Qualitative properties of solutions in the time differential dual-phase-lag model of heat conduction

Accepted Manuscript

Qualitative properties of solutions in the time differential dual-phase-lag model of heat conduction Stan Chirit¸a, ˘ Michele Ciarletta, Vincenzo Tibullo PII: DOI: Reference:

S0307-904X(17)30340-2 10.1016/j.apm.2017.05.023 APM 11775

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Applied Mathematical Modelling

Received date: Revised date: Accepted date:

29 May 2016 28 April 2017 11 May 2017

Please cite this article as: Stan Chirit¸a, ˘ Michele Ciarletta, Vincenzo Tibullo, Qualitative properties of solutions in the time differential dual-phase-lag model of heat conduction, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.05.023

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Highlights • Rich analysis of the time-differential dual-phase-lag model of heat conduction • Uniqueness with delay times τT > 0 and τq ≥ 0. • Appropriate continuous dependence estimates for 0 ≤ τq ≤ 2τT and 0 < 2τT < τq .

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• Theorem of influence domain: specific estimations for the speed of signal propagation.

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• Exponential decay estimates for the amplitude of the steady-state vibrations.

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Qualitative properties of solutions in the time differential dual-phase-lag model of heat conduction Stan Chirit¸˘ a

a,∗

, Michele Ciarlettac , Vincenzo Tibulloc

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a Faculty of Mathematics, Al. I. Cuza University of Ia¸ si, 700506 – Ia¸si, & Octav Mayer Mathematics Institute, Romanian Academy, 700505 – Ia¸si, Romania b Dipartimento di Ingegneria Industriale/DIIN, University of Salerno, 84084 - Fisciano (SA), Italia c Dipartimento di Matematica, University of Salerno, 84084 - Fisciano (SA), Italia

Abstract

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In this paper we study the time differential dual-phase-lag model of heat conduction incorporating the microstructural interaction effect in the fast-transient process of heat transport. We analyse the influence of the delay times upon some qualitative properties of the solutions of the initial boundary value problems associated to such a model. Thus, the uniqueness results are established under the assumption that the conductivity tensor is positive definite and the delay times τq and τT vary in the set {0 ≤ τq ≤ 2τT } ∪ {0 < 2τT < τq }. For the continuous dependence problem we establish two different estimates. The first one is obtained for the delay times with 0 ≤ τq ≤ 2τT , which agrees with the thermodynamic restrictions on the model in concern, and the solutions are stable. The second estimate is established for the delay times with 0 < 2τT < τq and it allows the solutions to have an exponential growth in time. The spatial behavior of the transient solutions and the steadystate vibrations is also addressed. For the transient solutions we establish a theorem of influence domain, under the assumption that the delay times are in {0 < τq ≤ 2τT } ∪ {0 < 2τT < τq }. While for the amplitude of the harmonic vibrations we obtain an exponential decay estimate of SaintVenant type, provided the frequency of vibration is lower than a critical value and without any restrictions upon the delay times.

Introduction

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Key Words: dual-phase-lag heat conduction model, uniqueness, continuous dependence, spatial behavior, transient solutions, steady-state vibrations

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Nowadays it is well-known in literature that the classical Fourier law of heat conduction leads to the so-called paradox of infinite speed of propagation. In fact, in the classical theory of diffusion the heat flux vector and the temperature gradient across a material volume are assumed to occur at the same instant of time. This immediate response leads to an infinite speed of heat propagation, implying that a thermal disturbance applied at a certain location will be felt instantaneously anywhere else in the body. There are many attempts to avoid such a paradox, but major part of all are based upon the introduction, within the context of constitutive equations, of an effect of a short memory related to the temperature and to the heat flux fields. The microstructural interactions in the heat transport require finite period of time to accomplish and these are in the range from picoseconds or femtoseconds (the delayed response for phonon scattering or for phonon-electron interactions) to seconds or even longer (such as the delayed response induced by the low-conducting pores in sand media)(Cf. Tzou [1]). In this connection Tzou ∗ Corresponding

author Email addresses: [email protected] (Stan Chirit¸˘ a ), [email protected] (Michele Ciarletta ), [email protected] (Vincenzo Tibullo )

Preprint submitted to Applied Mathematical Modelling

May 23, 2017

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[1], [2] introduces the dual-phase-lag model, allowing either the temperature gradient to precede the heat flux vector or the heat flux vector to precede the temperature gradient in the transient process. More precisely, he replaces the Fourier law of heat conduction with the following general constitutive equation qi (x, t + τq ) = −kij (x)T,j (x, t + τT ), (1)

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or, in particular, with a refined structure of the lagging response illustrated through expansions in terms of Taylor series with respect to the time. Here, τq ≥ 0 and τT ≥ 0 are the delay times, T,i represents the gradient of the temperature variation and kij is the conductivity tensor. As will be seen in a next paragraph, when a formal Taylor series is used, an appropriate approximation constitutive equation of (1) is assumed which, as combined with the energy equation, leads to a hyperbolic equation in terms of the temperature field and so the paradox of infinite speed is avoided. We want to draw attention on the fact that the lagging behavior as described by constitutive equations of type introduced by Tzou [1, 3, 4] is in well accord with the novel experiments for ultrafast pulse-laser heating on metal films made by Brorson et al. [5] and Qiu et al. [6]. In addition, Tzou [2], Chapter 4, examines the experimental results obtained by Bertman and Sandiford [7] and it is shown there that the additional delay in time between the heat flux vector and the temperature gradient is due to the finite time required for activating the low-temperature molecules to transport heat. Other applications are described in the book by Tzou [2]. We outline also here the bioheat transfer between tissues and blood during the nonequilibrium processes and the drug delivery in tumor cells involving species diffusion. As Tzou [2], pg. 376, concludes lagging behavior in pharmacodynamical and biological processes will become the focus during the next decade due to the greater uses of ultrafast laser and electromagnetic pulses in life sciences. Ongoing efforts include femtosecond-laser treatments of hard tissues (bones and teeth, for example) and soft tissues (melanoma, burned skins and/or breast cancer), as well as localized treatments of cancer cells via guided nanoparticles heated by electromegnetic waves. It is also worth noting that Antaki [8] uses the dual-phase-lag model of heat conduction to offer a new interpretation for experimental evidence of non-Fourier conduction in processed meat that was interpreted previously by Mitra et al. [9] with a hyperbolic equation of heat conduction. It was shown there that the dual-phase-lag model combines the wave features of hyperbolic conduction with a diffusion-like feature of the evidence not captured by the hyperbolic case and it is outlined that it accounts for the heterogeneous nature of the meat that is not accommodated by the classical Fourier model. What stated above is therefore certainly sufficient to consider of particular interest the investigation of the mathematical structure in revealing the fundamental characteristics of the lagging behavior as described by the time differential dual-phase-lag constitutive equations. The mixed mathematical formulation in terms of both temperature and heat flux vector is better approach in this connection. In this paper we study the dual-phase-lag model of heat conduction incorporating the microstructural interaction effects in the fast-transient process of heat transport. Tzou [1, 3, 4] (see also [2] and [10] and the references therein) proposed the following time differential constitutive law for the heat flux vector qi qi (x, t) + τq

1 ∂ 2 qi ∂qi (x, t) + τq2 2 (x, t) ∂t 2 ∂t

(2)

∂T,j = −kij (x)T,j (x, t) − τT kij (x) (x, t) . ∂t It was established by Fabrizio and Lazzari [11] that the restrictions imposed by thermodynamics on the constitutive equation (2), within the framework of a linear rigid conductor, implies that the delay times have to satisfy the inequality 0 ≤ τq ≤ 2τT . 3

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When the constitutive equation (2) is coupled with the energy equation −qi,i + %r = a

∂T , ∂t

(3)

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and under appropriate regularity assumptions, then we obtain the following governing equation of hyperbolic type for temperature field T    ∂T 1 ∂2 ∂ a + τq2 2 − %r 1 + τq ∂t 2 ∂t ∂t (4)     ∂ T,j . = kij 1 + τT ∂t ,i

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Such equation was studied intensively in literature in many papers (see, for example, [12]-[23]). Quintanilla [15] has shown that

the equation (4), together with suitable initial conditions for T , (∂T ) / (∂t) and ∂ 2 T / ∂t2 and appropriate boundary conditions in terms of T , leads to an exponentially stable system when 0 < τq < 2τT and to an unstable system when 0 < 2τT < τq . While in [12]-[14] the well-posedness problem is studied, provided some appropriate restrictions upon the parameters τq and τT are assumed. At this regard, Quintanilla [24, 25] performed a very interesting analysis about the well-posed problems of dual- and three-phase-lag models of heat conduction equation. We have to outline that some approximations of the Tzou’s theory have been studied recently by Quintanilla [26] and Amendola et al. [27]. However, there are many problems existing in practice that involve both temperature variation and heat flux in the boundary conditions. The lagging behavior in a heated sample with isothermal surfaces usually involves this type of boundary conditions. In such situations, solving the energy equation and the constitutive equation simultaneously is most convenient in avoiding the required more regularity of solutions and the complicated conversion between the heat flux vector and the temperature variation. In [28] the authors investigated the propagation of plane time harmonic waves and surface waves in the case of a homogeneous and isotropic dual-phase-lag thermoelastic material. Further, in [29] it is studied the well-posedness of the time differential dual-phase-lag model of a thermoelastic material where the elastic deformation is accompanied by thermal effects governed by a time differential equation for the heat flux with dual phase lags. Moreover, Chirit¸˘ a et al. [30] studied the compatibility of the three-phase-lag model of heat conduction and use the thermodynamic restrictions on the three delay times in order to establish the uniqueness and continuous dependence of the transient solutions. In the present paper we formulate the initial boundary value problem of heat conduction model based on the constitutive equation (2) and the basic energy equation (3), considering it is described by a differential system for the unknown couple {T, qi }. That means we consider initial conditions for T , qi and (∂qi ) / (∂t) and appropriate boundary conditions in terms of temperature variation and heat flux vector. Then we study the effects of the presence of the delay times upon the uniqueness and continuous data dependence results as well as the spatial behavior of the transient solutions and of the harmonic vibrations. The uniqueness results are established without any restrictions upon the delay times, except the case of the class of materials characterized by zero delay time of phase lag of the conductive temperature gradient and for which the delay time in the phase lag of heat flux vector is strictly positive. In such a case it should be expected to have an ill-posed model. Moreover, it was shown by Fabrizio and Lazzari [11] that the corresponding model (with τT = 0 and τq > 0) is incompatible with the thermodynamic principles. We also address the problem of continuous dependence of solutions with respect to the given data. To this aim we establish appropriate conservation laws and then we use the Gronwall’s inequality in order to establish two estimates describing the continuous dependence of solutions with respect to the prescribed initial data and with respect to the given supply terms. The first 4

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estimate is established for delay times satisfying the inequality 0 ≤ τq ≤ 2τT , which is in accord with the thermodynamic restriction established by Fabrizio and Lazzari [11]. The second estimate is established under the assumption 0 < 2τT < τq and it allows the solutions to have a growth exponential in time. Finally we proceed to study the spatial behavior of transient solutions and to this end we establish a domain of influence theorem, provided the delay times are in the set {0 < τq ≤ 2τT }∪{0 < 2τT < τq }. Explicit estimates are given, in terms of the constitutive thermal coefficients and delay times, for the speed of signal propagation for each of the two subsets of delay times. Moreover, for the steady-state vibrations we establish an exponential decaying estimate in terms of the amplitude vibration, provided the frequency is lower than a critical value and for delay times τq ≥ 0 and τT ≥ 0. Our results will be of importance for engineering in view of the fact that the related mathematical model of heat conduction is understood to be well posed in the sense of Hadamard if the corresponding boundary-initial value problem possesses a unique solution that depends continuously on the prescribed data. Knowing whether or not a solution is unique is important for numerical evaluation or for completeness of constructed by semi-inverse or similar methods. While the continuous data dependence is of practical and numerical importance. Physical measurements introduce unavoidable errors and these small errors have to influence the real solution in little measure. Formulation of the initial boundary value problem

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We assume that a regular region B is filled by an inhomogeneous and anisotropic material with dual phase lag times. Throughout this paper we consider the initial boundary value problem P defined by the field equations (2) and (3), the initial conditions

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T (x, 0) = T 0 (x), qi (x, 0) =

qi0 (x),

∂qi (x, 0) = q˙i0 (x), ∂t

(5) on

B,

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and the boundary conditions

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T (x, t) = ϑ(x, t) on Σ1 × [0, ∞), qi (x, t)ni = ξ(x, t) on

(6)

Σ2 × [0, ∞).

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Here T 0 (x), qi0 (x), q˙i0 (x) and ϑ (x, t), ξ (x, t) are prescribed smooth functions. Moreover, Σ1 and Σ2 are subsets of the boundary ∂B so that Σ1 ∪ Σ2 = ∂B and Σ1 ∩ Σ2 = ∅. By a solution of the initial boundary value problem P corresponding to the given data D = {r; T 0 , qi0 , q˙i0 ; ϑ, ξ} we mean the ordered array S = {T, qi } defined on B × [0, ∞) with the properties that T (x, t) ∈ C 1,1 (B × (0, ∞)), qi (x, t) ∈ C 1,2 (B × (0, ∞)) and which satisfy the field equations (2) and (3), the initial conditions (5) and the boundary conditions (6). In what follows we denote by P0 the initial boundary value problem P corresponding to the zero given data D = {r; T 0 , qi0 , q˙i0 ; ϑ, ξ} = 0. For further convenience we introduce the following operators Z t Z tZ s f 0 (t) = f (s)ds, f 00 (t) = f (z)dzds, etc.; (7) 0

0

gˆ(t) = g 00 (t) + τq g 0 (t) + 5

0

1 2 τ g(t); 2 q

(8)

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˜ = h0 (t) + τT h(t). h(t)

(9)

It is worth to note that if S = {T, qi } is a solution of the initial boundary value problem P corresponding to the given data D = {r; T 0 , qi0 , q˙i0 ; ϑ, ξ}, then Sˆ = {Tˆ, qˆi } is a solution of the initial boundary value problem Pˆ defined by the basic equations ∂ Tˆ = −ˆ qi,i + R, ∂t


qˆi (t) = −kij T,j00 (t) + τT T,j0 (t) + ϑi (t),

the initial conditions

1 Tˆ(x, 0) = τq2 T 0 (x), 2

(11)

(12)

1 ∂ qˆi (x, 0) = τq qi0 (x) + τq2 q˙i0 (x), ∂t 2

and the boundary conditions

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1 qˆi (x, 0) = τq2 qi0 (x), 2

(10)

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ˆ t) on Σ1 × [0, ∞), Tˆ(x, t) = ϑ(x, ˆ t) on qˆi (x, t)ni = ξ(x, where

(13)

Σ2 × [0, ∞),

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R(x, t) = %ˆ r(x, t) + a(t + τq )T 0 (x),

(14)

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  1 2 0 1 0 0 ϑi (x, t) = τT kij T,j (x) + τq qi (x) + τq q˙i (x) t + τq2 qi0 (x). 2 2

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Suppose now that the conductivity tensor kij is non-singular and let us denote by Kij its inverse tensor so that kij Kjk = Kij kjk = δik . (15)

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It is a straightforward task to verify that if S = {T, qi } is a solution of the initial boundary value problem P corresponding to the given data D = {r; T 0 , qi0 , q˙i0 ; ϑ, ξ}, then S˜ = {T˜, q˜i } is a solution of the initial boundary value problem P˜ defined by the basic equations

the initial conditions

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∂ T˜ = −˜ qi,i + R∗ , ∂t

  1 T˜,i (t) = −Kij qj0 (t) + τq qj (t) + τq2 q˙j (t) + ϕi , 2

(16)

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T˜(x, 0) = τT T 0 (x), q˜i (x, 0) = τT qi0 (x),

∂ q˜i (x, 0) = qi0 (x) + τT q˙i0 (x), ∂t

(18)

and the boundary conditions ˜ t) on Σ1 × [0, ∞), T˜(x, t) = ϑ(x, ˜ t) on q˜i (x, t)ni = ξ(x, 6

Σ2 × [0, ∞),

(19)

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where R∗ (x, t) = %˜ r(x, t) + aT 0 (x),   1 2 0 0 + Kij τq qj (x) + τq q˙j (x) . 2

Uniqueness results

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ϕi (x) =

τT T,i0 (x)

Throughout this section we consider the following constitutive hypotheses

(H1) kij is a positive definite tensor and hence km ξi ξi ≤ kij ξi ξj ≤ kM ξi ξi ,

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where km and kM are the lowest and the greatest eigenvalues of kij ;

(H2) a 6= 0; (H3) the delay times are such that

(H4) meas Σ1 6= 0.

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1 1 τq ≤ τT } ∪ {0 < τT < τq }; 2 2

(22)

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{0 ≤

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Theorem 1. Suppose that the hypotheses (H1) and (H3) hold true and, moreover, we assume that at least one of the hypotheses (H2) or (H4) is fulfilled. Then the initial boundary value problem P has at most one solution.

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Proof. In order to prove the uniqueness result it is sufficient to prove that the zero external given data, that is D = {r; T 0 , qi0 , q˙i0 ; ϑ, ξ} = 0, implies that the corresponding solution S = {T, qi } is vanishing on B × [0, ∞). That means we have to prove that the initial boundary value problem P0 has only the trivial solution. Thus, we consider here that S = {T, qi } is a solution of the initial boundary value problem P0 . It follows then that Sˆ = {Tˆ, qˆi } is a solution of the initial boundary value problem Pˆ with zero given data, denoted in what follows by Pˆ0 . Then, by means of an integration with respect to time variable of the equation (10), followed by the use of the zero initial data, it follows that aTˆ(t) = −

Z

t

qˆi,i (z)dz.

(23)

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Furthermore, we start with aTˆ(t + s)Tˆ(t − s) − aTˆ(t − s)Tˆ(t + s) = 0,

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for all s ∈ (0, t), t ≥ 0,

(24)

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which, by means of (23), implies Z  Z ˆ −T (t + s) B

t−s

0

qˆi,i (z)dz + Tˆ(t − s)

Z

t+s



qˆi,i (z)dz dv = 0.

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0

0

B

(26)

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If we use the divergence theorem and the zero boundary conditions (13), from (25) we get  Z t+s Z  Z t−s qˆi (z)dz dv = 0 qˆi (z)dz − Tˆ,i (t − s) Tˆ,i (t + s)

(25)

and hence, by replacing qˆi from the constitutive equation (11) and Tˆ,i by (8), we obtain Z   kij T,j000 (t − s)T,i00 (t + s) − kij T,j000 (t + s)T,i00 (t − s) dv B

Z

B

1 2 τ 2 q

Z

+ τT τq

Z

+

B

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  kij T,j000 (t − s)T,i (t + s) − kij T,j000 (t + s)T,i (t − s) dv 

Z

1 + τT τq2 2

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  kij T,j000 (t − s)T,i0 (t + s) − kij T,j000 (t + s)T,i0 (t − s) dv

(27)

 kij T,j00 (t − s)T,i0 (t + s) − kij T,j00 (t + s)T,i0 (t − s) dv

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  kij T,j00 (t − s)T,i (t + s) − kij T,j00 (t + s)T,i (t − s) dv = 0.

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+ τq

+ τq

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Moreover, we can write Z Z ∂ kij T,j000 (t − s)T,i000 (t + s)dv + τT τq kij T,j00 (t − s)T,i00 (t + s)dv ∂s B B Z

  kij T,j000 (t − s)T,i00 (t + s) + kij T,j000 (t + s)T,i00 (t − s) dv

1 2 τ 2 q

Z

B

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+

1 τT τq2 2

1 + τq2 2

Z

B

  kij T,j000 (t − s)T,i0 (t + s) + kij T,j000 (t + s)T,i0 (t − s) dv

Z

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+

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B

B

(28)

  kij T,j00 (t − s)T,i0 (t + s) + kij T,j00 (t + s)T,i0 (t − s) dv

kij T,j00 (t

−

s)T,i00 (t

+ s)dv



= 0,

so that an integration with respect to s on the interval [0, t] implies  Z Z d 000 000 2 τq kij T,j (t)T,i (t)dv + τq kij T,j000 (t)T,i00 (t)dv dt B B 1 + τT τq2 2 + τq



Z

B

kij T,j00 (t)T,i00 (t)dv

τq  τT − 2

Z

B



+

Z

B

kij T,j000 (t)T,i000 (t)dv

kij T,j00 (t)T,i00 (t)dv = 0. 8

(29)

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Thus, by twice integrations with respect to time variable, from (29) we obtain the following identity Z tZ Z tZ sZ kij T,j000 (z)T,i000 (z)dvdzds + τq kij T,j000 (s)T,i000 (s)dvds 0

τq2 + 2

B

Z

B

0

kij T,j000 (t)T,i000 (t)dv

1 + τT τq2 2

B

Z tZ 0

B

kij T,j00 (s)T,i00 (s)dvds

Z Z Z  τq  t s kij T,j00 (z)T,i00 (z)dvdzds = 0, + τq τT − 2 0 B 0 Let us first suppose that 0 ≤ τq ≤ 2τT .

(30)

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0

t ≥ 0.

(31)

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Then all the terms of the identity (30) are positive and hence we deduce

so that we have

kij T,j000 (t)T,i000 (t) = 0,

(32)

T,i000 (x, t) = 0 for all (x, t) ∈ B × (0, ∞).

(33)

T,i (x, t) = 0 for all (x, t) ∈ B × (0, ∞),

(34)

Thus, we have

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which, by using the constitutive equation (11), gives

qˆi (x, t) = 0 for all (x, t) ∈ B × (0, ∞).

(35)

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This last relation together with (8) gives qi (x, t) + τq

1 ∂ 2 qi ∂qi (x, t) + τq2 2 (x, t) = 0, ∂t 2 ∂t

(36)

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which, with the aid of the initial conditions

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implies

qi (x, 0) = 0,

q˙i (x, 0) = 0,

qi (x, t) = 0 for all (x, t) ∈ B × [0, ∞).

(37) (38)

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Furthermore, from the relations (3) and (38) we have a

∂T = 0. ∂t

(39)

Now, in view of the hypothesis (H2), the relation (39) furnishes ∂T =0 ∂t

(40)

and hence, by using the zero initial condition T (x, 0) = 0, we obtain T (x, t) = 0 for all (x, t) ∈ B × [0, ∞),

(41)

and therefore, the corresponding solution S = {T, qi } is vanishing on B × [0, ∞). Otherwise, if hypothesis (H4) holds true, then from relation (34) we deduce again the relation (41) and so we have that the corresponding solution S = {T, qi } is vanishing on B × [0, ∞). 9

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Let us now consider the case 0 < τT <

1 τq . 2

(42)

which leads to the following Gronwall type inequality  Z t 1 2 Ψ(t) ≤ Ψ(s)ds, − τT τq 0 with Ψ(t) =

Z tZ 0

Then the Gronwall’s lemma furnishes

B

t ≥ 0,

kij T,j00 (s)T,i00 (s)dvds.

and hence we have

T,i00 (x, t) = 0

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Ψ(t) = 0 for all t ≥ 0

(43)

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Then the identity (30) implies Z tZ Z tZ sZ τ 1 q 2 00 00 kij T,j00 (z)T,i00 (z)dvdzds, τT τq kij T,j (s)T,i (s)dvds ≤ τq − τT 2 2 0 B 0 B 0

for all (x, t) ∈ B × (0, ∞).

(44)

(45)

(46)

(47)

This last relation leads to the conclusion expressed in (34) and the analysis follows the way described in the proof of the case (31). Thus, the proof is complete. A similar result can be obtained if we replace the hypothesis a 6= 0 by a > 0. Thus, we have

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Theorem 2. Suppose that the hypotheses (H1) holds true, a > 0 and τT > 0 and τq ≥ 0. Then the initial boundary value problem P has at most one solution.

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Proof. In order to prove the result we use the time weighted method [35]. Thus, we use the time weight function e−αt , α ≥ 0 and the basic equations of the initial boundary value problem P0 to get the following time weighted conservation law Z Z h
1 t e−αs aTˆ2 (s) + τT + τq + ατq2 kij T,j00 (s)T,i00 (s) 2 0 B

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 Z 1 1 2 2 0 0 + τT τq kij T,j (s)T,i (s) dvds + τq e−αt kij T,j00 (t)T,i0 (t)dv 2 4 B s

Z

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Z tZ

+

0

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+



0

B

e

−αs



  α ˆ2 α α2 2 aT (z) + 1 + (τT + τq ) + τq kij T,j00 (z)T,i00 (z) 2 2 4

  α 1 τT τq2 + τq τT − τq dvdzds = 0, 4 2

(48)

t ≥ 0.

Further, it can be easily see that it is possible to choose the parameter α ≥ 0 such that   α 1 2 τT τq + τq τT − τq ≥ 0. 4 2

(49)

With this choice we observe that all the integral terms in the above time weighted identity are made positive and we can follow an analysis similar with that used in the proof of Theorem 1 to show that S = {T, qi } is vanishing on B × [0, ∞). Remark 3. We have established the uniqueness results described in the Theorem 1 and the Theoˆ but they can be proved as well by means of the problem P. ˜ We rem 2 by means of the problem P, leave this task to the reader. 10

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Continuous dependence results

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In this section we study the effects of the delay times upon the problem of continuous dependence of solutions with respect to the supply term and the initial given data. To this aim we consider S = {T, qi } be a solution of the initial boundary value problem P corresponding to the given data D = {r; T 0 , qi0 , q˙i0 ; 0, 0} and then it follows that S˜ = {T˜, q˜i } is a solution of the initial boundary value problem P˜ with zero boundary data. At this stage we introduce the following functional Z Z Z Z Z 1 t s 1 t Kij qi0 (z)qj0 (z)dvdzds, t ≥ 0 (50) aT˜2 (s)dvds + E(t) = 2 0 B 2 0 0 B and we note that the relations (16)-(19) furnish the following conservation law Z Z τT + τq t E(t) + Kij qj0 (s)qi0 (s)dvds 2 0 B

+

1 2

Z tZ 0

Z tZ 0

s

0

Z

1 Kij qi (s)qj (s)dvds + τq2 4 B

Z

B

Kij qj0 (t)qi0 (t)dv

AN US

1 + τT τq2 4

B

Kij qi0 (z)qj0 (z)dvdzds

+

t 2

Z

aT˜2 (0)dv +

B

Z tZ 0

0

s

Z

B

B

Z tZ 0

Z

B

Kij qi0 qj0 dv

R∗ (z)T˜(z)dvdzds +

PT + τT

1 τT τq2 2

ED

=

M

Z tZ sZ  1 Kij qi (z)qj (z)dvdzds + τq τT − τq 2 0 0 B

ϕi (x)qi0 (s)dvds,

Z tZ 0

s

0

Z

(51)



B

ϕi (x)qi0 (z)dvdzds

t ≥ 0.

CE

On the other side, we have

Km ξi ξi ≤ Kij ξi ξj ≤ KM ξi ξi ,

for all ξi ∈ R,

(52)

AC

where Km (x) and KM (x) are the lowest and the greatest eigenvalues of the positive definite tensor Kij . On the basis of the Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality, we have Z Z tZ sZ 1 t2 ϕi (x)ϕi (x)dv ϕi (x)qi0 (z)dvdzds ≤ 4 B ε1 K m 0 0 B (53) Z Z Z 1 t s 0 0 + ε1 Kij qi (z)qj (z)dvdzds, for every ε1 > 0, 2 0 0 B

and Z tZ 0

B

ϕi (x)qi0 (s)dvds

Z t 1 ≤ ϕi (x)ϕi (x)dv 2 B ε2 Km Z tZ 1 + ε2 Kij qi0 (s)qj0 (s)dvds, 2 0 B 11

(54) for every ε2 > 0.

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Setting ε1 = ε2 = 1 into the inequalities (53) and (54) and then using the result into identity (51), we obtain the following estimate Z tZ 0

Kij qi (s)qj (s)dvds

B

 Z tZ s Z 1 Kij qi (z)qj (z)dvdzds + τq τT − τq 2 0 B 0 t ≤ 2 +

t2 4

Z

1 aT (0)dv + τT τq2 2 B

Z

B

˜2

1 ϕi ϕi dv + Km

Z tZ 0

Z

Kij qi0 qj0 dv

B

s

0

Z

+ τT

Z

B

R∗ (z)T˜(z)dvdzds,

B

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1 τT τq2 4

E(t) +

1 ϕi ϕi dv Km



(55)

t ≥ 0.

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Then, the following theorem describes the continuous dependence of solutions of the problem P with respect to the supply term and with respect to the given initial data.

Theorem 4. Suppose that the constitutive hypotheses (H1) and (H3) hold true and a > 0. Then the solution S = {T, qi }(x, t) of the initial boundary value problem P depends continuously on the given data D = {r; T 0 , qi0 , q˙i0 ; 0, 0} for any finite time interval [0, S], S > 0. More precisely, for every t ∈ [0, S], we have the following results: (i) when

M

0 ≤ τq ≤ 2τT ,

the estimate

PT

ED

Z t p 1 g(s)ds E(t) ≤ √ 2 0  Z  1/2  Z S 1 τT 1 S2 2 2 0 0 ˜ + aT (0) + τT τq Kij qi qj + ϕi ϕi dv + ϕi ϕi dv , 2 B 2 Km 4 B Km holds true with

CE

g(t) =

while

Z t Z 0

B

 21 1 ∗2 R (s)dvds ; a

(56)

(57)

(58)

(ii) when

(59)

AC

0 < 2τT < τq ,

we have the estimate Z t p
1 F(t) ≤ √ exp σ 2 (t − s) g(s)ds 2 0  Z   1/2 Z
S 1 τT S2 1 + exp σ 2 t aT˜2 (0) + τT τq2 Kij qi0 qj0 + ϕi ϕi dv + ϕi ϕi dv , 2 B 2 Km 4 B Km (60)

with

1 F(t) = E(t) + τT τq2 4

Z tZ 0

Kij qi (s)qj (s)dvds,

B

12

σ2 =

1 2 − . τT τq

(61)

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Proof. Since a > 0 and in view of the constitutive hypothesis (H1) and (52), it follows that E, as defined by (50), can be considered as a measure for S in the sense that E(t) ≥ 0 for all t ≥ 0 and E(t) = 0 implies S = 0. Let us consider first the case (i). Then, a consequence of relation (55) is the following Gronwall type inequality Z t Z p S2 1 g(s) 2E(s)ds + E(t) ≤ ϕi ϕi dv 4 B Km (62) Z 0  Z Z S 1 1 2 0 0 2 ˜ Kij qi qj dv + τT ϕi ϕi dv . + aT (0)dv + τT τq 2 B 2 B B Km

and we note that (62) implies

p

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To integrate the integral inequality (62) we set Z t Z p S2 1 g(s) 2E(s)ds + ϕi ϕi dv Φ(t) = 4 B Km 0 Z 1/2 Z Z 1 1 S aT˜2 (0)dv + τT τq2 Kij qi0 qj0 dv + τT + ϕi ϕi dv . 2 B 2 B B Km E(t) ≤ Φ(t).

On the other side, from (63) we deduce

1 dΦ (t) ≤ √ g(t), dt 2

M

and hence, by an integration, we get

1 Φ(t) ≤ Φ(0) + √ 2

Z

(63)

(64)

(65)

t

g(s)ds.

(66)

0

CE

PT

ED

Concluding, we see that the relations (63) to (66) imply the estimate (57) and the proof of the case (i) is complete. Let us further consider the case (ii). Then the relations (55) and (61) imply # Z t" Z sZ τT τq2 2 F(t) ≤ 2σ Kij qi (z)qj (z)dvdz ds 4 0 0 B Z  Z Z t 1 1 2 2 0 0 ˜ + aT (0)dv + τT τq Kij qi qj dv + τT ϕi ϕi dv (67) 2 B 2 B B Km t2 + 4

Z

B

1 ϕi ϕi dv + Km

Z tZ 0

0

s

Z

B

R∗ (z)T˜(z)dvdzds,

t ≥ 0,

AC

and hence we obtain the following Gronwall type inequality Z t Z t Z p S2 1 F(t) ≤ 2σ 2 F(s)ds + g(s) 2F(s)ds + ϕi ϕi dv 4 K m 0 0 B Z  Z Z S 1 1 + aT˜2 (0)dv + τT τq2 Kij qi0 qj0 dv + τT ϕi ϕi dv . 2 B 2 B B Km

(68)

Further, if we set 

Z p S2 1 g(s) 2F(s)ds + ϕi ϕi dv 4 B Km 0 0 Z 1/2 Z Z S 1 1 + aT˜2 (0)dv + τT τq2 Kij qi0 qj0 dv + τT ϕi ϕi dv , 2 B 2 B B Km

Π(t) =

2σ 2

Z

t

F(s)ds +

Z

t

13

(69)

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then, from (68) we obtain

1 ˙ Π(t) − σ 2 Π(t) ≤ √ g(t), 2

(70)

which integrated provides the estimate (60) and the proof is complete.

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Remark 5. We can establish estimates similar to (57) and (60) by starting with a conservation ˆ law in terms of the initial boundary value problem P.

Remark 6. We have to outline that the estimate (57) corresponds to the case of a stable system, while the estimate (60) is of a new type in the sense that it allows the solutions to have an exponential growth in time and this should correspond to an unstable system (Cf. Quintanilla [15]). Spatial behavior results

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Throughout this section we study the influence of the delay times upon the spatial behavior of the transient and steady-state solutions within the context of the dual-phase-lag model of heat conduction. Transient solutions

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In this subsection we assume that B is the cylinder of length L + h and of generic section D, more precisely we assume that B = D × (−h, L), h > 0, L > 0. Moreover, we suppose that the support of the given data D = {r; T 0 , qi0 , q˙i0 ; ϑ, ξ} is included into the closed cylinder D × [−h, 0]. We are interested here into the behavior of the solution S = {T, qi } of the initial boundary value problem P with respect to the distance x3 from the support of loadings. To this aim we establish the following theorem of influence domain.

ED

Theorem 7. Suppose that kij is a positive definite tensor and assume that a > 0. Moreover, we suppose that the delay times belong to the set defined by {0 < τq ≤ 2τT } ∪ {0 < 2τT < τq } .

(71)

CE

PT

Let S = {T, qi } be a solution of the initial boundary value problem P, in the cylinder B = D × (−h, L), h > 0, L > 0, corresponding to the given data D = {r; T 0 , qi0 , q˙i0 ; ϑ, ξ} having the support included into the closed cylinder D × [−h, 0]. Then there exists a constant c > 0, depending on the thermal constitutive coefficients and on the delay times, so that S(x, t) = {T, qi }(x, t) = 0

for all

(x, t)

with

x3 > ct,

AC

A possible value for c is the following one s r κM 2τT 1 c0 = + am τq2 τT + τq

when 0 < τq ≤ 2τT ; while when 0 < 2τT < τq a value for c can be taken as s r 1 τT κM 5τq2 + 2τT2 − 6τq τT c1 = , τq am τT2 + 2τq2 − 3τT τq where am = minB a and κM = maxB kM .

14

t > 0.

(72)

(73)

(74)

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Proof. First of all we note that
qˆi = −kij T,j00 + τT T,j0 ,

for all

(x, t) ∈ (D × (0, L)) × (0, ∞),

(75)

so that, by means of the Cauchy-Schwarz inequality, we have

+ τT (kij qˆi qˆj )

1 2

krs T,r0 T,s0


21

≤ (kM qˆi qˆi )

1 2

h

krs T,r00 T,s00


21

(76)

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1
1 qˆi qˆi = −kij qˆi T,j00 + τT T,j0 ≤ (kij qˆi qˆj ) 2 krs T,r00 T,s00 2

+ τT

krs T,r0 T,s0

Further, by using the arithmetic-geometric mean inequality, from (76) we obtain     1 τT2 kij T,i0 T,j0 , qˆi qˆi ≤ kM (1 + ε1 ) kij T,i00 T,j00 + 1 + ε1


12 i

.

(77)

0

0

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for all (x, t) ∈ (D × (0, L)) × (0, ∞) and for every positive parameter ε1 . In order to study the spatial behavior of the solution S = {T, qi } into the cylinder D × [0, L] we introduce the following functional Z tZ sZ Iδ (x3 , t) = e−δz Tˆ(z)ˆ q3 (z)dadzds, x3 ∈ (0, L), t ≥ 0, (78) Dx3

ED

M

where δ ≥ 0 is a parameter whose values will be precisely given later. Further, we note that a direct differentiation with respect to the x3 variable, and the use of the basic relations (10)-(13), give Z Z h
∂Iδ 1 t − (x3 , t) = e−δs aTˆ2 (s) + τT + τq + δτq2 kij T,j00 (s)T,i00 (s) ∂x3 2 0 Dx3  Z τq2 1 e−δt kij T,j00 (t)T,i00 (t)da + τT τq2 kij T,j0 (s)T,i0 (s) dads + 2 4 Dx3

0

Z



  δa ˆ2 δ δ2 2 T (z) + 1 + (τT + τq ) + τq kij T,j00 (z)T,i00 (z) 2 2 4

(79)

PT

+

Z tZ

0

s

Dx3

e−δz

CE

    1 1 2 + τT τq2 δ − 2 − kij T,j0 (z)T,i0 (z) dadzds, 4 τT τq

AC

x3 ∈ (0, L), t ≥ 0. At this time we can see that −(∂Iδ )/(∂x3 ) can be made positive if we choose for δ the value δ = δ0 = 0 when 0 < τq ≤ 2τT and δ = δ1 = 2σ 2 when 0 < 2τT < τq . With these choices we deduce that Z Z h
∂Iδ 1 t − (x3 , t) ≥ e−δs aTˆ2 (s) + τT + τq + δτq2 kij T,j00 (s)T,i00 (s) ∂x3 2 0 Dx3 (80)  1 + τT τq2 kij T,j0 (s)T,i0 (s) dads ≥ 0, x3 ∈ (0, L), t ≥ 0, 2 where from now and in what follows we consider that δ has one of the two values δ0 = 0 and δ1 = 2σ 2 . 15

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Furthermore, we note that an integration in the relation (80), with respect to x3 variable on the interval [x3 , L], gives Z Z h
1 t e−δs aTˆ2 (s) + τT + τq + δτq2 kij T,j00 (s)T,i00 (s) Iδ (x3 , t) ≥ 2 0 Ωx3 (81)  1 + τT τq2 kij T,j0 (s)T,i0 (s) dvds ≥ 0, x3 ∈ (0, L), t ≥ 0, 2 where Ωx3 = D × [x3 , L]. This proves that Iδ (x3 , t) can be considered as a measure of the solution S = {T, qi }. On the other hand, by a direct differentiation with respect to time variable into relation (78), we have Z tZ ∂Iδ (x3 , t) = e−δs Tˆ(s)ˆ q3 (s)dads, (82) ∂t 0 Dx3

×



Z tZ 0

e−δs

Dx3


2

τT + τq + δτq

(

h i ε2 aTˆ2 (s) +

κM (1 + ε1 )
am ε2 τT + τq + δτq2

(83)

" #) 2  τ τ 2τ κ (1 + ε ) T T M 1 q kij T,i00 (s)T,j00 (s) + kij T,i0 (s)T,j0 (s) dads, am ε1 ε2 τq2 2

M

1 ≤ 2

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so that, by means of the Cauchy-Schwarz and the arithmetic-geometric mean inequalities and by using the estimate (77), we obtain   Z Z ∂Iδ 1 t 1 −δs 2 2 ˆ e ε2 am T (s) + qˆ (s) dads ∂t (x3 , t) ≤ 2 ε2 am 3 0 Dx3

ED

for all (x3 , t) ∈ (0, L) × (0, ∞) and for every positive parameters ε1 and ε2 . At this stage we equate the coefficients of the various energy terms in (83), that is we set

and hence we fix

κM (1 + ε1 ) 2τ κ (1 + ε1 )
= T M am ε1 ε2 τq2 am ε2 τT + τq + δτq2

PT

ε2 =


2τT τT + τq + δτq2 ε1 = , τq2

CE

ε2 =

r

κM am

s

2τT 1 + . τq2 τT + τq + δτq2

(84)

(85)

AC

With this choice, from the relations (80) and (83) we obtain the following first-order differential inequality ∂Iδ ∂Iδ (x , t) (86) ∂t 3 + c ∂x3 (x3 , t) ≤ 0, for all (x3 , t) ∈ (0, L) × (0, ∞), where

c = ε2 =

r

κM am

s

2τT 1 + . 2 τq τT + τq + δτq2

(87)

Further, the differential inequality (86) can be treated as in Chirit¸˘ a and Quintanilla [31] and Chirit¸˘a and Ciarletta [35] in order to deduce that Iδ (x3 , t) = 0

for all

(x3 , t) ∈ (ct, L) × (0, ∞),

and hence the relation (81) implies the conclusion (72) and the proof is complete. 16

(88)

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Steady-state vibrations Throughout this subsection we assume that the cylinder C = D × (0, L) is made of a material with the delay times τq ≥ 0, τT ≥ 0 and that the conductivity tensor is positive definite. The cylinder is free of heat supply and it is thermally insulated on its lateral surface and on the end situated in the plane x3 = L. The cylinder is subjected to a harmonic perturbation on its base x3 = 0 of the form T (x1 , x2 , 0, t) = h(x1 , x2 )eiωt ,

{T, qr }(x, t) = {θ, Qr }(x)eiωt ,

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(x1 , x2 ) ∈ D0 , t > 0, (89) √ where ω > 0 is the frequency of perturbation and i = −1 is the imaginary unit. Then inside of cylinder C we will have the following harmonic vibration (90)

Qr,r = −iωaθ, with the boundary conditions θ(x) = 0 on

for all x ∈ C,

(92)

(∂Dx3 × (0, L)) ∪ DL ,

(93)

M

and

AN US

where the amplitude {θ, Qr } of the vibration is a solution of the boundary value problem defined by the differential system   1 2 2 1 + iωτq − τq ω Qr = − (1 + iωτT ) krs θ,s , (91) 2

θ(x1 , x2 ) = h(x1 , x2 ),

(x1 , x2 ) ∈ D0 .

ED

We introduce the following functional Z   M (x3 ) = − (1 + iωτT ) k3s θ,s θ + (1 − iωτT ) k3s θ,s θ da,

(94)

x3 > 0,

(95)

Dx3

CE

PT

where a superposed bar denotes complex conjugate. Further, we note that the relations (91) and (95) imply      Z 1 1 M (x3 ) = 1 + iωτq − τq2 ω 2 Q3 θ + 1 − iωτq − τq2 ω 2 Q3 θ da, x3 > 0, (96) 2 2 Dx3

AC

and hence, by means of the divergence theorem and the use of relations (91) and (93), we get Z Z dM − (x3 ) = 2 krs θ,r θ,s da − 2τq ω 2 aθθda. (97) dx3 Dx3 Dx3 On the other side, in view of the lateral boundary condition in (93), we have Z Z θ,α θ,α da ≥ λ θθda, Dx3

(98)

Dx3

where λ is the lowest eigenvalue in the two-dimensional clamped membrane problem for the cross section Dx3 . Now, if we use the estimate (98) into relation (97), we deduce that  Z dM τq aM 2 − (x3 ) ≥ 2 1 − ω krs θ,r θ,s da, (99) dx3 λκm Dx3 17

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where aM = supC |a| and κm = inf C km . Further, we assume that the frequency of the vibration is such that s λκm 0<ω< , (100) τq aM

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and we note that an integration with respect to x3 variable over [x3 , L] and the use of the end boundary condition in (93) give Z  τq aM 2 (101) ω krs θ,r θ,s dv ≥ 0, M (x3 ) ≥ 2 1 − λκm Cx3

AN US

where Cx3 = D × (x3 , L). By means of the Cauchy-Schwarz and the arithmetic-geometric mean inequalities and by use of the inequality (98), from the relation (95) we obtain s Z (1 + τT2 ω 2 ) (k k ) (102) |M (x3 )| ≤ sup krs θ,r θ,s da. 3s 3s λκ2m Dx3 C Concluding, from the relations (100)-(102), we obtain the first-order differential inequality M (x3 ) + ν where ν=

dM ≤ 0, dx3

√

for all x3 ∈ (0, L),

r λ (1 + τT2 ω 2 ) sup (k3s k3s ). 2 (λκm − τq aM ω 2 ) C

(103)

(104)

M

When integrated, the differential inequality (103) furnishes the estimate 0 ≤ M (x3 ) ≤ M (0)e−

x3 ν

,

for all x3 ∈ (0, L),

(105)

ED

that expresses the exponential decay of the amplitude {θ, Qr } with respect to the distance x3 at the loaded base. Thus, we have the following theorem.

CE

PT

Theorem 8. Suppose that the hypothesis (H1) holds true. Then the spatial behavior of the amplitude {θ, Qr } of the harmonic vibration (90) is described by the inequality (105), provided the frequency ω is lower than the critical value s λκm ωc = , (106) τq aM

AC

being the measure of the amplitude M (x3 ) as results from (101), that is  Z ω2 M (x3 ) ≥ M ∗ (x3 ), M ∗ (x3 ) = 2 1 − 2 krs θ,r θ,s dv. ωc Cx3

(107)

Application

As an illustrative example we consider here a right circular cylinder of radius 1m, made of an isotropic and homogeneous copper material with delay times for which we have [32],[33]: a/k = 8066.8 s/m2 . Further, from [34], for a circle of radius r = 1m, we have λ = 5.7832 m−2 . Figure 1 (a) describes the influence of the delay time τq upon the critical value ωc . We outline that when τq = 0 we have no critical frequency and therefore the spatial decay estimate is valid for any frequencies of vibration. While when τq = 1 there exists a critical value equal to 0.026775 s−2 . The Figure 1 (b) proves the behavior of the decay rate factor ν1 with respect to the decay times τT = τq . It can be seen that this factor decays from the value 2.797 when τT = τq = 0 to the value 2.7957 when τT = τq = 3 s. 18

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1 Ν 2.7970 2.7968 2.7966 2.7964 2.7962 2.7960

0.10 0.08 0.06 0.04 0.0

0.2

0.4

0.6

0.8

1.0

Τq

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HbL

HaL Ωc

0.5 1.0 1.5 2.0 2.5 3.0

Τq

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1: The influence of the delay times upon the critical value ωc and upon the spatial decay rate.

Concluding remarks

AC

CE

PT

ED

M

In this paper we studied the dual-phase-lag model of heat conduction based on the constitutive equation (2), under various ranges of the delay times. The uniqueness and continuous data dependence problems are analyzed in the set {0 ≤ τq ≤ 2τT } ∪ {0 < 2τT < τq }. The proof of Theorem 1 follows the Lagrange identity method as used in [29], while the proof used to establish Theorem 2 is entirely new and it is based on the time-weighted function method, first time used in [35]. We expect that, in the next future, this last method will can be improved in order to treat the well-posedness question for time differential dual-phase-lag models which extend toward higher expansion orders. We also established the two continuous dependence estimates (57) and (60) for any compact interval [0, S], being the first one valid for 0 ≤ τq ≤ 2τT and the second one for 0 < 2τT < τq . We have to remark that the estimate (57) is of the same type like that in [29], while the estimate (60) is one new allowing the solutions to grow exponentially with respect to time variable. A version of this last method was used to establish continuous dependence results within the context of the three-phase-lag model [30]. However, there is an open problem for the class of materials characterized by zero delay time of phase lag of the conductive temperature gradient and for which the delay time in the phase lag of heat flux vector is strictly positive. In such a case it should be expected to have an ill-posed model. The estimate (57) together with the Theorem 2 proves that the time differential dual-phaselag model described by the equation (2) leads to an well-posed problem, provided 0 ≤ τq ≤ 2τT . Moreover, we have been able to establish uniqueness and continuous dependence results for the range 0 < 2τT < τq , when there is possible to have asymptotic instability. For the transient solutions we have established the theorem of influence domain as described by the relation (72), where the speed of signal propagation is estimated by the value (73) when the delay times satisfy the inequality 0 < τq ≤ 2τT and by the value (74) when the delay times satisfy the inequality 0 < 2τT ≤ τq . When τq = τT = 0 the thermal model with delay times reduces to the classical model of heat conduction and this case can be treated following the method developed in Chirit¸˘a and Ciarletta [35], Chirit¸˘ a [36] and Quintanilla [37]. When 0 = τq < 2τT our above analysis fails to describe the spatial behavior of the transient solutions. The results concerning the spatial behavior are entirely new and they characterize the two decay rates of disturbances. As regards to the steady-state solutions, we established an exponential decay estimate associated with the amplitude of vibration as described by (102), provided the frequency of the vibration is lower than the critical value given by (103) and the delay times satisfy τq ≥ 0 and τT ≥ 0. As it can be see the delay time τq influences the value of the critical frequency ωc , while τT and τq influence the decay rate of the amplitude vibrations. 19

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In closing, it is helpful to summarize the results presented in this paper as within the following Table 1. The analysis developed in the present paper is motivated by the awareness that the various microstructural interaction effects in the fast-transient process of heat transport are well captured by the considered dual-phase-lag model of heat conduction. From a mathematical point of view, this type of lagging behavior introduces a new type of initial boundary value problems that leads to new techniques and methods for analysing the properties of solutions. It is our strong belief that present analysis will constitute the first step for treating the more complex problems involved by the high orders of lagging behavior, adding even more high-order terms in the dual-phase-lag heat equation. Acknowledgments

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The authors are very grateful to the Associate Editor and to the anonymous reviewers for their valuable comments, which have led to an improvement of the present work. References

[1] D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, Journal of Heat Transfer 117 (1995) 8–16. [2] D. Y. Tzou, Macro- To Micro-Scale Heat Transfer: The Lagging Behavior, John Wiley & Sons, Chichester, 2015.

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[3] D. Y. Tzou, The generalized lagging response in small-scale and high-rate heating, International Journal of Heat and Mass Transfer 38 (1995) 3231–3234.

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[4] D. Y. Tzou, Experimental support for the lagging behavior in heat propagation, Journal of Thermophysics and Heat Transfer 9 (1995) 686–693. [5] S. D. Brorson, J. G. Fujimoto, E. P. Ippen, Femtosecond electron heat-transport dynamics in thin gold film, Physical Review Letters 59 (1987) 1962–1965.

PT

[6] T. Q. Qiu, T. Juhasz, C. Suarez, W. E. Bron, C. L. Tien, Femtosecond laser heating of multilayered metals. I. Analysis. II. Experiments, International Journal of Heat and Mass Transfer 37 (1994) 2789–2797; 2799–2808.

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Delay times Type of solutions

0 < τT < 12 τq

Uniqueness results

Uniqueness results

Continuous dependence estimates for models compatible with thermodynamics

Continuous dependence estimates for solutions with an exponential growth in time

Theorem of influence domain, provided τq > 0

Theorem of influence domain

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Exponential decay estimates of Saint-Venant type, provided the frequency of vibration is lower than a critical value

Table 1: Summary of results

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Steady state solutions

0 ≤ 21 τq ≤ τT

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Transient solutions

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