1 Introduction

The dual-phase-lag model incorporates the microstructural interaction effect in the fast-transient process of heat transport. It describes the finite time required for the various microstructural interactions to take place, including the phonon-electron interaction in metals, the phonon scattering in dielectric crystals, insulators, and semiconductors, and the activation of molecules at extremely low temperature, by the resulting phase-lag (time-delay) in the process of heat transport (see for example, Tzou [1]).

In 1995, Tzou [2–4] (see also [1, 5] and the references therein) proposed, instead of Fourier’s law, the following time differential constitutive law for the heat flux vector \(q_i\)

$$\begin{aligned}&q_i\left( {\mathbf {x}},t\right) +\tau _q \,\frac{\partial q_i}{\partial t}\left( {\mathbf {x,t}}\right) +\frac{1}{2}\, \tau _q^2\, \frac{\partial ^2 q_i}{\partial t^2}\left( {\mathbf {x}},t\right) \nonumber \\&\quad =-k_{ij}({\mathbf {x}})T_{,j}\left( {\mathbf {x}},t\right) -\tau _{T} k_{ij}({\mathbf {x}})\,\frac{\partial T_{,j}}{\partial t}\left( {\mathbf {x}},t\right) , \end{aligned}$$
(1)

where \(\tau _q \ge 0\) and \(\tau _T \ge 0\) are positive delay times and T is the temperature variation. When this constitutive equation is coupled with the energy equation

$$\begin{aligned} -q_{i,i}+\varrho r=a\,\frac{\partial T}{\partial t}, \end{aligned}$$
(2)

then we obtain the following governing equation of hyperbolic type for temperature field T

$$\begin{aligned}&\left( 1+\tau _q\, \frac{\partial }{\partial t}+\frac{1}{2} \, \tau _q^2\, \frac{\partial ^2}{\partial t^2}\right) \left( a\, \frac{\partial T}{\partial t} -\varrho r\right) \nonumber \\&\quad =\left[ k_{ij}\left( 1+\tau _T\, \frac{\partial }{\partial t}\right) T_{,j}\right] _{,i}. \end{aligned}$$
(3)

Such an equation was studied intensively in literature in many papers (see for example, [6–18]). In particular, it was established by Fabrizio and Lazzari [15] that the restrictions imposed by thermodynamics on the constitutive Eq. (1), within the framework of a linear rigid conductor, implies that the delay times have to satisfy the inequality \(0 \, \le \, \tau _q \, \le \, 2\tau _T\). Quintanilla and Racke [11] have shown that the Eq. (3), together with suitable initial and boundary conditions, leads to an exponentially stable system when \(0\,< \,\tau _q \,<\, 2\tau _T\) and to an unstable system when \(0\,< \,2\tau _T \,<\, \tau _q\). While in [6–8] is studied the well-posedness problem, provided some appropriate restrictions upon the parameters \(\tau _q\) and \(\tau _T\) are assumed.

Recently, the ability of the dual-phase-lag model as a new modified constitutive equation replacing the Fourier law to simulate the heat transport in some special cases such as micro/nanoscales [19, 20], ultra fast laser-pulsed processes [21, 22], living tissues [23], and Carbon nanotube [24] have been tested.

On the other side, when the mechanical deformation is accompanied by the thermal effects described by the constitutive Eq. (1) we are lead to a more complex mathematical problem who requires a special treatment. It is the main aim of the present paper to solve this requirement. In fact, in the present paper we formulate the initial boundary value problem of thermoelastic model based on the constitutive Eq. (1) and then we study the uniqueness and continuous data dependence results. In this connection we define two adequate initial boundary value problems associated with the problem in concern. Further, we use these auxiliary problems and the Lagrange identity method (see, e. g. [25, 26]) in order to establish uniqueness results under mild assumptions upon the thermoelastic coefficients and upon the delay times. Consequently, the uniqueness results are established under the same constitutive hypotheses like in the classical linear thermoelasticity, without any restrictions upon the delay times. 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.

Furthermore, we establish appropriate conservation laws and then we use the Gronwall’s inequality in order to establish some estimates describing the continuous dependence of solutions with respect to the prescribed initial data and with respect to the given supply terms. Such results are established under the assumption that \(0\le \tau _q \le 2\tau _T\), which is in accord with the thermodynamic restriction established by Fabrizio and Lazzari [15].

2 Mathematical model

We assume that a regular region B is filled by an inhomogeneous and anisotropic thermoelastic material with dual phase lag. The basic equations of the model in concern are:

  • the equations of motion

    $$\begin{aligned} t_{ji,j}+\varrho f_i=\varrho {\ddot{u}}_i, \end{aligned}$$
    (4)

  • the equation of energy

    $$\begin{aligned} \varrho T_0\, \dot{\eta }=- q_{i,i}+\varrho r, \end{aligned}$$
    (5)

    in \(B\times (0,\infty )\),

  • the constitutive equations

    $$\begin{aligned} t_{ij}= C_{ijkl}e_{kl}-\beta _{ij}T, \end{aligned}$$
    (6)
    $$\begin{aligned} \varrho \eta= \beta _{ij}e_{ij}+a T, \end{aligned}$$
    (7)
    $$\begin{aligned} q_i+\tau _q {\dot{q}}_i+\frac{1}{2} \tau _q^2 {\ddot{q}}_i= -k_{ij}T_{,j}-\tau _T k_{ij}{\dot{T}}_{,j}, \end{aligned}$$
    (8)

    in \(\overline{B}\times [0,\infty )\), and

  • the geometrical equations

    $$\begin{aligned} e_{ij}=\frac{1}{2}\left( u_{i,j}+u_{j,i}\right) , \end{aligned}$$
    (9)

    in \(\overline{B}\times [0,\infty )\). Here we have used the following notations: \(t_{ij}\) are the components of the stress tensor, \(\eta\) is the entropy per unit mass, \(q_{i}\) are the components of the heat flux vector, \(e_{ij}\) are the components of the strain tensor, \(\varrho\) is the mass density of the medium, \(u_{r}\) are the components of the displacement vector, T is the change in temperature from the constant ambient temperature \(T_{0}>0\), \(f_i\) are the components of the body force per unit mass, r is the heat supply per unit mass, a superposed dot denotes time differentiation and a coma denotes the partial differentiation with respect to the corresponding Cartesian coordinate. The constitutive coefficients \(C_{ijkl}\), \(\beta _{ij}\), a and \(k_{ij}\) are continuous differentiable functions on the spatial variable \(\mathbf {x}\) and have the following symmetries

    $$\begin{aligned} C_{ijkl}=C_{klij}=C_{jikl},\quad \beta _{ij}=\beta _{ji}. \end{aligned}$$
    (10)

    Furthermore, we will assume the symmetry of the conductivity tensor, that is

    $$\begin{aligned} k_{ij}=k_{ji}, \end{aligned}$$
    (11)

    and, moreover, we assume that it is a positive definite tensor. Then we can define \(K_{ij}\) so that

    $$\begin{aligned} k_{ij}K_{jk}=K_{ij}k_{jk}=\delta _{ik}. \end{aligned}$$
    (12)

    Then, the constitutive Eq. (8) can be written as

    $$\begin{aligned} T_{,i}+\tau _T{\dot{T}}_{,i}=-K_{ij}\left( q_j+\tau _q {\dot{q}}_j+\frac{1}{2}\, \tau _q^2 {\ddot{q}}_j\right) . \end{aligned}$$
    (13)

Throughout this paper we consider the initial boundary value problem \(\mathcal {P}\) defined by the field Eqs. (4) to (9), the initial conditions

$$\begin{aligned} u_i({\mathbf {x}},0)= u_i^0({\mathbf {x}}),\quad {\dot{u}}_i({\mathbf {x}},0)={\dot{u}}_i^{0}({\mathbf {x}}),\quad T({\mathbf {x}},0)=T^{0}({\mathbf {x}}),\nonumber \\ q_i({\mathbf {x}},0)= q_i^0({\mathbf {x}}),\quad {\dot{q}}_i({\mathbf {x}},0)={\dot{q}}_i^0({\mathbf {x}}),\quad \textit{on} \quad \overline{B}, \end{aligned}$$
(14)

and the boundary conditions

$$\begin{aligned} u_i({\mathbf {x}},t)= \omega _{i}({\mathbf {x}},t) \quad \textit{on}\quad \overline{\varSigma }_1\times [0,\infty ),\nonumber \\ t_{ji}({\mathbf {x}},t)n_j= \psi _{i}({\mathbf {x}},t)\quad \textit{on}\quad \varSigma _2\times [0,\infty ),\nonumber \\ T({\mathbf {x}},t)= \vartheta ({\mathbf {x}},t)\quad \textit{on}\quad \overline{\varSigma }_3\times [0,\infty ),\nonumber \\ q_i({\mathbf {x}},t)n_i= \xi ({\mathbf {x}},t)\quad \textit{on}\quad \varSigma _4\times [0,\infty ). \end{aligned}$$
(15)

Here \(u_{i}^{0}\left( {\mathbf {x}}\right)\), \({\dot{u}}_{i}^{0}\left( {\mathbf {x}} \right)\), \(T^{0}\left( {\mathbf {x}}\right)\), \(q_{i}^{0}\left( \mathbf {x}\right)\), \({\dot{q}}_i^0\left( \mathbf {x}\right)\) and \(\omega _{i}\left( {\mathbf {x}},t\right)\), \(\psi _{i}\left( {\mathbf {x}},t\right)\), \(\vartheta \left( {\mathbf {x}} ,t\right)\), \(\xi \left( {\mathbf {x}},t\right)\) are prescribed smooth functions. Moreover, \(\varSigma _{r}\) (\(r=1,2,3,4\)) are subsets of the boundary \(\partial B\) such that \(\overline{\varSigma }_{1} \cup \varSigma _{2}=\overline{\varSigma }_{3}\cup \varSigma _{4}=\partial B\) and \(\varSigma _{1}\cap \varSigma _{2}=\varSigma _{3}\cap \varSigma _{4}=\emptyset\).

By a solution of the initial boundary value problem \(\mathcal {P}\) corresponding to the given data \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0, T^0,q_i^0,{\dot{q}}_i^0;\omega _i,\psi _i,\vartheta ,\xi \}\) we mean the ordered array \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) defined on \(\overline{B}\times [0,\infty )\) with the properties that \(u_i({\mathbf {x}},t)\in C^{1,2}\left( B\times (0,\infty )\right)\), \(T({\mathbf {x}},t)\in C^{1,1}\;\left( B\times (0,\infty )\right)\), \(e_{ij}({\mathbf {x}},t)=e_{ji}({\mathbf {x}},t)\in C^{0,0}\left( B\times (0,\infty )\right)\), \(t_{ij}({\mathbf {x}},t)=t_{ji}({\mathbf {x}},t)\in C^{1,0}\left( B\times (0,\infty )\right)\), \(\eta ({\mathbf {x}},t)\in C^{0,1}\;\left( B\times (0,\infty )\right)\), \(q_{i}({\mathbf {x}},t)\in C^{1,2}\left( B\times (0,\infty )\right)\) and which satisfy the field Eqs. (4) to (9), the initial conditions (14) and the boundary conditions (15). For further convenience we denote by \(\mathcal {P}_0\) the initial boundary value problem \(\mathcal {P}\) corresponding to the zero given data \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0,T^0,q_i^0, {\dot{q}}_i^0;\omega _i,\psi _i,\vartheta,\xi \}=0\).

It is worth to note that if \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) is a solution of the initial boundary value problem \(\mathcal {P}\) corresponding to the given data \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0,T^0,q_i^0, {\dot{q}}_i^0;\omega _i,\psi _i,\vartheta ,\xi \}\), then we have

$$\begin{aligned}&\int _0^t\int _0^s q_i(z)dzds +\tau _q\int _0^t q_i(z)dz+\frac{1}{2} \, \tau _q^2 q_i(t)\nonumber \\&\quad =-k_{ij}\int _0^t\int _0^s T_{,j}(z)dzds -\tau _T k_{ij}\int _0^t T_{,j}(z)dz\nonumber \\&\qquad +\left( \tau _T k_{ij}T^0_{,j}+\tau _q q_i^0+\frac{1}{2} \, \tau _q^2{\dot{q}}_i^0\right) t+\frac{1}{2}\, \tau _q^2 q_i^0 \end{aligned}$$
(16)

and

$$\begin{aligned}&\int _0^tT_{,i}(z)dz+\tau _T T_{,i}(t)\nonumber \\&\quad =-K_{ij}\left[ \int _0^tq_j(z)dz+\tau _q q_j(t)+\frac{1}{2} \, \tau _q^2\, {\dot{q}}_j(t)\right] \nonumber \\&\qquad + \tau _TT_{,i}^0+K_{ij}\left( \tau _q q_j^0+\frac{1}{2}\, \tau _q^2 {\dot{q}}_j^0\right) . \end{aligned}$$
(17)

3 Some auxiliary operators

For further convenience, throughout this paper, we will use the following notations: for any continuous function f of time variable t, we denote by \(f'(t)\) the integral over [0, t] of that function, that is

$$\begin{aligned} f'(t)=\int _0^t f(z)dz, \quad f''(t)=\int _0^t\int _0^s f(z)dzds,\ldots , \end{aligned}$$
(18)

for any continuous function g(t) we will denote by \(g^{*}(t)\) the following function

$$\begin{aligned} g^{*}(t)=g''(t)+\tau _q g'(t)+\frac{1}{2} \, \tau _q^2 g(t), \end{aligned}$$
(19)

for any continuous function h(t) we will denote by \(\tilde{h}(t)\) the following function

$$\begin{aligned} \tilde{h}(t)=h'(t)+\tau _T h(t). \end{aligned}$$
(20)

Further, we note that

$$\begin{aligned} g^{*}(0)=\frac{1}{2}\, \tau _q^2 g(0),\quad \frac{ dg^{*}}{dt}(0)=\tau _q g(0)+\frac{1}{2}\, \tau _q^2{\dot{g}}(0), \end{aligned}$$
(21)

and

$$\begin{aligned} \tilde{h}(0)=\tau _T h(0),\quad \frac{d \tilde{h}}{dt}(0)=h(0)+\tau _T{\dot{h}}(0). \end{aligned}$$
(22)

Then, the relation (16) can be written in the following way

$$\begin{aligned} q_i^{*}(t)= -k_{ij}\left( T''_{,j}(t)+\tau _T T'_{,j}(t)\right) \nonumber \\&+\left( \tau _T k_{ij}T^0_{,j}+\tau _q q_i^0+\frac{1}{2}\, \tau _q^2{\dot{q}}_i^0\right) t+\frac{1}{2}\, \tau _q^2 q_i^0, \end{aligned}$$
(23)

while the relation (17) implies

$$\begin{aligned} \tilde{T}_{,i}(t)= -K_{ij}\left( q'_{j}(t)+\tau _qq_{j}(t)+\frac{1}{2}\, \tau _q^2{\dot{q}}_{j}(t)\right) \nonumber \\&+\tau _T T_{,i}^0+ K_{ij}\left( \tau _qq_{j}^0+\frac{1}{2}\, \tau _q^2{\dot{q}}_{j}^0\right) . \end{aligned}$$
(24)

The following results can be readily verified.

Lemma 1

Suppose that g is twice continuously differentiable. Then, we have

$$\begin{aligned} \left( \frac{dg}{dt}\right) '\left( t\right)= \frac{d g'}{dt}\left( t\right) -g(0),\\ \left( \frac{dg}{dt}\right)^{*}(t)= \frac{d g^{*}}{dt}(t)-\left( \tau _q +t\right) g(0),\nonumber \end{aligned}$$
(25)
$$\begin{aligned} \left( \frac{d^2g}{dt^2}\right)^{*}(t)= \frac{d^2 g^{*}}{dt^2}(t)-\left[ g(0)+\left( \tau _q +t\right) {\dot{g}}(0)\right] , \end{aligned}$$
(26)
$$\begin{aligned} \widetilde{\frac{dg}{dt}}(t)= \frac{d\tilde{g}}{dt}(t)-g(0),\quad \widetilde{\frac{d^2g}{dt^2}}(t)=\frac{d^2\tilde{g}}{dt^2}(t)-{\dot{g}}(0). \end{aligned}$$
(27)

Moreover, when \(g(0)=0\) and \({\dot{g}}(0)=0\) then, we have

$$\begin{aligned} \left( \frac{dg}{dt}\right) '\left( t\right)= \frac{d g'}{dt}\left( t\right) , \end{aligned}$$
(28)
$$\begin{aligned} \left( \frac{dg}{dt}\right) ^{*}(t)= \frac{d g^{*}}{dt}(t),\quad \left( \frac{d^2g}{dt^2}\right) ^{*}(t)=\frac{d^2 g^{*}}{dt^2}(t), \end{aligned}$$
(29)
$$\begin{aligned} \widetilde{\frac{dg}{dt}}(t)= \frac{d\tilde{g}}{dt}(t), \quad \widetilde{\frac{d^2g}{dt^2}}(t)=\frac{d^2\tilde{g}}{dt^2}(t). \end{aligned}$$
(30)

Lemma 2

If g is a continuous function satisfying

$$\begin{aligned} g^{*}(t)=0,\quad for\; all \quad t> 0, \end{aligned}$$
(31)

then we have

$$\begin{aligned} g(t)=0,\quad for\; all\quad t\ge 0. \end{aligned}$$
(32)

Proof

By setting

$$\begin{aligned} g''(t)=h(t), \end{aligned}$$
(33)

then the relation (31) can be written as

$$\begin{aligned} \frac{1}{2} \,\tau _q^2{\ddot{h}}(t)+\tau _q{\dot{h}}(t)+h(t)=0, \end{aligned}$$
(34)

and moreover, we have

$$\begin{aligned} h(0)=0,\quad {\dot{h}}(0)=0. \end{aligned}$$
(35)

Now it is easy to see that the Cauchy problem defined by the differential Eq. (34) and the initial conditions (35) has only the zero solution, that is

$$\begin{aligned} \int _0^t\int _0^sg(z)dzds=0, \quad for\; all\quad t\ge 0, \end{aligned}$$
(36)

and hence we get the conclusion expressed by relation (32) and the proof is complete.

In a similar way we can prove the following result:

Lemma 3

Suppose that g is a continuous function satisfying

$$\begin{aligned} \tilde{g}(t)=0,\quad for\; all \quad t> 0. \end{aligned}$$
(37)

Then, we have

$$\begin{aligned} g(t)=0,\quad \textit{for all}\quad t\ge 0. \end{aligned}$$
(38)

On the basis of the above results we can prove the following two theorems.

Lemma 4

Let \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) be a solution of the initial boundary value problem \(\mathcal {P}\) corresponding to the given data \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0,T^0,q_i^0, {\dot{q}}_i^0;\omega _i,\psi _i,\vartheta ,\xi \}\). Then \({\mathcal {S}}^*=\{u_{i}^*,T^*,e_{ij}^*,t_{ij}^*,\eta ^*,q_{i}^*\}\) satisfies the initial boundary value problem \({\mathcal {P}}^*\) defined by the basic equations

$$\begin{aligned} t^{*}_{ji,j}(t)+F_i(t)= \varrho \, \frac{ \partial ^2 u^{*}_{i}}{\partial t^2}(t), \end{aligned}$$
(39)
$$\begin{aligned} \varrho \, \frac{\partial \eta ^{*}}{\partial t}(t)= -\frac{1}{T_0}\, q^{*}_{i,i}(t)+R(t), \end{aligned}$$
(40)

in \(B\times (0,\infty )\),

$$\begin{aligned} t^{*}_{ij}(t)= C_{ijkl}e^{*}_{kl}(t)-\beta _{ij}T^{*}(t), \end{aligned}$$
(41)
$$\begin{aligned} \varrho \eta ^{*}(t)= \beta _{ij}e^{*}_{ij}(t)+a T^{*}(t), \end{aligned}$$
(42)
$$\begin{aligned} q_i^{*}(t)= -k_{ij}\left( T''_{,j}(t)+\tau _T T'_{,j}(t)\right) +Q_i^0(t), \end{aligned}$$
(43)

and

$$\begin{aligned} e^{*}_{ij}(t)=\frac{1}{2} \left( u^{*}_{i,j}(t)+u^{*}_{j,i}(t)\right) , \end{aligned}$$
(44)

in \(\overline{B}\times [0,\infty )\), the initial conditions

$$\begin{aligned} &u^{*}_i({\mathbf {x}},0)= \frac{1}{2} \, \tau _q^2 u_i^0({\mathbf {x}}),\nonumber \\ &\frac{\partial u^{*}_i}{\partial t}({\mathbf {x}},0)= \tau _qu_i^0({\mathbf {x}})+\frac{1}{2}\, \tau _q^2{\dot{u}}_i^0({\mathbf {x}}),\nonumber \\ &T^{*}({\mathbf {x}},0)= \frac{1}{2}\, \tau _q^2 T^0({\mathbf {x}}),\quad q^{*}_i({\mathbf {x}},0)=\frac{1}{2}\, \tau _q^2 q_i^0({\mathbf {x}}),\nonumber \\ &\frac{\partial q^{*}_i}{\partial t}({\mathbf {x}},0)= \tau _qq_i^0({\mathbf {x}})+\frac{1}{2}\, \tau _q^2{\dot{q}}_i^0({\mathbf {x}}),\quad \textit{on} \quad \overline{B}, \end{aligned}$$
(45)

and the boundary conditions

$$\begin{aligned} &u^{*}_i({\mathbf {x}},t)= \omega ^{*}_i({\mathbf {x}},t) \quad \textit{on}\quad \overline{\varSigma }_1\times [0,\infty ),\nonumber \\ & t^{*}_{ji}({\mathbf {x}},t)n_j= \psi ^{*}_i({\mathbf {x}},t)\quad \textit{on}\quad \varSigma _2\times [0,\infty ),\nonumber \\ & T^{*}({\mathbf {x}},t)= \vartheta ^{*}({\mathbf {x}},t)\quad \textit{on}\quad \overline{\varSigma }_3\times [0,\infty ),\nonumber \\ & q^{*}_i({\mathbf {x}},t)n_i= \xi ^{*}({\mathbf {x}},t)\quad on\quad \varSigma _4\times [0,\infty ), \end{aligned}$$
(46)

where

$$\begin{aligned} &F_i({\mathbf {x}},t)= \varrho f_i^*({\mathbf {x}},t)+\varrho \left[ \left( t+\tau _q\right) {\dot{u}}_i^0 ({\mathbf {x}})+u_i^0({\mathbf {x}})\right] ,\nonumber \\ &R({\mathbf {x}},t)= \frac{1}{T_0}\, \varrho r^*({\mathbf {x}},t)+ \left( t+\tau _q\right) \left[ \beta _{ij}u_{i,j}^0({\mathbf {x}})+aT^0({\mathbf {x}})\right] ,\nonumber \\ &Q_i^0({\mathbf {x}},t)= \left( \tau _T k_{ij}T^0_{,j}({\mathbf {x}})+\tau _q q_i^0({\mathbf {x}})+\frac{1}{2}\, \tau _q^2{\dot{q}}_i^0({\mathbf {x}})\right) t\nonumber \\&+\frac{1}{2} \, \tau _q^2 q_i^0({\mathbf {x}}). \end{aligned}$$
(47)

Lemma 5

Let \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) be a solution of the initial boundary value problem \(\mathcal {P}\) corresponding to the given data \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0,T^0,q_i^0,{\dot{q}}_i^0; \omega _i,\psi _i,\vartheta ,\xi \}\). Then \(\tilde{{\mathcal {S}}}=\{\tilde{u}_{i},\tilde{T},\tilde{e}_{ij},\tilde{t}_{ij}, \tilde{\eta },\tilde{q}_{i}\}\) satisfies the initial boundary value problem \(\tilde{\mathcal {P}}\) defined by the basic equations

$$\begin{aligned} \tilde{t}_{ji,j}(t)+G_i(t)= \varrho \, \frac{ \partial ^2 \tilde{u}_{i}}{\partial t^2}(t), \end{aligned}$$
(48)
$$\begin{aligned} \varrho \, \frac{\partial \tilde{\eta }}{\partial t}(t)= -\frac{1}{T_0}\, \tilde{q}_{i,i}(t)+P(t), \end{aligned}$$
(49)

in \(B\times (0,\infty )\),

$$\begin{aligned} \tilde{t}_{ij}(t)= C_{ijkl}\tilde{e}_{kl}(t)-\beta _{ij}\tilde{T}(t), \end{aligned}$$
(50)
$$\begin{aligned} \varrho \tilde{\eta }(t)= \beta _{ij}\tilde{e}_{ij}(t)+a \tilde{T}(t), \end{aligned}$$
(51)
$$\begin{aligned} \tilde{T}_{,i}(t)= -K_{ij}\left( q'_{j}(t)+\tau _qq_{j}(t)+\frac{1}{2}\, \tau _q^2{\dot{q}}_{j}(t)\right) +\varTheta ^0_i, \end{aligned}$$
(52)

and

$$\begin{aligned} \tilde{e}_{ij}(t)=\frac{1}{2} \left( \tilde{u}_{i,j}(t)+\tilde{u}_{j,i}(t)\right) , \end{aligned}$$
(53)

in \(\overline{B}\times [0,\infty )\), the initial conditions

$$\begin{aligned} \tilde{u}_i({\mathbf {x}},0) &\,=\, \tau _T u_i^0({\mathbf {x}}),\quad \frac{\partial \tilde{u}_i}{\partial t}({\mathbf {x}},0)=u_i^0({\mathbf {x}})+ \tau _T{\dot{u}}_i^0({\mathbf {x}}),\nonumber \\ & \quad \tilde{T}({\mathbf {x}},0)= \tau _T T^0({\mathbf {x}}), \quad \tilde{q}_i({\mathbf {x}},0)= \tau _T q_i^0({\mathbf {x}}),\nonumber \\ & \quad \frac{\partial \tilde{q}_i}{\partial t}({\mathbf {x}},0)= q_i^0({\mathbf {x}})+ \tau _T{\dot{q}}_i^0({\mathbf {x}}),\quad \textit{on} \quad \overline{B}, \end{aligned}$$
(54)

and the boundary conditions

$$\begin{aligned} &\tilde{u}_i({\mathbf {x}},t)= \tilde{\omega }_i({\mathbf {x}},t) \quad {on}\quad \overline{\varSigma }_1\times [0,\infty ),\nonumber \\ & \tilde{t}_{ji}({\mathbf {x}},t)n_j= \tilde{\psi }_i({\mathbf {x}},t)\quad {on}\quad \varSigma _2\times [0,\infty ),\nonumber \\ & \tilde{T}({\mathbf {x}},t)= \tilde{\vartheta }({\mathbf {x}},t)\quad {on}\quad \overline{\varSigma }_3\times [0,\infty ),\nonumber \\ &\tilde{q}_i({\mathbf {x}},t)n_i= \tilde{\xi }({\mathbf {x}},t)\quad {on}\quad \varSigma _4\times [0,\infty ), \end{aligned}$$
(55)

where

$$\begin{aligned} & G_i({\mathbf {x}},t)= \varrho \tilde{f}_i({\mathbf {x}},t)+\varrho {\dot{u}}_i^0({\mathbf {x}}),\nonumber \\ & P({\mathbf {x}},t)= \frac{1}{T_0}\, \varrho \tilde{r}({\mathbf {x}},t)+ \beta _{ij}u_{i,j}^0({\mathbf {x}})+aT^0({\mathbf {x}}),\nonumber \\ & \varTheta ^0_i({\mathbf {x}})= \tau _T T_{,i}^0({\mathbf {x}})+ K_{ij}\left( \tau _q q_{j}^0({\mathbf {x}})+\frac{1}{2}\, \tau _q^2{\dot{q}}_{j}^0({\mathbf {x}})\right) . \end{aligned}$$
(56)

4 Analysis of the initial boundary value problem \(\mathcal {P}\)

4.1 First method

We address first the uniqueness problem of solutions of the initial boundary value problem \(\mathcal {P}\). Thus, we have

Theorem 1

Suppose that \(meas\, {\varSigma _3}\ne 0\) and

$$\begin{aligned}&\varrho >0,\quad k_{ij}\xi _i\xi _j\ge k_0\xi _i\xi _i, \quad k_0>0,\quad {for\; all }\quad \xi _i,\nonumber \\&\{0\le \frac{1}{2} \, \tau _q\le \tau _T\}\cup \{0<\tau _T<\frac{1}{2} \, \tau _q\}. \end{aligned}$$
(57)

Then the initial boundary value problem \(\mathcal {P}\) has at most one solution.

Proof

In order to prove the uniqueness result it is sufficient to prove that the zero external given data, that is \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0,T^0,q_i^0, {\dot{q}}_i^0;\omega _i,\psi _i,\vartheta ,\xi \}=0\), implies that the corresponding solution \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\) \(\eta ,q_i\}\) is vanishing on \(\overline{B}\times [0,\infty )\). That means we have to prove that the initial boundary value problem \(\mathcal {P}_0\) has only the banal solution.

Thus, we consider here that \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) is a solution of the initial boundary value problem \(\mathcal {P}_0\). In view of the Lemma 4, it follows that \({\mathcal {S}}^*=\{u_i^*,T^*,e_{ij}^*,\) \(t_{ij}^*,\eta ^*,q_i^*\}\) is a solution of the initial boundary value problem \(\mathcal {P}^*\) with zero given data, denoted in what follows by \(\mathcal {P}_0^*\).

We proceed first to establish an identity of Lagrange type for the solutions of the initial boundary value problem \(\mathcal {P}_0^*\) associated with \(\mathcal {P}_0\). To this end we start with the following identity for all \(t>0,s\in (0,t)\)

$$\begin{aligned}&\frac{\partial }{\partial s}\left[ \varrho u^{*}_i(t+s)\frac{\partial u^{*}_{i}}{\partial t}(t-s)+\varrho \frac{\partial {u}^{*}_{i}}{\partial t}(t+s) u^{*}_{i}(t-s)\right] \nonumber \\&\quad =\varrho \left[ \frac{\partial ^2 u^{*}_{i}}{\partial t^2}(t+s)u^{*}_{i}(t-s)-u^{*}_{i}(t+s)\frac{\partial ^2 u^{*}_{i}}{\partial t^2}(t-s)\right] , \end{aligned}$$
(58)

which integrated with respect to \(\left( {\mathbf {x}},s\right)\) over \(B\times (0,t)\) and by using the zero initial conditions (45), gives

$$\begin{aligned} 2\int _B\varrho u^{*}_{i}(t){\dot{u}}^{*}_{i}(t)dv &= \int _0^t\int _B\varrho \left[ \frac{\partial ^2 u^{*}_{i}}{\partial s^2}(t-s)u^{*}_{i}(t+s)\right. \nonumber \\&\left. \quad -\,u^{*}_{i}(t-s)\frac{\partial ^2 u^{*}_{i}}{\partial s^2}(t+s)\right] dvds. \end{aligned}$$
(59)

We now use the relations (39) and (44), the divergence theorem and the zero boundary conditions (46), in order to transform (59) into

$$\begin{aligned} 2\int _B\varrho u^{*}_{i}(t){\dot{u}}^{*}_{i}(t)dv &= \int _0^t\int _B \left[ e^{*}_{ij}(t-s)t^{*}_{ij}(t+s)\right. \nonumber \\& \quad \left. -\,e^{*}_{ij}(t+s)t^{*}_{ij}(t-s)\right] dvds. \end{aligned}$$
(60)

In view of the constitutive Eq. (41), we obtain

$$\begin{aligned} 2\int _B\varrho u^{*}_{i}(t){\dot{u}}^{*}_{i}(t)dv & = \int _0^t\int _B\left[ T^{*}(t-s)\beta _{ij}e^{*}_{ij}(t+s)\right. \nonumber \\ & \quad\left. -T^{*}(t+s)\beta _{ij}e^{*}_{ij}(t-s)\right] dvds, \end{aligned}$$
(61)

so that, by means of the relation (42), we get

$$\begin{aligned} 2\int _B\varrho u^{*}_{i}(t){\dot{u}}^{*}_{i}(t)dv & = \int _0^t\int _B\left[ T^{*}(t-s)\varrho \eta ^{*}(t+s)\right. \nonumber \\&\quad \left. -\,T^{*}(t+s)\varrho \eta ^{*}(t-s)\right] dvds. \end{aligned}$$
(62)

At this time we integrate the Eq. (40) with respect to time variable and use the zero initial conditions (45) to obtain

$$\begin{aligned} \varrho \eta ^{*}(t)=-\frac{1}{T_0}\int _0^t q^{*}_{i,i}(z)dz, \end{aligned}$$
(63)

so that, with the aid of the divergence theorem and the zero boundary conditions (46), from (62) we deduce

$$\begin{aligned} 2\int _B\varrho u^{*}_{i}(t){\dot{u}}^{*}_{i}(t)dv &= \int _0^t\int _B \frac{1}{T_0} \left[ T^{*}_{,i}(t-s)\int _0^{t+s}q_i^{*}(z)dz\right. \nonumber \\& \quad \left. -T^{*}_{,i}(t+s)\int _0^{t-s}q_i^{*}(z)dz\right] dvds. \end{aligned}$$
(64)

We use the Eqs. (43) into (64), to obtain

$$\begin{aligned} & 2\int _B\varrho u^{*}_{i}(t){\dot{u}}^{*}_{i}(t)dv = \frac{\tau _T}{T_0}\int _0^t\int _B\left[ k_{ij}T_{,i}^{*}(t+s)\right. \nonumber \\&\quad \left. \cdot \int _0^{t-s}T'_{,j}(z)dz-k_{ij}T^{*}_{,i} (t-s)\int _0^{t+s}T'_{,j}(z)dz\right] dvds\nonumber \\&+\frac{1}{T_0}\int _0^t\int _B\left[ k_{ij}T^{*}_{,i}(t+s) \int _0^{t-s}T''_{,j}(z)dz\right. \nonumber \\&\left. -k_{ij}T^{*}_{,i}(t-s)\int _0^{t+s}T''_{,j}(z)dz\right] dvds. \end{aligned}$$
(65)

Further, we use the notations (18) and (19) in order to write

$$\begin{aligned}&k_{ij}T_{,i}^{*}(t+s)\int _0^{t-s}T'_{,j}(z)dz-k_{ij}T^{*}_{,i}(t-s)\nonumber \\&\cdot \int _0^{t+s}T'_{,j}(z)dz =k_{ij}T''_{,j}(t-s)\left[ T''_{,i}(t+s)+\,\tau _q T'_{,i}(t+s)\right. \nonumber \\&\qquad \left. +\frac{1}{2}\, \tau _q^2 T_{,i}(t+s)\right] -k_{ij}T''_{,j}(t+s)\left[ T''_{,i}(t-s)\right. \nonumber \\&\qquad \left. +\tau _q T'_{,i}(t-s)+\frac{1}{2}\, \tau _q^2 T_{,i}(t-s)\right] \nonumber \\&\quad = \frac{\partial }{\partial s}\left[ \tau _q k_{ij}T''_{,j}(t-s)T''_{,i}(t+s)\right] \nonumber \\&\qquad +\frac{\partial }{\partial s}\left[ \frac{1}{2}\, \tau _q^2k_{ij}T'_{,i}(t+s)T''_{,j}(t-s)\right. \nonumber \\&\qquad \left. +\frac{1}{2}\, \tau _q^2k_{ij}T'_{,i}(t-s)T''_{,j}(t+s)\right] , \end{aligned}$$
(66)

and

$$\begin{aligned}&k_{ij}T^{*}_{,i}(t+s)\int _0^{t-s}T''_{,j}(z)dz-k_{ij}T^{*}_{,i}(t-s)\nonumber \\&\qquad \cdot \int _0^{t+s}T''_{,j}(z)dz= k_{ij}T'''_{,j}(t-s)\left[ T''_{,i}(t+s)\right. \nonumber \\&\qquad \left. +\tau _q T'_{,i}(t+s)+\frac{1}{2}\, \tau _q^2 T_{,i}(t+s)\right] -k_{ij}T'''_{,j}(t+s)\nonumber \\&\qquad \cdot \left[ T''_{,i}(t-s)+\tau _q T'_{,i}(t-s)+\frac{1}{2}\, \tau _q^2 T_{,i}(t-s)\right] \nonumber \\&\quad = \frac{\partial }{\partial s}\left[ k_{ij}T'''_{,j}(t-s)T'''_{,i}(t+s)\right] +\tau _q\frac{\partial }{\partial s}\left[ k_{ij}T''_{,j}(t+s)\right. \nonumber \\&\qquad \left. \cdot T'''_{,i}(t-s)+k_{ij}T''_{,j}(t-s)T'''_{,i}(t+s)\right] \nonumber \\&\qquad +\frac{1}{2}\, \tau _q^2\frac{\partial }{\partial s}\left[ k_{ij}T'_{,i}(t+s)T'''_{,j}(t-s)+k_{ij}T'_{,i}(t-s)\right. \nonumber \\&\qquad \left. \cdot T'''_{,j}(t+s)\right] +\frac{1}{2}\, \tau _q^2\frac{\partial }{\partial s}\left[ k_{ij}T''_{,i}(t+s)T''_{,j}(t-s)\right] . \end{aligned}$$
(67)

Finally, by replacing the relations (66) and (67) into identity (65), we obtain the following Lagrange identity

$$\begin{aligned}&\frac{d}{dt}\left\{ \int _B\varrho u^{*}_{i}(t) u^{*}_{i}(t)dv+\frac{\tau _T\tau _q^2}{2T_0}\int _Bk_{ij}T''_{,j}(t)T''_{,i}(t)dv\right. \nonumber \\&\quad \left. +\frac{\tau _q}{T_0}\int _Bk_{ij}T'''_{,j}(t)T'''_{,i}(t)dv+\frac{\tau _q^2}{T_0}\int _B k_{ij}T''_{,j}(t)T'''_{,i}(t)dv\right\} \nonumber \\&\quad +\frac{1}{T_0}\int _B k_{ij}T'''_{,j}(t)T'''_{,i}(t)dv\nonumber \\&\quad +\frac{\tau _q}{2T_0}\left( 2\tau _T-\tau _q\right) \int _B k_{ij}T''_{,i}(t)T''_{,j}(t)dv=0. \end{aligned}$$
(68)

Further, we integrate (68) twice with respect to time variable and use the zero initial conditions in order to obtain

$$\begin{aligned}&\int _0^t\int _B\varrho u^{*}_i(s)u^{*}_{i}(s)dvds+\frac{\tau _q}{T_0}\int _0^t\int _B k_{ij}T'''_{,i}(s)\nonumber \\&\quad \cdot T'''_{,j}(s)dvds+\frac{\tau _q^2}{2T_0}\int _B k_{ij}T'''_{,i}(t)T'''_{,j}(t)dv\nonumber \\&\quad +\frac{1}{T_0}\int _0^t\int _0^s\int _Bk_{ij}T'''_{,i}(z)T'''_{,j}(z)dvdzds\nonumber \\&\quad +\frac{\tau _T\tau ^2_q}{2T_0}\int _0^t\int _B k_{ij}T''_{,i}(s)T''_{,j}(s)dvds+\frac{\tau _q}{2T_0}\left( 2\tau _T-\tau _q\right) \nonumber \\&\quad \cdot \int _0^t\int _0^s\int _B k_{ij}T''_{,i}(z)T''_{,j}(z)dvdzds=0. \end{aligned}$$
(69)

Let us suppose first that

$$\begin{aligned} 0\le \frac{1}{2} \tau _q\le \tau _T. \end{aligned}$$
(70)

Then it follows that all integral terms in (69) are positive and hence (69) implies that

$$\begin{aligned} u^{*}_i({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ), \end{aligned}$$
(71)

and

$$\begin{aligned} T'''_{,i}({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ). \end{aligned}$$
(72)

The last relation implies

$$\begin{aligned} T_{,i}({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ), \end{aligned}$$
(73)

and since \(meas\, {\varSigma _3}\ne 0\), we can deduce that

$$\begin{aligned} T({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ). \end{aligned}$$
(74)

While the relation (71) and the Lemma 2 give

$$\begin{aligned} u_i({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ). \end{aligned}$$
(75)

If we substitute (73) into relation (43) then we get

$$\begin{aligned} q^{*}_{i}({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ), \end{aligned}$$
(76)

and therefore, by means of the Lemma 2, we have

$$\begin{aligned} q_i({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ). \end{aligned}$$
(77)

Consequently, we have \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}=0\) and so we have the uniqueness result.

Let us now consider the case when

$$\begin{aligned} 0< \tau _T < \frac{1}{2} \tau _q. \end{aligned}$$
(78)

Then the identity (69) implies that

$$\begin{aligned}&\frac{\tau _T\tau ^2_q}{2T_0}\int _0^t\int _B k_{ij}T''_{,i}(s)T''_{,j}(s)dvds\nonumber \\&\quad \le \frac{\tau _q}{2T_0}\left( \tau _q-2\tau _T\right) \int _0^t\int _0^s\int _B k_{ij}T''_{,i}(z)T''_{,j}(z)dvdzds, \end{aligned}$$
(79)

and hence we have

$$\begin{aligned} \varPhi (t)\le \left( \frac{1}{\tau _T}-\frac{2}{\tau _q}\right) \int _0^t\varPhi (s)ds,\quad t\in (0,\infty ), \end{aligned}$$
(80)

with

$$\begin{aligned} \varPhi (t)= \int _0^t\int _B k_{ij}T''_{,i}(s)T''_{,j}(s)dvds. \end{aligned}$$
(81)

By the Gronwall’s lemma, from the inequality (80) we deduce that

$$\begin{aligned} \varPhi (t)=0, \quad t\in (0,\infty ), \end{aligned}$$
(82)

and hence

$$\begin{aligned} T_{,i}({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ), \end{aligned}$$
(83)

and since \(meas\, {\varSigma _3}\ne 0\), we can obtain that

$$\begin{aligned} T({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ). \end{aligned}$$
(84)

Further, the identity (69) implies that

$$\begin{aligned} u^{*}_i({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ), \end{aligned}$$
(85)

and by using an argument like in the above case we obtain the uniqueness result again. Thus, the proof is complete.

Let us now address the question of continuous dependence of solutions of the initial boundary value problem \(\mathcal {P}\) with respect to the given data. To this aim we consider the solution \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) satisfying the initial boundary value problem \(\mathcal {P}\) corresponding to the data \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0,T^0,q_i^0,{\dot{q}}_i^0;0,0,0,0\}\) and we introduce the following functional

$$\begin{aligned} {\mathcal {E}}(t)= \frac{1}{2}\,\int _0^t\int _B\left[ \varrho \frac{\partial u_i^*}{\partial s}(s)\frac{\partial u_i^*}{\partial s}(s)+C_{ijkl}e^*_{ij}(s)e^*_{kl}(s)\right. \nonumber \\&\left. +\,a\left( T^*(s)\right) ^2\right] dvds+\frac{\tau _T+\tau _q}{2T_0} \,\int _0^t\int _Bk_{ij}T''_{,j}(s)T''_{,i}(s)dvds\nonumber \\&+\frac{1}{T_0}\, \int _0^t\int _0^s\int _Bk_{ij}T''_{,j}(z)T''_{,i}(z)dvdzds\nonumber \\&+\frac{\tau _q^2}{4T_0}\, \int _Bk_{ij}T''_{,j}(t)T''_{,i}(t)dv\nonumber \\&+\frac{\tau _T\tau _q^2}{4T_0}\, \int _0^t\int _Bk_{ij}T'_{,j}(s)T'_{,i}(s)dvds\nonumber \\&+\frac{\tau _q}{T_0}\left( \tau _T-\frac{\tau _q}{2}\right) \, \int _0^t\int _0^s\int _Bk_{ij}T'_{,j}(z)T'_{,i}(z)dvdzds, \end{aligned}$$
(86)

for all \(t\ge 0\). It is a straightforward task to verify the following result.

Lemma 6

Assume that the conductivity tensor \(k_{ij}\) is a positive definite tensor, the elasticity tensor is a positive semi-definite tensor and moreover, the following constitutive hypotheses hold

$$\begin{aligned} \varrho >0, \quad a>0, \quad 0\le \tau _q \le 2\tau _T. \end{aligned}$$
(87)

Then \(\mathcal {E}(t)\) can be considered as a measure of \({\mathcal {S}}=\{u_i,T, e_{ij},t_{ij},\eta ,q_i\}\) in the sense that \(\mathcal {E}(t)\ge 0\) for all \(t\ge 0\) and \(\mathcal {E}(t)=0\) for all \(t\ge 0\) implies that \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\) \(\eta ,q_i\}=0\).

The continuous dependence of solutions of the initial boundary value problem \(\mathcal {P}\), with respect to the initial data and the given body supplies, is described by the following result.

Theorem 2

Suppose that the constitutive hypotheses of Lemma 6 hold true and moreover, \(meas\, \varSigma _4=0\). Let \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) be a solution of the initial boundary value problem \(\mathcal {P}\) corresponding to the given data \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0,T^0,q_i^0,{\dot{q}}_i^0;0,0,0,0\}\). Then, for any finite time \(S>0\) and for every \(t\in [0,S]\), we have the following estimate describing the continuous dependence of solution with respect to the initial given data and with respect to the given supply terms

$$\begin{aligned} \sqrt{{\mathcal {E}}(t)}\le \left\{ \frac{S}{2}\, \int _B\left[ \varrho \frac{\partial u_i^*}{\partial t}(0)\frac{\partial u_i^*}{\partial t}(0)+C_{ijkl}e^*_{ij}(0)e^*_{kl}(0)\right. \right. \nonumber \\&\left. \left. +a\left( T^*(0)\right) ^2\right] dv\right\} ^{\frac{1}{2}} +\frac{1}{\sqrt{2}}\int _0^tb(s)ds, \end{aligned}$$
(88)

where

$$\begin{aligned} b(t)=\left( \int _0^t\int _B\left[ \frac{1}{\varrho }F_i(s)F_i(s) +\frac{1}{a}R^+(s)^2\right] dvds\right) ^{\frac{1}{2}}, \end{aligned}$$
(89)

and

$$\begin{aligned} R^+({\mathbf {x}},t)=\frac{1}{T_0} \left[ R({\mathbf {x}},t)-Q^0_{i,i}({\mathbf {x}},t)\right] . \end{aligned}$$
(90)

Proof

In view of Lemma 4 it follows that \({\mathcal {S}}^*=\{u_i^*,T^*,e_{ij}^*,t_{ij}^*,\eta ^*,q_i^*\}\) is a solution of the initial boundary value \(\mathcal {P}^*\) with \(\omega _i^*=0\), \(\psi _i^*=0\), \(\vartheta ^*=0\) and \(\xi ^*=0\). Then the relations (39)–(47) lead to the following identity

$$\begin{aligned}&\frac{d}{dt} \int _B \frac{1}{2}\, \left[ \varrho \, \frac{\partial u_i^*}{\partial t}(t)\frac{\partial u_i^*}{\partial t}(t)+C_{ijkl}e^*_{ij}(t)e^*_{kl}(t)\right. \nonumber \\&\quad \left. +a\left( T^*(t)\right) ^2\right] dv=\int _B\left[ F_i(t)\frac{\partial u^*_i}{\partial t}(t)+\frac{1}{T_0}R(t)T^*(t)\right] dv\nonumber \\&\quad +\int _B\frac{T^*_{,i}(t)}{T_0}\left[ -k_{ij}\left( T''_{,j}(t)+\tau _T T'_{,j}(t)\right) +Q^0_i(t)\right] dv. \end{aligned}$$
(91)

By using the divergence theorem and the zero boundary condition on \(\partial B\) we get

$$\begin{aligned} \int _B\frac{T^*_{,i}(t)}{T_0}Q^0_i(t)dv=-\, \int _B\frac{T^*(t)}{T_0}Q^0_{i,i}(t)dv. \end{aligned}$$
(92)

Moreover, we replace \(T^*_{,i}(t)\) by means of (19) in order to obtain

$$\begin{aligned}&-\, \frac{1}{T_0}\int _B k_{ij}T^*_{,i}(t)\left[ T''_{,j}(t)+\tau _T T'_{,j}(t)\right] dv\nonumber \\&\quad =\, -\, \frac{d}{dt}\left\{ \frac{\tau _T\tau _q^2}{4T_0}\int _B k_{ij}T'_{,j}(t)T'_{,i}(t)dv\right. \nonumber \\&\qquad \left. +\, \frac{\tau _T+\tau _q}{2T_0}\int _B k_{ij}T''_{,j}(t)T''_{,i}(t)dv\right. \nonumber \\&\qquad \left. +\frac{d}{dt}\left( \frac{\tau _q^2}{4T_0}\int _B k_{ij}T''_{,j}(t)T''_{,i}(t)dv\right) \right\} \nonumber \\&\qquad -\, \frac{\tau _q}{2T_0}\left( 2\tau _T-\tau _q\right) \int _B k_{ij}T'_{,j}(t)T'_{,i}(t)dv\nonumber \\&\qquad -\, \frac{1}{T_0}\int _B k_{ij}T''_{,j}(t)T''_{,i}(t)dv. \end{aligned}$$
(93)

Thus, by replacing the relations (92) and (93) into (91) and by integrating twice with respect to time variable, we are led to the following conservation law

$$\begin{aligned} {\mathcal {E}}(t) &= \frac{t}{2}\, \int _B\left[ \varrho \frac{\partial u_i^*}{\partial t}(0)\frac{\partial u_i^*}{\partial t}(0)+C_{ijkl}e^*_{ij}(0)e^*_{kl}(0)\right. \nonumber \\& \quad \left. +\, a\left( T^*(0)\right) ^2\right] dv+\int _0^t\int _0^s\int _B\left[ F_i(z)\frac{\partial u_i^*}{\partial z}(z)\right. \nonumber \\& \quad \left. +\,R^+(z)T^*(z)\right] dvdzds, \end{aligned}$$
(94)

for all \(t\in [0,S]\). By means of the Cauchy–Schwarz inequality, from (94) we obtain

$$\begin{aligned} {\mathcal {E}}(t) \,&\le\, \frac{S}{2}\, \int _B\left[ \varrho \frac{\partial u_i^*}{\partial t}(0)\frac{\partial u_i^*}{\partial t}(0)+C_{ijkl}e^*_{ij}(0)e^*_{kl}(0)\right. \nonumber \\& \quad \left. +\,a\left( T^*(0)\right) ^2\right] dv+\int _0^tb(s)\left[ \int _0^s\int _B\left( \varrho \frac{\partial u_i^*}{\partial z}(z)\frac{\partial u_i^*}{\partial z}(z)\right. \right. \nonumber \\& \quad \left. \left. +\,a\left( T^*(z)\right) ^2\right) dvdz\right] ^{\frac{1}{2}}ds, \end{aligned}$$
(95)

so that we have the following Gronwall inequality

$$\begin{aligned} {\mathcal {E}}(t) & \le \frac{S}{2}\, \int _B\left[ \varrho \frac{\partial u_i^*}{\partial t}(0)\frac{\partial u_i^*}{\partial t}(0)+C_{ijkl}e^*_{ij}(0)e^*_{kl}(0)\right. \nonumber \\& \quad \left. +\,a\left( T^*(0)\right) ^2\right] dv+\int _0^tb(s)\sqrt{2{\mathcal {E}}(s)}ds,\quad t\in [0,S]. \end{aligned}$$
(96)

By integrating the differential inequality (96) we obtain the estimate (88) and the proof is complete. \(\square\)

4.2 Second method

In this section we use the Lemma 5 in order to prove the uniqueness and the continuous dependence of solutions of the initial boundary value problem \(\mathcal {P}\). We first give a proof of the Theorem 1 based on the Lemma 5. In this aim we consider \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) be a solution of the initial boundary value problem \(\mathcal {P}_0\). In view of the Lemma 5, it follows that \(\tilde{\mathcal {S}}=\{\tilde{u}_i,\tilde{T},\tilde{e}_{ij}, \tilde{t}_{ij},\tilde{\eta },\tilde{q}_i\}\) is a solution of the initial boundary value problem \(\tilde{\mathcal {P}}\) with zero given data, denoted in what follows by \(\tilde{\mathcal {P}}_0\). Starting with a Lagrange identity of type (58) for the problem \(\tilde{\mathcal {P}}_0\), we obtain the following identity

$$\begin{aligned} \int _B2\varrho \tilde{u}_i(t)\dot{\tilde{u}}_i(t)dv & = \int _0^t \int _B\frac{1}{T_0}\left[ \tilde{T}_{,i}(t-s)\right. \nonumber \\& \quad \left. \cdot \int _0^{t+s}\tilde{q}_i(z)dz-\tilde{T}_{,i}(t+s) \int _0^{t-s}\tilde{q}_i(z)dz\right] dvds, \end{aligned}$$
(97)

which, when we replace \(\tilde{T}_{,i}\) from relation (52) and we substitute \(\tilde{q}_i\) by means of (20), gives

$$\begin{aligned} &\int _B2\varrho \tilde{u}_i(t)\dot{\tilde{u}}_i(t)dv = \frac{1}{T_0}\int _0^t\int _B\left[ K_{ij}q''_i(t-s)q'_j(t+s)\right. \nonumber \\& \quad \left. -K_{ij}q''_i(t+s)q'_j(t-s)\right] dvds\nonumber \\& \quad +\frac{\tau _q}{T_0}\int _0^t\int _B\left[ K_{ij}q''_i(t-s)q_j(t+s)\right. \nonumber \\& \quad\left. -\,K_{ij}q''_i(t+s)q_j(t-s)\right] dvds\nonumber \\& \quad +\frac{\tau _q^2}{2T_0}\int _0^t\int _B\left[ K_{ij}q''_i(t-s)\dot{q}_j(t+s)\right. \nonumber \\& \quad \left. -K_{ij}q''_i(t+s)\dot{q}_j(t-s)\right] dvds\nonumber \\& \quad +\frac{\tau _q\tau _T}{T_0}\int _0^t\int _B\left[ K_{ij}q'_i(t-s)q_j(t+s)\right. \nonumber \\& \quad \left. -K_{ij}q'_i(t+s)q_j(t-s)\right] dvds\nonumber \\& \quad +\frac{\tau _q^2\tau _T}{2T_0}\int _0^t\int _B\left[ K_{ij}q'_i(t-s)\dot{q}_j(t+s)\right. \nonumber \\& \quad \left. -K_{ij}q'_i(t+s)\dot{q}_j(t-s)\right] dvds. \end{aligned}$$
(98)

Further, we observe that

$$\begin{aligned}&K_{ij}q''_i(t-s)q'_j(t+s)-K_{ij}q''_i(t+s)q'_j(t-s)\nonumber \\&\quad =\frac{\partial }{\partial s}\left[ K_{ij}q''_i(t-s)q''_j(t+s)\right] , \end{aligned}$$
(99)
$$\begin{aligned}&K_{ij}q''_i(t-s)q_j(t+s)-K_{ij}q''_i(t+s)q_j(t-s)\nonumber \\&\quad =\frac{\partial }{\partial s}\left[ K_{ij}q''_i(t-s)q'_j(t+s)\right. \nonumber \\&\qquad \left. +K_{ij}q''_i(t+s)q'_j(t-s)\right] , \end{aligned}$$
(100)
$$\begin{aligned}&K_{ij}q''_i(t-s)\dot{q}_j(t+s)-K_{ij}q''_i(t+s)\dot{q}_j(t-s)\nonumber \\&\quad =\frac{\partial }{\partial s}\left[ K_{ij}q''_i(t-s)q_j(t+s)\right. \nonumber \\&\qquad \left. +K_{ij}q''_i(t+s)q_j(t-s)+K_{ij}q'_i(t-s)q'_j(t+s)\right] , \end{aligned}$$
(101)
$$\begin{aligned}&K_{ij}q'_i(t-s)q_j(t+s)-K_{ij}q'_i(t+s)q_j(t-s)\nonumber \\&\quad =\frac{\partial }{\partial s}\left[ K_{ij}q'_i(t-s)q'_j(t+s)\right] , \end{aligned}$$
(102)
$$\begin{aligned}&K_{ij}q'_i(t-s)\dot{q}_j(t+s)-K_{ij}q'_i(t+s)\dot{q}_j(t-s)\nonumber \\&\quad =\frac{\partial }{\partial s}\left[ K_{ij}q'_i(t-s)q_j(t+s)\right. \nonumber \\&\qquad \left. +K_{ij}q'_i(t+s)q_j(t-s)\right] . \end{aligned}$$
(103)

Finally, by substituting the relations (99)–(103) into (98) and then by integrating twice with respect to time variable the result, we get

$$\begin{aligned}&\int _0^t\int _B\varrho \tilde{u}_i(s)\tilde{u}_i(s)dvds\nonumber \\&\quad +\frac{1}{T_0}\int _0^t\int _0^s\int _BK_{ij}q''_i(z)q''_j(z)dvdzds\nonumber \\&\quad +\frac{\tau _q}{T_0}\int _0^t\int _BK_{ij}q''_i(s)q''_j(s)dvds\nonumber \\&\quad +\frac{\tau _q^2}{2T_0}\int _BK_{ij}q''_i(t)q''_j(t)dv\nonumber \\&\quad +\frac{\tau _T\tau _q^2}{2T_0}\int _0^t\int _BK_{ij}q'_i(s)q'_j(s)dvds+\frac{\tau _q}{2T_0}\left( 2\tau _T-\tau _q\right) \nonumber \\&\quad \cdot \int _0^t\int _0^s\int _BK_{ij}q'_i(z)q'_j(z)dvdzds=0. \end{aligned}$$
(104)

Under the assumption (70) all terms of the identity (104) are positive so that we conclude that

$$\begin{aligned} \tilde{u}_i({\mathbf {x}},t)=0\quad {in}\quad \overline{B}\times [0,\infty ), \end{aligned}$$
(105)

and

$$\begin{aligned} q''_i({\mathbf {x}},t)=0 \quad {in}\quad \overline{B}\times [0,\infty ). \end{aligned}$$
(106)

By means of Lemma 3, from (105) we readily obtain the conclusion (75), while (106) implies that

$$\begin{aligned} q_i({\mathbf {x}},t)=0 \quad {in}\quad \overline{B}\times [0,\infty ), \end{aligned}$$
(107)

and hence, by means of the constitutive Eq. (52), we obtain

$$\begin{aligned} \tilde{T}_{,i}({\mathbf {x}},t)=0 \quad \textit{in}\quad \overline{B}\times [0,\infty ). \end{aligned}$$
(108)

This last relation when coupled with the Lemma 3 implies the relation (74) and then we can obtain the uniqueness result described in Theorem 1.

When the relation (78) holds true, the identity (104) implies the inequality (80) where now \(\varPhi\) is defined by

$$\begin{aligned} \varPhi (t)=\int _0^t\int _BK_{ij}q'_i(s)q'_j(s)dvds, \end{aligned}$$
(109)

and therefore, we can follow the same way as in the proof of Theorem 1 to prove the uniqueness result.

Let us now address the continuous dependence question. Let \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) be a solution of the initial boundary value problem \(\mathcal {P}\) corresponding to the data \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0,T^0,q_i^0,{\dot{q}}_i^0;0,0,0,0\}\). In view of Lemma 5 it follows that \(\tilde{\mathcal {S}}=\{\tilde{u}_{i},\tilde{T},\tilde{e}_{ij},\tilde{t}_{ij},\tilde{\eta },\tilde{q}_{i}\}\) satisfies the initial boundary value problem \(\tilde{\mathcal {P}}\) with \(\tilde{\omega }_i=0\), \(\tilde{\psi }_i=0\), \(\tilde{\vartheta }=0\) and \(\tilde{\xi }=0\). Under these conditions we have the following conservation law

$$\begin{aligned} {\mathcal {F}}(t) &= \int _0^t\int _0^s\int _B\left\{ G_i(z)\frac{\partial \tilde{u}_i}{\partial z}(z)+P(z)\tilde{T}(z)\right. \nonumber \\&\left. +\,\frac{1}{T_0}\left[ q'_i(z)+\tau _T q_i(z)\right] \varTheta ^0_i\right\} dvdzds\nonumber \\&+\,\frac{t}{2}\, \int _B\left[ \varrho \frac{\partial \tilde{u}_i}{\partial t}(0)\frac{\partial \tilde{u}_i}{\partial t}(0)+C_{ijkl}\tilde{e}_{ij}(0)\tilde{e}_{kl}(0)\right. \nonumber \\&\left. +\,a\left( \tilde{T}(0)\right) ^2+\frac{\tau _T\tau _q^2}{2T_0}\, K_{ij}q_i(0)q_j(0)\right] dv, \end{aligned}$$
(110)

for all \(t\ge 0\), where

$$\begin{aligned} {\mathcal {F}}(t)= \frac{1}{2}\, \int _0^t\int _B\left[ \varrho \frac{\partial \tilde{u}_i}{\partial s}(s)\frac{\partial \tilde{u}_i}{\partial s}(s)+C_{ijkl}\tilde{e}_{ij}(s)\tilde{e}_{kl}(s)\right. \nonumber \\&\left. +\,a\left( \tilde{T}(s)\right) ^2\right] dvds+\frac{\tau _T+\tau _q}{2T_0}\int _0^t\int _BK_{ij}q'_i(s)q'_j(s)dvds\nonumber \\&+\frac{\tau _T\tau _q^2}{4T_0}\int _0^t\int _BK_{ij}q_i(s)q_j(s)dvds+\frac{\tau _q^2}{4T_0}\int _BK_{ij}q'_i(t)\nonumber \\&\cdot q'_j(t)dv+\frac{1}{T_0}\int _0^t\int _0^s\int _BK_{ij}q'_i(z)q'_j(z)dvdzds\nonumber \\&+\frac{\tau _q}{T_0}\left( \tau _T-\frac{\tau _q}{2}\right) \int _0^t\int _0^s\int _BK_{ij}q_i(z)q_j(z)dvdzds. \end{aligned}$$
(111)

Theorem 3

Suppose that the constitutive hypotheses of Lemma 6 hold true. Let \({\mathcal {S}}=\{u_i,T,e_{ij},t_{ij},\eta ,q_i\}\) be a solution of the initial boundary value problem \(\mathcal {P}\) corresponding to the given data \({\mathcal {D}}=\{f_i,r;u_i^0,{\dot{u}}_i^0,T^0,q_i^0,{\dot{q}}_i^0;\) \(0,0,0,0\}\). Then \(\mathcal {F}(t)\) can be considered as a measure for \(\mathcal {S}\) and moreover, for any finite time \(S>0\) and for every \(t\in [0,S]\), we have the following estimate

$$\begin{aligned} & \sqrt{{\mathcal {F}}(t)}\le \left[ \frac{S}{2}\, \int _B\left[ \varrho \frac{\partial \tilde{u}_i}{\partial t}(0)\frac{\partial \tilde{u}_i}{\partial t}(0)+C_{ijkl}\tilde{e}_{ij}(0)\tilde{e}_{kl}(0)\right. \right. \nonumber \\ & \quad \left. \left. +\,a\left( \tilde{T}(0)\right) ^2+\frac{\tau _T \tau _q^2}{2T_0}K_{ij}q_i(0)q_j(0)\right] dv\right] ^{\frac{1}{2}}\nonumber \\& \quad +\frac{1}{\sqrt{2}}\int _0^t\beta (s)ds, \end{aligned}$$
(112)

where

$$\begin{aligned} \beta (t)= \left( \int _0^t\int _B\left[ \frac{1}{\varrho }G_i(s)G_i(s) +\frac{1}{a}\left( P(s)\right) ^2\right. \right. \nonumber \\&\left. \left. +\frac{1}{T_0K_0}\left( \frac{1}{\tau _T+\tau _q}+ \frac{4\tau _T}{\tau ^2_q}\right) \varTheta _i^0\varTheta _i^0\right] dvds\right) ^{\frac{1}{2}}, \end{aligned}$$
(113)

and \(K_0\) is related to the lower eigenvalue of the tensor \(K_{ij}\).

Proof

By using the Cauchy–Schwarz inequality, from the identity (110) we obtain the following inequality

$$\begin{aligned} {\mathcal {F}}(t)\le \frac{S}{2}\int _B\left[ \varrho \frac{\partial \tilde{u}_i}{\partial t}(0)\frac{\partial \tilde{u}_i}{\partial t}(0)+C_{ijkl}\tilde{e}_{ij}(0)\tilde{e}_{kl}(0)\right. \nonumber \\&\left. +\,a\left( \tilde{T}(0)\right) ^2+\frac{\tau _T\tau _q^2}{2T_0} K_{ij}q_i(0)q_j(0)\right] dv\nonumber \\&+\int _0^t\beta (s)\sqrt{2{\mathcal {F}}(s)}ds, \end{aligned}$$
(114)

which, when integrated, leads to the estimate (112) and the proof is complete. \(\square\)

5 Concluding remarks

Our analysis establishes the uniqueness and continuous data dependence of solutions of the initial boundary value problems within the context of the time differential model of dual-phase-lag thermoelastic theory. The key point of our proof consists to introduce the two initial boundary value problems \(\mathcal {P}^*\) and \(\tilde{\mathcal {P}}\) inspired by the two operators involved into basic constitutive Eq. (1). The uniqueness results are established either when the conducting media is such that \(\tau _T>\tau _q\) that is when the heat flux vector precedes the temperature gradient in the time-history, implying that the heat flux vector is the cause while the temperature gradient is the effect of the heat flow as well as for media with \(\tau _q>\tau _T\), when the temperature gradient becomes the cause while the heat flux vector becomes the effect. It can be seen that our uniqueness theorems cannot cover the case when

$$\begin{aligned} \tau _T=0, \quad \tau _q>0. \end{aligned}$$
(115)

In fact, in this case the identity (69) becomes

$$\begin{aligned}&\int _0^t\int _B\varrho u^{*}_i(s)u^{*}_{i}(s)dvds\nonumber \\&\quad +\frac{\tau _q}{T_0}\int _0^t\int _B k_{ij}T'''_{,i}(s)T'''_{,j}(s)dvds\nonumber \\&\quad + \frac{\tau _q^2}{2T_0}\int _B k_{ij}T'''_{,i}(t)T'''_{,j}(t)dv\nonumber \\&\quad +\frac{1}{T_0}\int _0^t\int _0^s\int _Bk_{ij}T'''_{,i}(z)T'''_{,j}(z)dvdzds\nonumber \\&\quad - \frac{\tau _q^2}{2T_0}\int _0^t\int _0^s\int _B k_{ij}T''_{,i}(z)T''_{,j}(z)dvdzds=0, \end{aligned}$$
(116)

and it is no clear how we can handle it to get the uniqueness of solution. Namely, the last integral term in the identity (116) is negative one and it cannot be convenably related to the any other integral terms in order to get the uniqueness. Such a case remains an open problem and it is expected that it can lead to an ill-posed model. In fact, the case was studied by Fabrizio and Franchi [16] and it was concluded that such a model is not compatible with the Second Law of Thermodynamics. However, when \(\tau _T=0\), \(\tau _q>0\) and \(\tau _q^2\) can be neglected (that is for the Cattaneo–Maxwell model or the Lord-Shulman model), the identity (116) yet furnishes the uniqueness result.

The uniqueness of solutions of the mixed initial boundary value problem in the linear theory of thermoelasticity with dual phase-lags was established by Kothari and Mukhopadhyay [27] under more restrictive assumptions upon the thermoelastic coefficients (the elasticity tensor \(C_{ijkl}\) is positive definite and the specific heat is strictly positive) and upon the delay times (\(0<\tau _q<2\tau _T\)).

The continuous dependence of solutions with respect to the initial given data and given supply terms is described by the estimates of Theorems 2 and 3. They are established under the assumption \(0\le \tau _q\le 2\tau _T\) which agrees with the thermodynamic restrictions found by Fabrizio and Lazzari in [15]. We have to remark that the continuous dependence results can by obtained by means of the Lagrange identity method, as described in Sect. 4, under relaxed conditions upon the delay times, provided some suitable classes of solutions are considered. Finally, we have to mention that the conditions under which the continuous dependence results are established into Theorems 2 and 3 can be relaxed by replacing, for example, the condition that \(a>0\) by \(meas \, \varSigma _3\ne 0\).

Concluding, we can see that the thermodynamic restrictions found by Fabrizio and Lazzari [15] allow to prove the well position of the related initial boundary value problems.