1 Introduction
In [1], the authors derived a theory for a thermopiezoelectric body in which the second gradient of the displacement field and the second gradient of the electric potential are included in the set of independent constitutive variables. They obtained the thermodynamic restrictions and constitutive equations using the entropy production inequality proposed by Green and Laws in [3]. To overcome the problem of propagation of thermal waves with infinite velocity, inherent in Fourier’s law, Green and Laws use an entropy production inequality in which a new thermal function appears. This new formulation, in addition to overcoming the paradox of Fourier’s law, leads to a symmetric thermal conductivity tensor. Some theorems for this theory were derived in [2] and [4], in the case of centrosymmetric materials, using a reformulation of the problem with the help of the time convolution product.
The earliest works dealing with nonsimple elastic materials date back to Toupin [5, 6] and Mindlin [7]. Nonsimple materials are the ones whose constitutive functions depends on the second (and possibly higher) gradient of the dispacement. Toupin and Gazis [8] applied a theory of nonsimple materials, up to the second gradient, to the problem of surface deformation of a crystal. They showed that initial stresses and hypershear stresses in a uniform crystal give rise to a deformation of a thin boundary layer near a free surface, such as that observed in electron diffraction experiments. Some theories for nonsimple thermoelastic materials can be found in several articles (see, for example, Refs. [9‐17]).
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In addition, Kalpakides and Agiasofitou [13] established a theory of electroelasticity that includes both the strain gradient and the electric field gradient. In fact, considering the second gradient of the electric potential implies the presence of quadrupolar polarization in the continuum model, which is of practical interest for problems related to surface effects. The problem of electromagnetic field interaction with the motion of elastic solids has been the subject of important investigations (see, for example, Refs. [18‐24] and the literature cited therein). Some crystals, for example the quartz, when subjected to stress, become electrically polarized. Conversely, an external electromagnetic field can produce deformation in a piezoelectric crystal. These two effects are called direct and inverse piezoelectric effect, respectively.
Continuous data dependence is of practical and numerical importance, given that physical measurements introduce unavoidable errors [25]. Without continuous dependence, a small difference in initial or given data can lead to completely different solution of the initial-boudary value problem. Continuous dependence problems were discussed in [26‐28] and express the possibility of making the solution of the problem vary, continuously, from initial data, boundary data and external data.
In Sect. 2, we begin by summarizing the fundamental equations based on the linear theory of nonsimple thermopiezoelectric materials established in [1], defining a mixed initial-boundary value problem under nonhomogeneous initial conditions and presenting a reciprocity relation involving two processes at different times that extends that shown in [2]. In Sect. 3, by using a key relation involving external data, initial data and boundary data, we establish two continuous dependence theorems.
2 Basic equations and preliminaries
We consider a body that at some instant occupies the region \({\mathcal {B}}\) of the Euclidean three-dimensional space and is bounded by the piecewise smooth surface \(\partial {\mathcal {B}}\). The motion of the body is referred to the reference configuration \({\mathcal {B}}_0\) and a fixed system of rectangular Cartesian axes \(Ox_i\) (\(i = 1, 2, 3\)). We shall employ the usual summation and differentiation conventions: Latin subscripts are understood to range over the integers (\(1, 2, 3\)), summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate. In what follows we use a superposed dot to denote partial differentiation with respect to time \(t\). We consider the linear theory of thermopiezoelectricity of non simple materials established in [1] adopting the entropy production inequality proposed by Green and Laws [3] (in particular, Sect. 2), in local formwhere the function \(\phi\) is a suitable defined strictly positive thermal function. The fundamental system of field equations consists of the equations of motionthe equation for the quasi-static electric fieldthe energy equation in the linear approximationthe constitutive equationswhere \(c=a^{(44)}\), and the geometric equationswhere \(\tau _{ji}=t_{ji}+\mu _{kji,k}\) and \(\sigma _{i}=D_{i}+Q_{ji,j}\).
$$\rho \phi \dot{\eta } \ge - q_{{i,i}} + \rho h + \frac{1}{\phi }q_{i} \phi _{{,i}} ,$$
$$\begin{aligned} \tau _{ji,j}-\mu _{kji,kj}+\rho f_{i}=\rho \ddot{u}_{i}, \end{aligned}$$
(1)
$$\begin{aligned} \sigma _{i,i}-Q_{ji,ji}=g, \end{aligned}$$
(2)
$$\begin{aligned} \rho T_0{\dot{\eta }}=-q_{i,i}+\rho h, \end{aligned}$$
(3)
$$\begin{aligned} \tau _{ij}&= a_{ijkl}^{(11)}e_{kl} + a_{ijklh}^{(12)}\kappa _{klh} + a_{ijk}^{(13)}E_k \\& \quad + a_{ijkl}^{(17)}V_{kl} + a_{ij}^{(14)} ( \theta + \beta {\dot{\theta }} ), \\ \mu _{ijk}&= a_{lhijk}^{(12)}e_{lh} + a_{ijklhm}^{(22)}\kappa _{lhm} + a_{ijkl}^{(23)}E_l\\& \quad + a_{ijklh}^{(27)}V_{lh} + a_{ijk}^{(24)} ( \theta + \beta {\dot{\theta }} ), \\ \quad -\sigma _i&= a_{jki}^{(13)}e_{jk} + a_{jkli}^{(23)}\kappa _{jkl} + a_{ij}^{(33)}E_j\\&\quad + a_{ijk}^{(37)}V_{jk} + a_i^{(34)} ( \theta + \beta {\dot{\theta }} ), \\ -Q_{ij}&= a_{klij}^{(17)}e_{kl} + a_{klmij}^{(27)}\kappa _{klm} + a_{kij}^{(37)}E_k\\& \quad + a_{ijkl}^{(77)}V_{kl} + a_{ij}^{(47)} ( \theta + \beta {\dot{\theta }}), \\ \quad -\rho \eta&= a_{ij}^{(14)}e_{ij} + a_{ijk}^{(24)}\kappa _{ijk} + a_i^{(34)}E_i\\& \quad + a_{ij}^{(47)}V_{ij} + c ( \theta + \beta {\dot{\theta }} ) +\frac{1}{\beta }\!\left( \gamma {\dot{\theta }} + a_i^{(56)}g_i \right) , \\ \quad -\frac{q_i}{T_0}&= \frac{1}{\beta } \left( a_i^{(56)}{\dot{\theta }} + a_{ij}^{(66)}g_j \right) , \end{aligned}$$
(4)
$$\begin{aligned} \begin{alignedat}{2}&e_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}),&\kappa _{ijk}=u_{k,ij},\\&E_i=-\varphi _{,i}, \qquad V_{ij}=-\varphi _{,ij},&g_i = \theta _{,i}, \end{alignedat} \end{aligned}$$
(5)
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We denote with \(t_{ij}\) the stress tensor, \(\mu _{ijk}\) the hyperstress tensor, \(\rho\) the reference mass density, \(f_i\) the external body force per unit mass, \(u_i\) the displacement vector field, \(D_i\) the displacement electric vector field, \(q_i\) the heat flux vector, \(Q_{ij}\) the electric quadrupole, \(g\) the density of free charges, \(\varphi\) the electric potential, \(\eta\) the entropy per unit mass, \(h\) the heat supply per unit mass and unit time and \(\theta\) the difference of the absolute temperature \(T\) from the absolute temperature in the reference configuration \(T_{0}>0\) (i.e. \(\theta =T-T_{0}\)). Following [29, 30], for the traction vector \(t_i\), the hypertraction vector \(\mu _{ji}\), the heat flux \(q\) and the the generalized surface charge density \(Q_{i}\) we havewhere \(n_i\) is the outward unit normal vector to the boundary surface \(\partial {\mathcal {B}}\). The constitutive coefficients satisfy the following symmetry relations:For sake of simplicity we set \(k=\gamma /\beta\), \(k_{i}=\frac{1}{\beta }a_{i}^{(56)}\) and \(k_{ij}=\frac{1}{\beta }a_{ij}^{(66)}\). From the entropy inequality the following restriction has been obtainedThen it is necessary that \(k\) be non negative and the thermal conductivity tensor \(k_{ij}\) be positive semi-definite.
$$\begin{aligned} \begin{alignedat}{2}&t_i = t_{ji} n_j, \quad&\mu _{ji}&= \mu _{kji}n_k,\\&q = q_i n_i, \quad&{\dot{Q}}_i&= {\dot{Q}}_{ji}n_j, \end{alignedat} \end{aligned}$$
$$\begin{aligned} \begin{alignedat}{2}&a_{ijkl}^{(11)} = a_{jikl}^{(11)} = a_{klij}^{(11)},\quad{} & {} a_{ijk}^{(13)} = a_{jik}^{(13)}, \\&a_{ijklh}^{(12)} = a_{jiklh}^{(12)} = a_{ijlkh}^{(12)}, \quad{} & {} a_{ij}^{(14)} = a_{ji}^{(14)},\\&a_{ijklhm}^{(22)} = a_{jiklhm}^{(22)} = a_{lhmijk}^{(22)}, \quad{} & {} a_{ij}^{(15)} = a_{ji}^{(15)},\\&a_{ijkl}^{(17)} = a_{jikl}^{(17)} = a_{klij}^{(17)}, \quad{} & {} a_{ijkl}^{(23)} = a_{jikl}^{(23)},\\&a_{ijkhl}^{(27)} = a_{jikhl}^{(27)} = a_{ijklh}^{(27)},\quad{} & {} a_{ijk}^{(24)} = a_{jik}^{(24)},\\&a_{ijkl}^{(77)} = a_{jikl}^{(77)} = a_{klij}^{(77)}, \quad{} & {} a_{ijk}^{(25)} = a_{jik}^{(25)},\\&a_{ij}^{(57)} = a_{ji}^{(57)},{} & {} a_{ijkl}^{(26)} = a_{jikl}^{(26)},\\&a_{ij}^{(66)} = a_{ji}^{(66)}, \quad{} & {} a_{ij}^{(33)} = a_{ji}^{(33)},\\&a_{ijk}^{(67)} = a_{ikj}^{(67)},\quad{} & {} a_{ijk}^{(37)} = a_{ikj}^{(37)},\\&\quad{} & {} a_{ij}^{(47)} = a_{ji}^{(47)}. \end{alignedat} \end{aligned}$$
(6)
$$\begin{aligned} k{\dot{\theta }}^2+2k_i{\dot{\theta }}g_{i}+k_{ij}g_ig_j\ge 0,\qquad \forall \,{\dot{\theta }},g_i. \end{aligned}$$
(7)
Following [6, 7], we consider \(P_i\), \(R_i\), \(\Lambda\) and \(H\) defined in such a way that the total rate of work of the surface forces over the smooth surface \(\partial {\mathcal {P}}\) can be expressed in the formwhere \({\mathcal {P}}\subseteq {\mathcal {B}}\) is a part of the body and \(\partial {\mathcal {P}}\) its boundary. We haveandwhere \({\mathcal {D}} \equiv n_i \partial / \partial x_i\) is the normal derivative andis the surface gradient. Now, we denote withthe solutions of the mixed initial-boundary value problem \(\Pi\) defined by Equations (1)–(5) with the restrictions (7), where the coefficients satisfy the symmetries (6), the following initial conditionsin \(\bar{{\mathcal {B}}}\) and the following boundary conditionsIn the following, instead of \(\vartheta ^0\) we will use \(\eta ^0\) defined as the initial value of the entropy obtained from (4)\(_5\) evaluated at \(t=0\). In the previous initial and boundary conditions \(u_i^0\), \(v_i^0\), \(\theta ^0\), \(\eta ^0\), \({\hat{u}}_i\), \({\hat{d}}_i\), \({\hat{\varphi }}\), \({\hat{\xi }}\), \({\hat{\theta }}\), \({\hat{P}}_i\), \({\hat{R}}_i\), \({\hat{\Lambda }}\), \({\hat{H}}\) and \({\hat{q}}\) are prescribed functions and the surfaces \(S_i\) and \(\Sigma _i\) such thatwhere the closure is relative to \(\partial {\mathcal {B}}\). The (external) data of the mixed initial-boundary value problem in concern areFollowing [2], we define an ordered array of functionsas an admissible process on \(\bar{{\mathcal {B}}}\times I\) with the following properties We say that \(\pi\) is a process corresponding to the supply terms \((f_{i},g,h)\) if \(\pi\) is an admissible process that satisfies the fundamental system of field Eqs. (1)–(5) with the restriction (7) on \({\mathcal {B}}\times I\), where the coefficients must satisfy the symmetries (6) and where \(I=[0,t_1]\) is a bounded time interval. Then, if a process \(\pi\) satisfies the initial conditions (8) and the boundary conditions (9), we identify it as a solution of the mixed initial-boundary value problem \(\Pi\). Henceforth, the dependence on \({\textbf{x}}\) is implicit, while the dependence on the time variable might be explicit. We consider the following notations for any function \(f\)
$$\begin{aligned} \begin{aligned} \int _{\partial {\mathcal {P}}}&\left( t_{ki}{\dot{u}}_i + \mu _{kji}{\dot{u}}_{i,j} - {\dot{D}}_k\varphi - {\dot{Q}}_{ki}\varphi _{,i} \right) n_k~da\\&=\int _{\partial {\mathcal {P}}} \left( P_i{\dot{u}}_i + R_i{\mathcal {D}}{\dot{u}}_i - {\dot{\Lambda }}\varphi - {\dot{H}}{\mathcal {D}}\varphi \right) da \end{aligned} \end{aligned}$$
$$\begin{aligned}&P_i = ( \tau _{ji} - \mu _{kji,k} ) n_j - {\mathcal {D}}_j ( \mu _{kji} n_k ) + ( {\mathcal {D}}_l n_l ) \mu _{kji} n_k n_j, \\&\Lambda = ( \sigma _j - Q_{kj,k} ) n_j - {\mathcal {D}}_j ( Q_{kj} n_k ) + ( {\mathcal {D}}_l n_l ) Q_{kj} n_k n_j, \end{aligned}$$
$$\begin{aligned} R_i = \mu _{kji} n_k n_j, \qquad H = Q_{kj} n_k n_j, \end{aligned}$$
$$\begin{aligned} {\mathcal {D}}_i \equiv ( \delta _{ij} - n_i n_j ) \frac{\partial }{\partial x_j} \end{aligned}$$
$$\begin{aligned} {\mathcal {U}} = ( u_i, \varphi , \theta ) \end{aligned}$$
$$\begin{array}{*{20}l} {u_{i} (0) = u_{i}^{0} ,} \hfill & {\dot{u}_{i} (0) = v_{i}^{0} ,} \hfill \\ {\theta (0) = \theta ^{0} ,} \hfill & {\dot{\theta }(0) = \vartheta ^{0} ,} \hfill \\ \end{array}$$
(8)
$$\begin{array}{*{20}l} {u_{i} = \hat{u}_{i} \;{\text{on}}\;S_{1} \times I,} \hfill & {P_{i} = \hat{P}_{i} \;{\text{on}}\;\Sigma _{1} \times I,} \hfill \\ {{\mathcal{D}}u_{i} = \hat{d}_{i} \;{\text{on}}\;S_{2} \times I,} \hfill & {R_{i} = \hat{R}_{i} \;{\text{on}}\;\Sigma _{2} \times I,} \hfill \\ {\varphi = \hat{\varphi }\;{\text{on}}\;S_{3} \times I,} \hfill & {\Lambda = \hat{\Lambda }\;{\text{on}}\;\Sigma _{3} \times I,} \hfill \\ {{\mathcal{D}}\varphi = \hat{\xi }\;{\text{on}}\;S_{4} \times I,} \hfill & {H = \hat{H}\;{\text{on}}\;\Sigma _{4} \times I,} \hfill \\ {\theta = \hat{\theta }\;{\text{on}}\;S_{5} \times I,} \hfill & {q = \hat{q}\;{\text{on }}\Sigma _{5} \times I.} \hfill \\ \end{array}$$
(9)
$$\begin{aligned} {\bar{S}}_i \cup \Sigma _i = \partial {\mathcal {B}}, \quad \quad S_i \cap \Sigma _i = \varnothing , \qquad i=1, \ldots , 5, \end{aligned}$$
$$\begin{aligned} \begin{aligned} \Gamma = \big \{&f_i, g, h, u_i^0, v_i^0, \theta ^0, \eta ^0,\\&{\hat{u}}_i, {\hat{d}}_i, {\hat{\varphi }}, {\hat{\xi }}, {\hat{\theta }}, {\hat{P}}_i, {\hat{R}}_i, {\hat{\Lambda }}, {\hat{H}}, {\hat{q}} \big \}. \end{aligned} \end{aligned}$$
$$\begin{aligned}\begin{aligned} \pi =(&u_{i},\varphi ,\theta ,e_{ij},\kappa _{ijk},E_{i},V_{ij},\\ {}&g_i,\tau _{ij},\mu _{ijk},\sigma _{i},Q_{ij},\eta ,q_{i}) \end{aligned} \end{aligned}$$
1.
\(u_{i}\in C^{4,2}(\bar{{\mathcal {B}}}\times I)\), \(\varphi \in C^{4,0}(\bar{{\mathcal {B}}}\times I)\), \(\theta \in C^{2,2}(\bar{{\mathcal {B}}}\times I)\), \(e_{ij},V_{ij}\in C^{2,1}(\bar{{\mathcal {B}}}\times I)\), \(\eta \in C^{0,1}(\bar{{\mathcal {B}}}\times I),\) \(\kappa _{ijk},E_{i},\mu _{ijk},Q_{ij}\in C^{2,0}(\bar{{\mathcal {B}}}\times I)\), \(g_i,\tau _{ij},q_{i}\in C^{1,0}(\bar{{\mathcal {B}}}\times I)\);
2.
\({e_{ij}=e_{ji}}\), \(\kappa _{ijk}=\kappa _{jik}\), \(\,V_{ij}=V_{ji}\), \(\tau _{ji}=\tau _{ij}\), \(\mu _{kji}=\mu _{jki}\), \(Q_{ji}=Q_{ij}\) on \(\bar{{\mathcal {B}}}\times I\).
$$\begin{aligned} Af=f+\beta {\dot{f}},\qquad {\bar{f}}(t)=\int _{0}^{t}f(\tau )d\tau ,\qquad \forall t\in I. \end{aligned}$$
Lemma 1
The proof of this lemma is immediate.
We consider the body \({\mathcal {B}}\) subjected to two different external data systemswith \(\alpha =1,2\), and we denote the corresponding solutions of the mixed initial-boundary problem value with
$$\begin{aligned}\begin{aligned} \Gamma ^{(\alpha )} = \biggl (&f_{i}^{(\alpha )}, g^{(\alpha )}, h^{(\alpha )}, u_{i}^{0(\alpha )}, v_{i}^{0(\alpha )}, \theta ^{0(\alpha )}, \eta ^{0(\alpha )}, \\&{\hat{u}}_{i}^{(\alpha )}, {\hat{d}}_{i}^{(\alpha )}, {\hat{\varphi }}^{(\alpha )}, {\hat{\xi }}^{(\alpha )}, {\hat{\theta }}^{(\alpha )},\\&{\hat{P}}_{i}^{(\alpha )}, {\hat{R}}_{i}^{(\alpha )}, {\hat{\Lambda }}^{(\alpha )}, {\hat{H}}^{(\alpha )}, {\hat{q}}^{(\alpha )} \biggr ), \end{aligned} \end{aligned}$$
$$\begin{aligned}\begin{aligned} \pi ^{(\alpha )} = (&u_{i}^{(\alpha )}, \varphi ^{(\alpha )}, \theta ^{(\alpha )}, e_{ij}^{(\alpha )}, \kappa _{ijk}^{(\alpha )}, E_{i}^{(\alpha )}, V_{ij}^{(\alpha )},\\&g_i^{(\alpha )}, \tau _{ij}^{(\alpha )}, \mu _{ijk}^{(\alpha )}, \sigma _{i}^{(\alpha )}, Q_{ij}^{(\alpha )}, \eta ^{(\alpha )}, q_{i}^{(\alpha )}). \end{aligned} \end{aligned}$$
Theorem 2
Suppose that the symmetry relations (6) hold. Let for all \(r,s\in I\) and \(\alpha ,\beta \in \{1,2\}\). Then
$$\begin{aligned} \begin{aligned}&\Omega _{\alpha \beta }(r,s)=\int _{{\mathcal {B}}}\biggl [\rho f_{i}^{(\alpha )}(r)u_{i}^{(\beta )}(s)-g^{(\alpha )}(r)\varphi ^{(\beta )}(s)\\&-\frac{1}{T_{0}}{\mathcal {R}}^{(\alpha )}(r)A\theta ^{(\beta )}(s)\biggr ]dv+\int _{\partial {\mathcal {B}}}\biggl [P_{i}^{(\alpha )}(r)u_{i}^{(\beta )}(s)\\&+R_{i}^{(\alpha )}(r){\mathcal {D}}u_{i}^{(\beta )}(s)+\Lambda ^{(\alpha )}(r)\varphi ^{(\beta )}(s)\\&\qquad +H^{(\alpha )}(r){\mathcal {D}}\varphi ^{(\beta )}(s)+\frac{1}{T_{0}}{\bar{q}}^{(\alpha )}(r)A\theta ^{(\beta )}(s)\biggr ]da\\&-\int _{{\mathcal {B}}}\biggl [k{\dot{\theta }}^{(\alpha )}(r)\theta ^{(\beta )}(s)+k_ig_i^{(\alpha )}(r)A\theta ^{(\beta )}(s)\\&\qquad +\rho \ddot{u}_{i}^{(\alpha )}(r)u_{i}^{(\beta )}(s)+\frac{1}{T_0}{\bar{q}}_i^{(\alpha )}(r)Ag_i^{(\beta )}(s)\biggr ]dv \end{aligned} \end{aligned}$$
(10)
$$\begin{aligned} \Omega _{\alpha \beta }(r,s)=\Omega _{\beta \alpha }(s,r). \end{aligned}$$
(11)
Proof
We introduce the following functionsandTaking into account the constitutive equations (4)\(_{1-5}\) and the symmetry relations (6) we havefor all \(r,s\in I\) and \(\alpha ,\beta \in \{1,2\}\). On the other hand, in view of Eqs. (1), (2), (6), (12) and Lemma 1, we can writeTaking into account Eqs. (10), (13), (15) and using the divergence theorem, we easily obtainFrom the symmetry relation (14) we arrive to the thesis. \(\square\)
$$\begin{aligned} \begin{aligned}&\Gamma _{\alpha \beta }(r,s)= \tau _{ji}^{(\alpha )}(r)e_{ij}^{(\beta )}(s)+\mu _{kji}^{(\alpha )}(r)\kappa _{kji}^{(\beta )}(s)\\&\qquad -\sigma _{i}^{(\alpha )}(r)E_{i}^{(\beta )}(s)-Q_{ji}^{(\alpha )}(r)V_{ij}^{(\beta )}(s)\\&\qquad -\rho \eta ^{(\alpha )}(r)A\theta ^{(\beta )}(s),\qquad \forall r,s\in I, \end{aligned} \end{aligned}$$
(12)
$$\begin{aligned} \begin{aligned}&I_{\alpha \beta }(r,s)=\Gamma _{\alpha \beta }(r,s)\\&\qquad -\Bigl (k_ig_i^{(\alpha )}(r)A\theta ^{(\beta )}(s)+k{\dot{\theta }}^{(\alpha )}(r)\theta ^{(\beta )}(s)\Bigr ). \end{aligned} \end{aligned}$$
(13)
$$\begin{aligned} I_{\alpha \beta }(r,s)=I_{\beta \alpha }(s,r), \end{aligned}$$
(14)
$$\begin{aligned} \begin{aligned} \Gamma _{\alpha \beta }&(r,s)=\Biggl [t_{ji}^{(\alpha )}(r)u_{i}^{(\beta )}(s)+\mu _{kji}^{(\alpha )}(r)u_{i,k}^{(\beta )}(s)\\&+D_j^{(\alpha )}(r)\varphi ^{(\beta )}(s)+Q_{ji}^{(\alpha )}(r)\varphi _{,i}^{(\beta )}(s)\\&+\frac{1}{T_0}{\bar{q}}_j^{(\alpha )}(r)A\theta ^{(\beta )}(s)\Biggr ]_{,j}+\rho f_i^{(\alpha )}(r)u_i^{(\beta )}(s)\\&-\rho \ddot{u}_i^{(\alpha )}(r)u_i^{(\beta )}(s)-g^{(\alpha )}(r)\varphi ^{(\beta )}(s)\\&-\frac{1}{T_0}{\bar{q}}_i^{(\alpha )}(r)Ag_i^{(\beta )}(s)-\frac{1}{T_0}{\mathcal {R}}^{(\alpha )}(r)A\theta ^{(\beta )}(s). \end{aligned} \end{aligned}$$
(15)
$$\begin{aligned} \int _{{\mathcal {B}}} I_{\alpha \beta }(r,s)\,dv=\Omega _{\alpha \beta }(r,s). \end{aligned}$$
3 Continuous dependence theorems
In this section we prove that the result established in the last section leads to two continuous dependence theorems of the solution of the mixed problem \(\Pi\) on the external body forces, electric charge density and heat supply \(\{f_i,g,h\}\), called supply terms, and on the initial data \(\{u^0_i, v^0_i,\theta ^0,\eta ^0\}\).
In all this section, we restrict our attention to the case of centrosymmetric materials, which present a symmetry with respect to the inversion of axes, and this leads to the vanishing of all tensors of odd order, see eq. (6) of [2]. Proceeding as in the proof of the uniqueness theorem in [2], the following theorem can be proved.
Theorem 3
Suppose that the symmetry relations (6) hold. Let \(\pi\) be a solution corresponding to the external data system \(\Gamma\) and letfor all \(r,s\in I\). Then
$$\begin{aligned} \begin{aligned} J(r,s)&=\int _{{\mathcal {B}}}\biggl [\rho f_{i}(r)u_{i}(s)-g(r)\varphi (s)\\&-\frac{1}{T_{0}}{\mathcal {R}}(r)A\theta (s)\biggr ]dv+\int _{\partial {\mathcal {B}}}\biggl [P_{i}(r)u_{i}(s)\\&+R_{i}(r){\mathcal {D}}u_{i}(s)+\Lambda (r)\varphi (s)\\&+H(r){\mathcal {D}}\varphi (s)+\frac{1}{T_{0}}{\bar{q}}(r)A\theta (s)\biggr ]da \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} \frac{d}{dt}&\Bigl (\int _{{\mathcal {B}}}\Bigl [\rho u_{i}u_{i}+\beta k_{ij}{\bar{g}}_i{\bar{g}}_j\Bigr ]\,dv\\&\quad +\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [k\theta ^{2}+ k_{ij}{\bar{g}}_i{\bar{g}}_j\Bigr ]\,dvd\tau \Bigr )\\&=\int _{0}^{t}[J(t-\tau ,t+\tau )-J(t+\tau ,t-\tau )]\,d\tau \\&\quad +\int _{{\mathcal {B}}}[\rho u_{i}^{0}{\dot{u}}_{i}(2t)+\rho v_{i}^{0}u_{i}(2t)\\&\qquad + k\theta ^{0}\theta (2t) +\beta k_{ij}g_i^0{\bar{g}}_j(2t)]\,dv. \end{aligned} \end{aligned}$$
(16)
3.1 Continuous dependence on supply terms
First, we consider the dependence of the solution on supply terms. We consider two external data systems which differ only by the external body force, electric charge density and heat supplywith \(\alpha =1,2,\) and let \(\pi ^{(\alpha )}\) be two solutions of the initial-boundary value problems corresponding to \(\Gamma ^{(\alpha )}\) respectively. We defineThen, the difference of the two solutions \(\pi =\pi ^{(2)}-\pi ^{(1)}\) represents a solution of the mixed problem \(\Pi\) for null boundary data, null initial data and for the external body forces, electric charge density and heat supply in (17). By the application of Eq. (16) for the solution \(\pi\) we deducefor every \(t\in [0,t_1/2]\). Furthermore, for the next results we will assume that the constitutive coefficient \(\beta\) satisfies the following assumptionWe shall use the identity (18) in order to prove the following continuous dependence theorem.
$$\begin{aligned} \begin{aligned} \Gamma ^{(\alpha )}=\{&f_i^{(\alpha )}, g^{(\alpha )}, h^{(\alpha )}, u_i^0, v_i^0, \theta ^0, \eta ^0,\\&{\hat{u}}_i, {\hat{d}}_i, {\hat{\theta }}, {\hat{\varphi }}, {\hat{\xi }}, {\hat{P}}_i, {\hat{R}}_i, {\hat{q}}, {\hat{\Lambda }}, {\hat{H}}\}, \end{aligned}\end{aligned}$$
$$\begin{aligned} \begin{alignedat}{2}&f_i{} & {} =f_i^{(2)}-f_i^{(1)},\\&g{} & {} =g^{(2)}-g^{(1)},\\&h{} & {} =h^{(2)}-h^{(1)}. \end{alignedat} \end{aligned}$$
(17)
$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\Bigl (\int _{{\mathcal {B}}}\Bigl [\rho u_{i}u_{i}+\beta k_{ij}{\bar{g}}_i{\bar{g}}_j\Bigr ]\,dv\\&\quad +\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [k\theta ^{2}+ k_{ij}\bar{g_i}\bar{g_j}\Bigr ]\,dvd\tau \Bigr )\\&=\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [\rho f_{i}(t-\tau )u_{i}(t+\tau ) -\rho f_{i}(t+\tau )u_{i}(t-\tau )\\&\qquad -g(t-\tau )\varphi (t+\tau ) +g(t+\tau )\varphi (t-\tau )\Bigr ]\,dvd\tau \\&\quad +\frac{1}{T_0}\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [\rho {\bar{h}}(t+\tau )A\theta (t-\tau )\\&\qquad -\rho {\bar{h}}(t-\tau )A\theta (t+\tau )\Bigr ]\, dvd\tau \end{aligned} \end{aligned}$$
(18)
$$\begin{aligned}\exists \, \beta _0>0:\,|\beta |\le \beta _0\quad \forall {\textbf{x}}\in {\mathcal {B}}.\end{aligned}$$
Theorem 4
We suppose that there exists \(t^*\in (0,t_1)\) and some positive constants \(B\), \(C\), \(D\) and \(E\) such thatThen, the following inequality holds:for every \(t\in [0,t^*/2].\)
$$\begin{aligned} \int _0^{t^*}\!\!\!\int _{{\mathcal {B}}} \rho u_iu_i\, dvd\tau \le B^2,\quad \int _0^{t^*}\!\!\!\int _{{\mathcal {B}}} \varphi ^2 dvd\tau \le C^2, \end{aligned}$$
(19)
$$\begin{aligned} \int _0^{t^*}\!\!\!\int _{{\mathcal {B}}} \theta ^2\, dvd\tau \le T_0 D^2,\quad \int _0^{t^*}\!\!\!\int _{{\mathcal {B}}} {\dot{\theta }}^2\, dvd\tau \le T_0 E^2. \end{aligned}$$
(20)
$$\begin{aligned} \begin{aligned}&\int _{{\mathcal {B}}}\Bigl [\rho u_{i}u_{i}+\beta k_{ij}{\bar{g}}_i{\bar{g}}_j\Bigr ]\,dv+\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [k\theta ^{2}+ k_{ij}\bar{g_i}\bar{g_j}\Bigr ]\,dvd\tau \\&\le Bt^*\Biggl [\int _0^{t^*}\!\!\!\int _{{\mathcal {B}}}\rho f_if_i\,dvd\tau \Biggr ]^{\frac{1}{2}}+Ct^*\Biggl [\int _0^{t^*}\!\!\!\int _{{\mathcal {B}}}g^2\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\quad +(D+\beta _0 E)t^*\Biggl [\int _0^{t^*}\!\!\!\int _{{\mathcal {B}}}\frac{1}{T_0}(\rho {\bar{h}})^2\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(21)
Proof
By using the Cauchy–Schwartz inequality and (19) we obtainfor every \(t\in [0,t^*/2]\). In the same way, we find the inequalityfor every \(t\in [0,t^*/2]\). From (22) and (23) we getfor every \(t\in [0,t^*/2]\). Similarly, we obtainfor every \(t\in [0,t^*/2]\). In the same way, we find the inequalityfor every \(t\in [0,t^*/2]\). From (25) and (26) we getfor every \(t\in [0,t^*/2]\). For the last term of Eq. (18) we haveOn the other hand, by the application of the Cauchy-Schwartz inequality and (20)\(_1\), we havefor every \(t\in [0,t^*/2]\). Similarly, we obtainfor every \(t\in [0,t^*/2]\). From the relations (28) and (29), we get\(\forall t\in [0,t^*/2]\). If we substitute the estimates given by Eqs. (24), (27), and (30) into the identity (18) and we integrate the relation thus obtained with respect to the time variable from \(0\) to \(t\), then we obtain that Eq. (21) holds, for every \(t\in [0,t^*/2]\). The proof is complete. \(\square\)
$$\begin{aligned} \begin{aligned}&\int _{0}^{t}\!\int _{{\mathcal {B}}}\rho f_i(t-\tau )u_i(t+\tau )\,dvd\tau \\&\le \Biggl [\int _{0}^{t}\!\int _{{\mathcal {B}}}\rho f_i(t-\tau )f_i(t-\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\quad \times \Biggl [\int _{0}^{t}\!\int _{{\mathcal {B}}}\rho u_i(t+\tau )u_i(t+\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\le B\Biggl [\int _{0}^{t^*}\!\!\!\int _{{\mathcal {B}}}\rho f_if_i\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(22)
$$\begin{aligned} \begin{aligned} -&\int _{0}^{t}\!\int _{{\mathcal {B}}}\rho f_i(t+\tau )u_i(t-\tau )\,dvd\tau \\&\le B\Biggl [\int _{0}^{t^*}\!\!\!\int _{{\mathcal {B}}}\rho f_if_i\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(23)
$$\begin{aligned} \begin{aligned}&\int _{0}^{t}\!\int _{{\mathcal {B}}}\rho [f_i(t-\tau )u_i(t+\tau )\\&\qquad \quad \,-f_i(t+\tau )u_i(t-\tau )]\,dvd\tau \\&\le 2B\Biggl [\int _{0}^{t^*}\!\!\!\int _{{\mathcal {B}}}\rho f_if_i\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(24)
$$\begin{aligned} \begin{aligned}&\int _0^t\!\int _{{\mathcal {B}}}\varphi (t-\tau )g(t+\tau )\,dvd\tau \\&\le \Biggl [\int _{0}^{t}\!\int _{{\mathcal {B}}} \varphi ^2(t-\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\quad \times \Biggl [\int _{0}^{t}\!\int _{{\mathcal {B}}}g^2(t+\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\le C\Biggl [\int _{0}^{t^*}\!\!\!\int _{{\mathcal {B}}}g^2\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(25)
$$\begin{aligned} \begin{aligned} -&\int _0^t\!\int _{{\mathcal {B}}}\varphi (t+\tau )g(t-\tau )\,dvd\tau \\&\le C\Biggl [\int _{0}^{t^*}\!\!\!\int _{{\mathcal {B}}}g^2\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(26)
$$\begin{aligned} \begin{aligned}&\int _0^t\!\int _{{\mathcal {B}}}[\varphi (t-\tau )g(t+\tau )-\varphi (t+\tau )g(t-\tau )]\,dvd\tau \\&\le 2C\Biggl [\int _{0}^{t^*}\!\!\!\int _{{\mathcal {B}}}g^2\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(27)
$$\begin{aligned} \begin{aligned}&\frac{1}{T_0}\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [\rho {\bar{h}}(t+\tau )A\theta (t-\tau )\\&\qquad \quad \,-\rho {\bar{h}}(t-\tau )A\theta (t+\tau )\Bigr ]\, dvd\tau \\&=\frac{1}{T_0}\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [\rho {\bar{h}}(t+\tau )\theta (t-\tau )\\&\qquad \qquad \,-\rho {\bar{h}}(t-\tau )\theta (t+\tau )\Bigr ]\, dvd\tau \\&+\frac{1}{T_0}\int _{0}^{t}\!\int _{{\mathcal {B}}}\beta \Bigl [\rho {\bar{h}}(t+\tau ){\dot{\theta }}(t-\tau )\\&\qquad \qquad \,-\rho {\bar{h}}(t-\tau ){\dot{\theta }}(t+\tau )\Bigr ]\, dvd\tau . \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned}&\frac{1}{T_0}\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [\rho {\bar{h}}(t+\tau )\theta (t-\tau )-\rho {\bar{h}}(t-\tau )\theta (t+\tau )\Bigr ]\, dvd\tau \\&\le \Biggl [\int _{0}^{t}\!\int _{{\mathcal {B}}}\frac{1}{T_0}\theta ^2(t-\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\quad \times \Biggl [\int _{0}^{t}\!\int _{{\mathcal {B}}}\frac{1}{T_0}(\rho {\bar{h}})^2(t+\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&+\Biggl [\int _{0}^{t}\!\int _{{\mathcal {B}}}\frac{1}{T_0}\theta ^2(t+\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\quad \times \Biggl [\int _{0}^{t}\!\int _{{\mathcal {B}}}\frac{1}{T_0}(\rho {\bar{h}})^2(t-\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\le 2D\Biggl [\int _{0}^{t^*}\!\!\!\int _{{\mathcal {B}}}\frac{1}{T_0}(\rho {\bar{h}})^2(t-\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(28)
$$\begin{aligned} \begin{aligned} \frac{1}{T_0}&\int _{0}^{t}\!\int _{{\mathcal {B}}}\beta \Bigl [\rho {\bar{h}}(t+\tau ){\dot{\theta }}(t-\tau )\\&\qquad \,-\rho {\bar{h}}(t-\tau ){\dot{\theta }}(t+\tau )\Bigr ]\, dvd\tau \\&\le 2\beta _0 E\Biggl [\int _{0}^{t^*}\!\!\!\int _{{\mathcal {B}}}\frac{1}{T_0}(\rho {\bar{h}})^2(t-\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(29)
$$\begin{aligned} \begin{aligned}&\frac{1}{T_0}\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [\rho {\bar{h}}(t+\tau )A\theta (t-\tau )\\&\qquad \,-\rho {\bar{h}}(t-\tau )A\theta (t+\tau )\Bigr ]\, dvd\tau \\&\le 2(D+\beta _0E)\Biggl [\int _{0}^{t^*}\!\!\!\int _{{\mathcal {B}}}\frac{1}{T_0}(\rho {\bar{h}})^2(t-\tau )\,dvd\tau \Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(30)
3.2 Continuous dependence on initial data
Let us now investigate the continuous dependence of the solution on initial data. We denote by \(\pi ^{(\alpha )}\) the solutions corresponding, respectively, to two external data systems which differ only by the initial datawith \(\alpha =1,2\). Moreover we setThe difference of the two solutions \(\pi =\pi ^{(2)}-\pi ^{(1)}\) satisfies the mixed problem \(\Pi\) for the case when the supply terms and boundary data are all zero, while the initial data are given by (31).
$$\begin{aligned}\begin{aligned} \Gamma ^{(\alpha )}=\{&f_i, g, h, u_i^{0(\alpha )}, v_i^{0(\alpha )}, \theta ^{0(\alpha )}, \eta ^{0(\alpha )},\\&{\hat{u}}_i, {\hat{d}}_i, {\hat{\theta }}, {\hat{\varphi }}, {\hat{\xi }}, {\hat{P}}_i, {\hat{R}}_i, {\hat{q}}, {\hat{\Lambda }}, {\hat{H}}\}, \end{aligned}\end{aligned}$$
$$\begin{aligned} \begin{alignedat}{2}&u_i^0=u_i^{0(2)}-u_i^{0(1)},\quad&v_i^0&=v_i^{0(2)}-v_i^{0(1)},\\&\theta ^0=\theta ^{0(2)}-\theta ^{0(1)},\quad&\eta ^0&=\eta ^{0(2)}-\eta ^{0(1)}. \end{alignedat} \end{aligned}$$
(31)
If we apply Eq. (16) to the solution \(\pi\), then we obtainfor every \(t\in (0,t_1/2)\). The identity (32) will help us to prove the next continuous dependence theorem.
$$\begin{aligned} \begin{aligned} \frac{d}{dt}&\Bigl (\int _{{\mathcal {B}}}\Bigl [\rho u_{i}u_{i}+\beta k_{ij}{\bar{g}}_i{\bar{g}}_j\Bigr ]\,dv\\&\qquad +\int _{0}^{t}\!\int _{{\mathcal {B}}}\Bigl [k\theta ^{2}+ k_{ij}\bar{g_i}\bar{g_j}\Bigr ]\,dvd\tau \Bigr )\\&=\int _{{\mathcal {B}}}[\rho u_{i}^{0}{\dot{u}}_{i}(2t)+\rho v_{i}^{0}u_{i}(2t)\\&\qquad + k\theta ^{0}\theta (2t) +\beta k_{ij}g_i^0{\bar{g}}_j(2t)]\,dv\\&\quad +\int _{0}^{t}\!\int _{{\mathcal {B}}}\rho \eta ^{0}[A\theta (t-\tau )-A\theta (t+\tau )]\,dvd\tau , \end{aligned} \end{aligned}$$
(32)
Theorem 5
We suppose that there exists \(t^*\in (0,t_1)\) and some positive constants \(F\), \(G\) and \(K\) such thatThen, the following inequality holds:for every \(t\in [0,t^*/2].\)
$$\begin{aligned} \int _0^{t^*}\!\!\!\int _{{\mathcal {B}}} \big [\rho u_iu_i+ \beta k_{ij}{\bar{g}}_i{\bar{g}}_j+k\theta ^2\big ]\, dvd\tau \le F^2, \end{aligned}$$
(33)
$$\begin{aligned} \int _0^{t^*}\!\!\!\int _{{\mathcal {B}}} \theta ^2\, dvd\tau \le G^2,\,\quad \int _0^{t^*}\!\!\!\int _{{\mathcal {B}}} \rho u_iu_i dvd\tau \le K^2. \end{aligned}$$
(34)
$$\begin{aligned} \begin{aligned}&\int _0^t\int _{{\mathcal {B}}}\Bigl [\rho u_{i}u_{i}+\beta k_{ij}{\bar{g}}_i {\bar{g}}_j\Bigr ]\,dv d\tau \\&\quad +\int _{0}^{t}\int _{{\mathcal {B}}}(t-\tau )\Bigl [k\theta ^{2}+ k_{ij}\bar{g_i}\bar{g_j}\Bigr ]\,dvd\tau \\&\le F(t^{*})^{3/2}\Biggl [\int _{{\mathcal {B}}}\Big [\rho v_i^0v_i^0+\beta k_{ij}g_i^0g_j^0+k(\theta ^0)^2\big ]\,dv\Biggr ]^{\frac{1}{2}}\\&\quad +\Bigg \{\big (\sqrt{2}t^*+\beta _0\big )G(t^*)^{3/2}\\&\qquad \quad \,+\beta _0(t^*)^2\Biggl [\int _{{\mathcal {B}}}(\theta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}\Bigg \}\Biggl [\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}\\&\quad +K(t^*)^{3/2}\Biggl [\int _{{\mathcal {B}}}\rho u_i^0u_i^0\,dv\Biggr ]^{\frac{1}{2}}+\frac{t^*}{2}\int _{{\mathcal {B}}}\rho u_i^0u_i^0 \,dv, \end{aligned} \end{aligned}$$
(35)
Proof
By integration of the relation (32) with respect to the time variable from \(0\) to \(t\), we getfor every \(t\in (0,t_1/2)\). Thanks to the Cauchy-Schwartz inequality we arrive the to following relationsandfor every \(t\in [0,t^*/2]\). On the other hand, using again the Cauchy-Schwartz inequality we deduce the relationthe relationand the relationfor every \(t\in [0,t^*/2]\).
$$\begin{aligned} \begin{aligned}&\int _{{\mathcal {B}}}\Bigl [\rho u_{i}u_{i}+\beta k_{ij}{\bar{g}}_i{\bar{g}}_j\Bigr ]\,dv\\&+\int _{0}^{t}\int _{{\mathcal {B}}}\Bigl [k\theta ^{2}+k_{ij}\bar{g_i}\bar{g_j}\Bigr ]\,dvd\tau \\&=\frac{1}{2}\int _{{\mathcal {B}}}\big [\rho u_i^0u_i(2t)+\rho u_i^0u_i^0 \big ]\,dv\\&+\int _0^t\int _{{\mathcal {B}}}\big [\rho v_{i}^{0}u_{i}(2\tau )+\beta k_{ij}g_i^0{\bar{g}}_j(2\tau )\\&+k\theta ^0\theta (2\tau )\big ]\,dvd\tau +\int _0^t\int _{{\mathcal {B}}}\rho \eta ^0\beta [\theta ^0-\theta (2\tau )]\,dvd\tau \\&+\int _0^t\int _0^{\xi }\int _{{\mathcal {B}}}\rho \eta ^0[\theta (\xi -\tau )-\theta (\xi +\tau )]\,dvd\tau d\xi , \end{aligned} \end{aligned}$$
(36)
$$\begin{aligned} \begin{aligned}&\int _0^t\int _{{\mathcal {B}}}\big [\rho v_{i}^{0}u_{i}(2\tau )+\beta k_{ij}g_i^0{\bar{g}}_j(2\tau )+k\theta ^0\theta (2\tau )\big ]\,dvd\tau \\&\le \Biggl [\frac{1}{2}\int _0^{2t}\!\!\!\int _{{\mathcal {B}}}\big [\rho u_iu_i+\beta k_{ij}{\bar{g}}_i{\bar{g}}_j+k\theta ^2\big ]\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\quad \times \Biggl [\int _0^{t}\!\!\int _{{\mathcal {B}}}\Big [\rho v_i^0v_i^0+\beta k_{ij}g_i^0g_j^0+k(\theta ^0)^2\big ]\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\le \frac{Ft^{\frac{1}{2}}}{\sqrt{2}}\Biggl [\int _{{\mathcal {B}}}\Big [\rho v_i^0v_i^0+\beta k_{ij}g_i^0g_j^0+k(\theta ^0)^2\big ]\,dv\Biggr ]^{\frac{1}{2}}\\&\le F(t^*)^{\frac{1}{2}}\Biggl [\int _{{\mathcal {B}}}\Big [\rho v_i^0v_i^0+\beta k_{ij}g_i^0g_j^0+k(\theta ^0)^2\big ]\,dv\Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(37)
$$\begin{aligned} \begin{aligned}&\int _0^t\!\int _0^{\xi }\!\int _{{\mathcal {B}}}\rho \eta ^0[\theta (\xi -\tau )-\theta (\xi +\tau )]\,dv d\tau d\xi \\&\le \Biggl [\int _0^t \!\int _0^{\xi }\!\int _{{\mathcal {B}}}\theta ^2(\xi -\tau )\,dvd\tau d\xi \Biggr ]^{\frac{1}{2}}\\&\quad \times \Biggl [\int _0^t \!\int _0^{\xi }\!\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dvd\tau d\xi \Biggr ]^{\frac{1}{2}}\\&+\Biggl [\int _0^t \!\int _0^{\xi }\!\int _{{\mathcal {B}}}\theta ^2(\xi +\tau )\,dvd\tau d\xi \Biggr ]^{\frac{1}{2}}\\&\quad \times \Biggl [\int _0^t \!\int _0^{\xi }\!\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dvd\tau d\xi \Biggr ]^{\frac{1}{2}}\\&\le 2\Biggl [\int _0^t\!\int _0^{\xi }\!\int _{{\mathcal {B}}} \theta ^2\, dvd\tau d\xi \Biggr ]^{\frac{1}{2}}\\&\quad \times \Biggl [\int _0^t\!\int _0^{\xi }\!\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dvd\tau d\xi \Biggr ]^{\frac{1}{2}}\\&\le \sqrt{2}Gt^{3/2}\Biggl [\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}\\&\le \sqrt{2}G(t^*)^{3/2}\Biggl [\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(38)
$$\begin{aligned} \begin{aligned}&\frac{1}{2}\int _0^t\!\int _{{\mathcal {B}}}\rho u_i(2\tau )u_i^0\,dvd\tau \\&\le \frac{1}{2}\Biggl [\frac{1}{2}\int _0^{2t}\!\!\int _{{\mathcal {B}}}\rho u_iu_i\,dvd\tau \Biggr ]^{\frac{1}{2}}\times \Biggl [\int _0^{t}\!\int _{{\mathcal {B}}}\rho u_i^0u_i^0\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\le \frac{Kt^{\frac{1}{2}}}{2\sqrt{2}}\Biggl [\int _{{\mathcal {B}}}\rho u_i^0u_i^0\,dv\Biggr ]^{\frac{1}{2}}\le K(t^*)^{1/2}\Biggl [\int _{{\mathcal {B}}}\rho u_i^0u_i^0\,dv\Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(39)
$$\begin{aligned} \begin{aligned}&-\int _0^t\!\int _{{\mathcal {B}}} \rho \eta ^0\beta \theta (2\tau )\,dvd\tau \\&\le \Biggl [\frac{1}{2}\int _0^{2t}\!\!\int _{{\mathcal {B}}}\theta ^2\,dvd\tau \Biggr ]^{\frac{1}{2}}\times \Biggl [\int _0^{t}\!\int _{{\mathcal {B}}}(\rho \eta ^0\beta )^2\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\le \frac{\beta _0 Gt^{1/2}}{\sqrt{2}}\Biggl [\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}\\&\le \beta _0 G(t^*)^{1/2}\Biggl [\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(40)
$$\begin{aligned} \begin{aligned}&\int _0^t\!\int _{{\mathcal {B}}}\rho \eta ^0\theta ^0\beta \,dvd\tau \\&\le \beta _0 \Biggl [\int _0^{t}\!\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dvd\tau \Biggr ]^{\frac{1}{2}}\times \Biggl [\int _0^{t}\!\int _{{\mathcal {B}}}(\theta ^0)^2\,dvd\tau \Biggr ]^{\frac{1}{2}}\\&\le \beta _0 t\Biggl [\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}\times \Biggl [\int _{{\mathcal {B}}}(\theta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}\\&\le \beta _0 t^*\Biggl [\int _{{\mathcal {B}}}(\rho \eta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}\times \Biggl [\int _{{\mathcal {B}}}(\theta ^0)^2\,dv\Biggr ]^{\frac{1}{2}}, \end{aligned} \end{aligned}$$
(41)
4 Conclusion
In this paper we considered the linear theory of thermopiezoelectric nonsimple materials as established in [1] in the general case of non-centrosymmetric materials. We defined a mixed initial-boundary value problem under non-homogeneous initial conditions as presented in [2]. Starting from a reciprocity relation which involves two processes at different times, two continuous dependence results were established, in the case of centrosymmetric materials. The results shown in this article add another element to the analysis of the well-posedness of the problem of thermopiezoelectric materials. Further developments of this theory could be related to the study of wave propagation in isotropic materials and to the analysis, from a numerical point of view, of the discrete stability and of error estimate results using a finite element method and some implicit scheme, under the Green and Laws model.
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