behavior of the thermoelastic plates with second sound
url https://nasplib.isofts.kiev.ua/handle/123456789/106745
work_keys_str_mv AT fastovskatb onthelongtimebehaviorofthethermoelasticplateswithsecondsound