Let us specify that the main hypotheses necessary to obtain the proposed results are the following:

i) *c*, the function of the capacity of heat, and *ϱ*, the density of mass, are positive with regards to the variable of position, i.e.,

$$ c(x)\ge c_{0}>0,\qquad \varrho (x)\ge \varrho _{0}>0, \quad x\in \Omega ; $$

ii) all tensors of thermoelastic coefficients are bounded.

iii) the conductivity of heat tensor, \(k_{ij}\), is positively definite, i.e., \(\exists k_{0}>0\) so that

$$\begin{aligned} \kappa _{mn}\xi _{m}\xi _{n}\ge k_{0} \xi _{m}\xi _{m},\quad \forall \xi _{m}. \end{aligned}$$

(9)

Let us designate by \(c^{*}\) the biggest value of the proper values for the matrix of components \(c_{mn}\) and, also, by \(k^{*}\) the smallest of the proper values of the matrix of components \(\kappa _{mn}\). Then, for any vector \(\xi _{m}\), from Eq. (10), we obtain the following estimate:

$$\begin{aligned} \kappa _{mn}\xi _{m}\xi _{n}\ge k_{1} \vert c_{mn} \xi _{m}\xi _{n} \vert ,\quad k_{1}= \frac{k^{*}}{c^{*}}. \end{aligned}$$

(10)

Now, we consider that \(k_{m}\) is the minimum value of the eigenvalues for the matrix of components \(\kappa _{mn}\), and \(k^{M}\) is the maximum value for the same matrix.

If we use the notation \(k_{2}=k^{M}/k_{m}\) and consider any function *τ*, which satisfies the relation \(\tau (0)=0\), we can deduce the next Poincaré type estimate:

$$\begin{aligned} k_{2} t^{2} \int _{0}^{t}\kappa _{mn}\dot{\tau }_{,m}\dot{\tau }_{,n} \,ds \ge \frac{\pi ^{2}}{4} \int _{0}^{t}\kappa _{mn}\tau _{,m}\tau _{,n} \,ds. \end{aligned}$$

(11)

This estimate is useful in obtaining our main results.

To complete the mixed initial-boundary value problem in the above context, we adjoin to the system of Eqs. (8) the following initial values:

$$\begin{aligned} \begin{gathered} v_{m}(0,x)=v_{m}^{0}(x), \qquad \dot{v}_{m}(0,x)=v_{m}^{1}(x),\qquad \phi _{m}(0,x)= \phi _{m}^{0}(x), \\ \dot{\phi }_{m}(0,x)=\phi _{m}^{1}(x),\qquad \theta (0,x)=\theta ^{0}(x),\qquad \dot{\theta }(0,x)=\theta ^{1}(x), \quad x\in \Omega , \end{gathered} \end{aligned}$$

(12)

and the boundary conditions:

$$\begin{aligned} \begin{gathered} v_{m}(t,x)=\tilde{v}_{m}(t,x),\qquad \phi _{m}(t,x)=\tilde{\phi }_{m}(t, x), \\ \theta (t,x)=\tilde{\theta }(t,x),\quad (t,x)\in [0,t_{0})\times \partial \Omega . \end{gathered} \end{aligned}$$

(13)

We will denote by \({\mathcal{P}}\) the mixed problem consisting of differential Eqs. (8), the initial values (12), and the boundary relations (13).

Let us consider two systems of loads,

$$ \bigl(F_{m}^{(\nu )}, G_{m}^{(\nu )}, r^{(\nu )} \bigr), \quad \nu =1,2,$$

and denote by

$$ \bigl(v_{m}^{(\nu )}, \phi _{m}^{(\nu )}, \tau ^{(\nu )} \bigr), \quad \nu =1,2$$

the solutions that correspond to each previous loads.

We will use the notations:

$$\begin{aligned} \begin{gathered} v_{m}=v_{m}^{(2)}-v_{m}^{(1)}, \qquad \phi _{m}=\phi _{m}^{(2)}-\phi _{m}^{(1)}, \qquad \tau =\tau ^{(2)}-\tau ^{(1)}, \\ {\mathcal{F}}_{m}=F_{m}^{(2)}-F_{m}^{(1)}, \qquad {\mathcal{G}}_{m}=G_{m}^{(2)}-G_{m}^{(1)}, \qquad {\mathcal{R}}=r^{(2)}-r^{(1)} \end{gathered} \end{aligned}$$

(14)

relative to the difference of the two solutions and of the two loads, respectively.

As such, because of the linearity, the ordered array \((v_{m},\phi _{m}, \tau )\) is also a solution to the mixed problem \({\mathcal{P}}\), which now corresponds to:

– the partial differential equations:

$$\begin{aligned} \begin{gathered} \varrho \ddot{v}_{m}= (A_{mnkl}e_{kl}+ B_{mnkl}\varepsilon _{kl} )_{,n}- (a_{mn}\theta )_{,n}+\varrho {\mathcal{F}}_{m}, \\ \begin{aligned} I_{mn}\ddot{\phi }_{m}={}& (B_{mnkl}e_{kl}+ C_{mnkl}\varepsilon _{kl} )_{,m}- (b_{mn}\theta )_{,m} \\ &{}+\varepsilon _{mnj} ( A_{mjkl}e_{kl}+ B_{mjkl}\varepsilon _{kl}-a_{mj} \theta )+ \varrho {\mathcal{G}}_{n}, \end{aligned} \\ c\dot{\theta }=-a_{mn}\dot{e}_{mn}- b_{mn} \dot{\varepsilon }_{mn}+ (\kappa _{mn}\theta _{,n} )_{,m}+\varrho {\mathcal{R}}. \end{gathered} \end{aligned}$$

(15)

– the null initial values:

$$\begin{aligned} v_{m}(0,x) = \dot{v}_{m}(0,x) = \phi _{m}(0,x) = \dot{\phi }_{m}(0,x) = \theta (0,x) = \dot{\theta }(0,x) = 0, \quad \forall x\in \Omega , \end{aligned}$$

(16)

– the null relations to the limit:

$$\begin{aligned} v_{m}(t,x) = \phi _{m}(t,x) = \theta (t,x) = 0,\quad \forall (t,x) \in [0,t_{0})\times \partial \Omega . \end{aligned}$$

(17)

The following two theorems are devoted to some estimations that will help us prove the main results.

### Theorem 1

*If* \((v_{m},\phi _{m}, \tau )\) *is a solution to the mixed problem consisting of Eqs*. (15) *and conditions* (16), (17), *then we have the following identity*:

$$\begin{aligned}& \int _{\Omega } \bigl(\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m} \dot{\phi }_{n}+c\theta ^{2}+ A_{mnkl}e_{mn}e_{kl} +2B_{mnkl}e_{mn}\varepsilon _{kl}+ C_{mnkl}\varepsilon _{mn} \varepsilon _{kl} \bigr)\,dV \\& \quad {} - \int _{0}^{t} \int _{\Omega } (\varrho {\mathcal{F}}_{m} \dot{v}_{m}+\varrho {\mathcal{G}}_{m}\dot{\phi }_{m} + \varrho {\mathcal{R}} \theta ) \,dV \,ds + \int _{0}^{t} \int _{\Omega } \kappa _{mn} \theta _{,m} \theta _{,n} \,dV\,ds = 0. \end{aligned}$$

(18)

### Proof

It is easy to prove this estimation if we take into account the equation of energy and consider the homogeneous Dirichlet boundary relations and zero initial values. □

The Lagrange identity, which we prove in the following theorem, is a useful tool in obtaining the Hölder type stability.

### Theorem 2

*For any solution* \((v_{m},\phi _{m}, \tau )\) *of the mixed problem consisting of Eqs*. (15) *and conditions* (16), (17), *then we have the following identity*:

$$\begin{aligned}& \int _{\Omega } (\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m} \dot{\phi }_{n}+c_{mn}\tau _{,m}\tau _{,n} ) \,dV+ \int _{0}^{t} \int _{\Omega }\kappa _{mn}\theta _{,m} \theta _{,n} \,dV\,ds \\& \quad = \int _{0}^{t} \int _{\Omega } \varrho ({\mathcal{F}}_{m} \dot{v}_{m} + {\mathcal{G}}_{m}\dot{\phi }_{m} + {\mathcal{R}}\theta ) \,dV \,ds \\& \qquad {}+ \frac{1}{2} \int _{0}^{t} \int _{\Omega } \varrho \bigl[{\mathcal{F}}_{m}(s) \dot{v}_{m}(2t - s) - {\mathcal{F}}_{m}(2t - s) \dot{v}_{m}(s) \bigr] \,dV \,ds \\& \qquad {}+\frac{1}{2} \int _{0}^{t} \int _{\Omega }\varrho \bigl[{\mathcal{G}}_{m}(s) \dot{\phi }_{m}(2t-s)- {\mathcal{G}}_{m}(2t-s)\dot{\phi }_{m}(s) \bigr] \,dV \,ds \\& \qquad {} + \frac{1}{2} \int _{0}^{t} \int _{\Omega } \varrho \bigl[{\mathcal{R}}(2t-s) \theta (s)- { \mathcal{R}}_{ij}(s) \theta (2t-s) \bigr]\,dV \,ds. \end{aligned}$$

(19)

### Proof

Based on the basic rule of deriving a product of functions, it is easy to show that the following relations take place:

$$\begin{aligned} \begin{gathered} \frac{d}{\,ds} \bigl[\varrho \dot{v}_{m}(s) \dot{v}_{m}(2t-s) \bigr]= \varrho \ddot{v}_{m}(s) \dot{v}_{m}(2t-s)- \varrho \dot{v}_{m}(s) \ddot{v}_{m}(2t-s), \\ \frac{d}{\,ds} \bigl[ I_{mn}\dot{\phi }_{m}(s) \dot{\phi }_{n}(2t-s) \bigr]= I_{mn}\ddot{\phi }_{m}(s)\dot{\phi }_{n}(2t-s)- I_{mn} \dot{ \phi }_{m}(s)\ddot{\phi }_{n}(2t-s), \\ \frac{d}{\,ds} \bigl[ \theta (s)\theta (2t-s) \bigr]=c \dot{\theta }(s) \theta (2t-s)- c \theta (s)\dot{\theta }(2t-s). \end{gathered} \end{aligned}$$

(20)

By summing up these three equalities, term by term we obtain an equality in which we consider the differential Eqs. (15). Then, we introduce the constitutive Eqs. (3), and the obtained relation is integrated into cylinder \([0,t]\times \Omega \). If we take into account that we have zero initial values and null boundary relations, we get the following identity:

$$\begin{aligned}& \int _{\Omega } \bigl(\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m} \dot{\phi }_{n}+ c_{mn}\tau _{,m}\tau _{,n}-c\theta ^{2} \\& \qquad {}-A_{mnkl}e_{mn}e_{kl} -2B_{mnkl}e_{mn} \varepsilon _{kl}- C_{mnkl} \varepsilon _{mn} \varepsilon _{kl} \bigr)\,dV \\& \quad = \int _{0}^{t} \int _{\Omega }\varrho \bigl[{\mathcal{F}}_{m}(s) \dot{v}_{m}(2t-s)- {\mathcal{F}}_{m}(2t-s) \dot{v}_{m}(s) \bigr]\,dV \,ds \\& \qquad {}+ \int _{0}^{t} \int _{\Omega } \varrho \bigl({\mathcal{G}}_{m}(s) \dot{\phi }_{m}(2t-s)- {\mathcal{G}}_{m}(2t-s)\dot{\phi }_{m}(s) \bigr) \,dV \,ds \\& \qquad {}-\frac{1}{2} \int _{0}^{t} \int _{\Omega }\varrho \bigl[{\mathcal{R}}(2t-s) \theta (s)- { \mathcal{R}}(s) \theta (2t-s) \bigr]\,dV \,ds. \end{aligned}$$

(21)

Finally, using the identities (18) and (21), we ge the desired estimate (19). □

Our first main result will be proven in the next theorem. The inequality that will be obtained ensures the stability for the solutions to the problem \({\mathcal{P}}\) of Hölder type, in relation to the external loads.

### Theorem 3

*We suppose the the assumptions i*)-*iii*) *are satisfied and consider a solution* \((v_{m},\phi _{m}, \tau )\) *to the mixed problem* \({\mathcal{P}}\), *but consisting of Eqs*. (15), *conditions* (16), *and* (17). *Then*, *we have the next inequality*:

$$\begin{aligned}& \int _{\Omega } (\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m} \dot{\phi }_{n}+ c_{mn}\tau _{,m}\tau _{,n} ) \,dV+ \int _{0}^{t} \int _{\Omega }\kappa _{mn}\theta _{,m} \theta _{,n} \,dV\,ds \\& \quad \le \frac{3}{2}\sqrt{t_{0}}M \biggl[ \int _{0}^{t_{0}} \int _{\Omega } \varrho \bigl({\mathcal{F}}_{m}{ \mathcal{F}}_{m}+{\mathcal{G}}_{m}{\mathcal{G}}_{m}+{ \mathcal{R}}^{2} \bigr)\,dV \,ds \biggr]^{1/2}, \end{aligned}$$

(22)

*which takes place for any* \(t\in [0,t_{0}/2 ]\).

*The positive constant*
*M*
*is chosen so that*

$$\begin{aligned} \sup_{t\in [0, t_{0}]} \int _{\Omega } \bigl(\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m}\dot{\phi }_{n}+c\theta ^{2} \bigr)\,dV \le M^{2}. \end{aligned}$$

(23)

### Proof

In order to estimate the integrals from the identity (19), we will use a simple inequality of the form:

$$\begin{aligned}& \sqrt{a}\sqrt{p}+\sqrt{b}\sqrt{q}+\sqrt{c}\sqrt{r} \\& \quad \le \sqrt{a+b}\sqrt{p+q}+ \sqrt{a+c}\sqrt{p+r}+ \sqrt{b+c}\sqrt{q+r}, \end{aligned}$$

which is true for the real numbers \(a, b, c, p, q, r\ge 0\).

Based on this elementary inequality and considering the first integral of (19), we immediately obtain the estimation:

$$\begin{aligned}& \int _{0}^{t} \int _{\Omega }\varrho [{\mathcal{F}}_{m} \dot{v}_{m}+{ \mathcal{G}}_{m}\dot{\phi }_{m}+ {\mathcal{R}}\theta ] \,dV \,d\tau \\& \quad \le \biggl[ \int _{0}^{t} \int _{\Omega }\varrho {\mathcal{F}}_{m}{ \mathcal{F}}_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \biggl[ \int _{0}^{t} \int _{\Omega } \varrho \dot{v}_{m} \dot{v}_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \\& \qquad {}+ \biggl[ \int _{0}^{t} \int _{\Omega }\varrho {\mathcal{G}}_{m}{ \mathcal{G}}_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \biggl[ \int _{0}^{t} \int _{\Omega } \varrho \dot{\phi }_{m}\dot{\phi }_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \\& \qquad {}+ \biggl[ \int _{0}^{t} \int _{\Omega }\varrho {\mathcal{R}}^{2} \,dV \,ds \biggr]^{\frac{1}{2}} \biggl[ \int _{0}^{t} \int _{\Omega }\varrho \theta ^{2} \,dV \,ds \biggr]^{\frac{1}{2}} \\& \quad \le \biggl[ \int _{0}^{t} \int _{\Omega } \varrho \bigl({\mathcal{F}}_{m}{ \mathcal{F}}_{m} + {\mathcal{G}}_{m}{ \mathcal{G}}_{m} + { \mathcal{R}}^{2} \bigr) \,dV \,ds \biggr]^{ \frac{1}{2}} \\& \qquad {}\times \biggl[ \int _{0}^{t} \int _{\Omega } \varrho \bigl(\dot{v}_{m} \dot{v}_{m} + \dot{\phi }_{m}\dot{\phi }_{m} + \theta ^{2} \bigr) \,dV \,ds \biggr]^{ \frac{1}{2}} . \end{aligned}$$

(24)

Similarly, the second integral in (19) leads to the following estimate:

$$\begin{aligned}& \int _{0}^{t} \int _{\Omega }\varrho \bigl[{\mathcal{F}}_{m}(s) \dot{v}_{m}(2t-s)-{ \mathcal{F}}_{m}(2t-s) \dot{v}_{m}(s) \bigr]\,dV\,ds \\& \quad \le \biggl[ \int _{0}^{t} \int _{\Omega }\varrho {\mathcal{F}}_{m}{ \mathcal{F}}_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \biggl[ \int _{t}^{2t} \int _{\Omega } \varrho \dot{v}_{m} \dot{v}_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \\& \qquad {}+ \biggl[ \int _{t}^{2t} \int _{\Omega }\varrho {\mathcal{F}}_{m}{ \mathcal{F}}_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \biggl[ \int _{0}^{t} \int _{\Omega } \varrho \dot{v}_{m} \dot{v}_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \\& \quad \le \biggl[ \int _{0}^{2t} \int _{\Omega }\varrho {\mathcal{F}}_{m}{ \mathcal{F}}_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \biggl[ \int _{0}^{2t} \int _{\Omega } \varrho \dot{v}_{m} \dot{v}_{m} \,dV \,ds \biggr]^{\frac{1}{2}}. \end{aligned}$$

(25)

With an analogous procedure, using the third integral from (19), we deduce:

$$\begin{aligned}& \int _{0}^{t} \int _{\Omega }\varrho \bigl[{\mathcal{G}}_{m}(s) \dot{\phi }_{m}(2t-s)-{ \mathcal{G}}_{m}(2t-s)\dot{\phi }_{m}(s) \bigr] \,dV\,ds \\& \quad \le \biggl[ \int _{0}^{2t} \int _{\Omega }\varrho {\mathcal{G}}_{m}{ \mathcal{G}}_{m} \,dV \,ds \biggr]^{\frac{1}{2}} \biggl[ \int _{0}^{2t} \int _{\Omega } \varrho \dot{\phi } \dot{\phi }_{m} \,dV \,ds \biggr]^{\frac{1}{2}}. \end{aligned}$$

(26)

Ultimately, the fourth integral in (19) leads to the following estimate:

$$\begin{aligned}& \int _{0}^{t} \int _{\Omega }\varrho \bigl[{\mathcal{R}}(2t-s)\theta (s)-{ \mathcal{R}}(s)\theta (2t-s) \bigr]\,dV\,ds \\& \quad \le \biggl[ \int _{0}^{2t} \int _{\Omega }\varrho {\mathcal{R}}^{2} \,dV \,ds \biggr]^{\frac{1}{2}} \biggl[ \int _{0}^{2t} \int _{\Omega }\varrho \theta ^{2} \,dV \,ds \biggr]^{\frac{1}{2}}. \end{aligned}$$

(27)

Now, we take into account all estimations from (24), (25), (26), and (27), so that considering the identity (19), we obtain the inequality:

$$\begin{aligned}& \int _{\Omega } (\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m} \dot{\phi }_{n}+ c_{mn}\tau _{,m}\tau _{,n} ) \,dV+ \int _{0}^{t} \int _{\Omega }\kappa _{mn}\theta _{,m} \theta _{,n} \,dV\,d\tau \\& \quad \le \frac{3}{2} \biggl[ \int _{0}^{2t} \int _{\Omega } \varrho \bigl({\mathcal{F}}_{m}{ \mathcal{F}}_{m} + {\mathcal{G}}_{m}{ \mathcal{G}}_{m} + {\mathcal{R}}^{2} \bigr) \,dV \,ds \biggr]^{ \frac{1}{2}} \\& \qquad {}\times \biggl[ \int _{0}^{2t} \int _{\Omega } \bigl( \varrho \dot{v}_{m} \dot{v}_{m} + I_{mn}\dot{\phi }_{m}\dot{ \phi }_{n} + \varrho \theta ^{2} \bigr) \,dV \,ds \biggr]^{ \frac{1}{2}} . \end{aligned}$$

(28)

Finally, we consider the estimate (23) so that from (28), we observe that the estimate (22) takes place for any \(t\in [0,t_{0}]\), which concludes the demonstration of this theorem. □

### Remark

It is easy to see that the estimation (22) with *M* from (23) assures the stability of the solution in the sense of the Hölder regarding the supply terms.

Finally, we want to study the stability of solutions, in the sense of the Hölder regarding the initial values. In this regard, we will take the partial differential Eqs. (8) without the charges, that is, \(F_{m}=0\), \(G_{m}=0\), \(r=0\), i.e., in its homogeneous form. Let us consider two solutions \((v_{m}^{(1)}, \phi _{m}^{(1)}, \tau ^{(1)} )\), \((v_{m}^{(2)}, \phi _{m}^{(2)}, \tau ^{(2)} )\) of (8), corresponding to equal boundary relations but to different initial values. Then, the difference of these two solutions \((v_{m}, \phi _{m}, \tau )\) satisfies the system (16) with null loads, that is, \({\mathcal{F}}_{m}=0\), \({\mathcal{G}}_{m}=0\), \({\mathcal{R}}=0\), also, with null to the limit values (17), but the initial values of the following form:

$$\begin{aligned} \begin{gathered} v_{m}(x, 0)=v_{m}^{(2)}(x, 0)-v_{m}^{(1)}(x, 0)=v_{m}^{0}(x), \\ \dot{v}_{m}(x, 0)=\dot{v}_{m}^{(2)}(x, 0)- \dot{v}_{m}^{(1)}(x, 0)=w_{m}^{0}(x), \\ \phi _{m}(x, 0) = \phi _{m}^{(2)}(x, 0) - \phi _{m}^{(1)}(x, 0) = \phi _{m}^{0}(x), \\ \dot{\phi }_{m}(x, 0) = \dot{\phi }_{m}^{(2)}(x, 0) - \dot{\phi }_{m}^{(1)}(x, 0) = \psi _{m}^{0}(x), \\ \tau (x, 0)=\tau ^{(2)}(x, 0)- \tau ^{(1)}(x, 0)=\tau ^{0}(x), \\ \dot{\tau }(x, 0)=\dot{\tau }^{(2)}(x, 0)- \dot{\tau }^{(1)}(x, 0)= \theta ^{0}(x). \end{gathered} \end{aligned}$$

(29)

### Theorem 4

*We suppose that the assumptions i*)-*iii*) *are satisfied and consider the difference of the two solutions* \((v_{m},\phi _{m}, \tau )\) *to the mixed problem* \({\mathcal{P}}\), *but consisting of the homogeneous Eqs*. (15), *null boundary conditions*, *and the initial conditions in the form* (30). *Then the stability of the solution is ensured*, *in the sense of the Hölder regarding the initial values*.

### Proof

Recalling the form of the internal energy:

$$\begin{aligned} W(t) =& \int _{\Omega } \bigl(\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn} \dot{\phi }_{m}\dot{\phi }_{n}+c\theta ^{2}+c_{mn}\tau _{,m}\tau _{,n} \\ &{}+A_{mnkl}e_{mn}e_{kl}+ 2B_{mnkl}e_{mn} \varepsilon _{kl}+ C_{mnkl} \varepsilon _{mn} \varepsilon _{kl} \bigr)\,dV \\ &{}+2 \int _{0}^{t} \int _{\Omega }\kappa _{mn}\theta _{,m} \theta _{,n}\,dV\,ds, \end{aligned}$$

(30)

we can deduce the following law of conservation:

$$\begin{aligned} W(t)=W(0), \end{aligned}$$

(31)

where

$$\begin{aligned} W(0) =& \int _{\Omega } \bigl(\varrho w_{m}^{0} w_{m}^{0}+I_{mn} \psi _{m}^{0} \psi _{n}^{0}+c \bigl(\theta ^{0} \bigr)^{2}+ c_{mn}\tau ^{0}_{,m} \tau ^{0}_{,n} \\ &{}+A_{mnkl}e_{mn}^{0}e_{kl}^{0} +2B_{mnkl}e_{mn}^{0} \varepsilon _{kl}^{0}+ C_{mnkl}\varepsilon _{mn}^{0}\varepsilon _{kl}^{0} \bigr) \,dV. \end{aligned}$$

(32)

However, we consider only the solutions to the homogeneous system of partial differential Eqs. (15), so that, with help of the Lagrange identity, we obtain the following identity:

$$\begin{aligned}& \int _{\Omega } \bigl(\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m} \dot{\phi }_{n}+c_{mn}\tau _{,m}\tau _{,n}-c\theta ^{2} \\& \qquad {}-A_{mnkl}e_{mn}e_{kl} -2B_{mnkl}e_{mn} \varepsilon _{kl}- C_{mnkl} \varepsilon _{mn} \varepsilon _{kl} \bigr) \,dV \\& \quad = \int _{\Omega } \bigl[\varrho w_{m}^{0} \dot{v}_{m}(2t)+I_{mn}\psi _{m}^{0} \dot{\phi }_{n}(2t)+c_{mn}\tau ^{0}_{,m} \tau _{,n}(2t) -c\theta ^{0} \theta (2t) \\& \qquad {}-A_{mnkl}e_{mn}^{0}e_{kl}(2t) -2B_{mnkl}e_{mn}^{0} \varepsilon _{kl}(2t)- C_{mnkl}\varepsilon _{mn}^{0} \varepsilon _{kl}(2t) \bigr] \,dV. \end{aligned}$$

(33)

Now, we take into account the expression (30) for the energy \(W(t)\) and (32) for \(W(0)\). As such, taking into account (31) and (33), we obtain the identity:

$$\begin{aligned}& \int _{\Omega } (\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m} \dot{\phi }_{n}+c_{mn}\tau _{,m}\tau _{,n} ) \,dV+ \int _{0}^{t} \int _{\Omega } \kappa _{mn}\theta _{,m} \theta _{,n} \,dV\,ds \\& \quad = \frac{E(0)}{2} + \frac{1}{2} \int _{\Omega } \bigl[\varrho w_{m}^{0} \dot{v}_{m}(2t) + I_{mn}\psi _{m}^{0} \dot{\phi }_{n}(2t) + c_{mn} \tau ^{0}_{,m} \tau _{,n}(2t) \\& \qquad {} - A_{mnkl}e_{mn}^{0}e_{kl}(2t) - 2B_{mnkl}e_{mn}^{0} \varepsilon _{kl}(2t) - C_{mnkl}\varepsilon _{mn}^{0} \varepsilon _{kl}(2t) - c\theta ^{0}\theta (2t) \bigr] \,dV. \end{aligned}$$

(34)

It is not difficult to obtain the next inequalities:

$$\begin{aligned}& \biggl\vert \int _{\Omega }\varrho w_{m}^{0} \dot{v}_{m}(2t) \,dV \biggr\vert \le \biggl( \int _{\Omega }\varrho w_{m}^{0} w_{m}^{0} \,dV \biggr)^{ \frac{1}{2}} \biggl( \int _{\Omega }\varrho \dot{v}_{m} (2t) \dot{v}_{m}(2t) \,dV \biggr)^{\frac{1}{2}}, \\& \biggl\vert \int _{\Omega }I_{mn}\psi _{m}^{0} \dot{\phi }_{n}(2t) \,dV \biggr\vert \le \biggl( \int _{\Omega }I_{mn}\psi _{m}^{0} \psi _{n}^{0} \,dV \biggr)^{ \frac{1}{2}} \biggl( \int _{\Omega }I_{mn}\dot{\phi }_{m}(2t) \dot{\phi }_{n}(2t) \,dV \biggr)^{\frac{1}{2}}, \\& \biggl\vert \int _{\Omega } c\theta ^{0}\theta (2t) \,dV \biggr\vert \le \biggl( \int _{\Omega }c \bigl(\theta ^{0} \bigr)^{2} \,dV \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega }c\theta ^{2}(2t) \,dV \biggr)^{\frac{1}{2}}, \\& \biggl\vert \int _{\Omega } c_{mn} \tau _{,m}^{0} \tau _{,n}(2t) \,dV \biggr\vert \le k_{3} \biggl( \int _{\Omega }\tau _{,m}^{0} \tau _{,m}^{0} \,dV \biggr)^{ \frac{1}{2}} \biggl( \int _{\Omega }\tau _{,n}(2t)\tau _{,n}(2t) \,dV \biggr)^{\frac{1}{2}}, \\& \biggl\vert \int _{\Omega }A_{mnkl} e_{mn}^{0} e_{kl}(2t) \,dV \biggr\vert \le k_{4} \biggl( \int _{\Omega }e_{mn}^{0}e_{mn}^{0} \,dV \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega }e_{mn} (2t) e_{mn} (2t)\,dV \biggr)^{\frac{1}{2}}, \\& \biggl\vert \int _{\Omega }B_{mnkl} e_{mn}^{0} \varepsilon _{kl}(2t) \,dV \biggr\vert \le k_{5} \biggl( \int _{\Omega }e_{mn}^{0}e_{mn}^{0} \,dV \biggr)^{ \frac{1}{2}} \biggl( \int _{\Omega }\varepsilon _{kl} (2t) \varepsilon _{kl} (2t)\,dV \biggr)^{\frac{1}{2}}, \\& \biggl\vert \int _{\Omega }C_{mnkl} \varepsilon _{mn}^{0} \varepsilon _{kl}(2t) \,dV \biggr\vert \le k_{6} \biggl( \int _{\Omega }\varepsilon _{mn}^{0} \varepsilon _{mn}^{0} \,dV \biggr)^{\frac{1}{2}} \biggl( \int _{\Omega } \varepsilon _{kl} (2t) \varepsilon _{kl} (2t)\,dV \biggr)^{\frac{1}{2}}, \end{aligned}$$

(35)

in which \(k_{m}\) takes the values 3, 4, 5, 6.

These constants are expressed only with the help of the thermoelastic tensor situated in the same line. As an example, the constant \(k_{5}\) depends only on the themoelastic tensor \(B_{mnkl}\).

Now, we take an arbitrary \(t\in [0,t_{0}/2]\) and suppose that

$$\begin{aligned}& \sup_{t\in [0, t_{0}]} \int _{\Omega } \bigl(\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m}\dot{\phi }_{n}+ k_{3} \tau _{,m}\tau _{,m} \\& \quad {}+N_{1}e_{mn}e_{mn} +N_{2} \varepsilon _{mn}\varepsilon _{mn}+ c \theta ^{2} \bigr)\,dV\le M_{1}^{2}, \end{aligned}$$

(36)

in which the constants \(N_{1}>0\), \(N_{2}>0\) and \(N_{3}>0\) can be expressed as functions of \(k_{3}, k_{4},\ldots, k_{6}\).

Then, with the help of (34), we deduce the estimate:

$$\begin{aligned}& \int _{\Omega } (\varrho \dot{v}_{m} \dot{v}_{m}+I_{mn}\dot{\phi }_{m} \dot{\phi }_{n}+ c_{mn}\tau _{,m}\tau _{,n} ) \,dV \\& \quad {}+ \int _{0}^{t} \int _{\Omega }\kappa _{mn}\theta _{,m} \theta _{,n} \,dV\,ds \le \frac{1}{2}E(0) +\frac{t_{0}}{2}M_{1} \sqrt{M_{2}}, \end{aligned}$$

(37)

in which the constant \(M_{2}\) has the expression:

$$\begin{aligned} M_{2} =& \int _{\Omega } \bigl(\varrho w_{m}^{0} w_{m}^{0}+I_{mn} \psi _{m}^{0} \psi _{n}^{0}+k_{3} \tau _{,m}^{0}\tau _{,m}^{0} \\ &{}+N_{1}e_{mn}^{0}e_{mn}^{0}+ N_{2}\varepsilon _{mn}^{0} \varepsilon _{mn}^{0} + c \bigl(\theta ^{0} \bigr)^{2} \bigr)\,dV. \end{aligned}$$

(38)

With this, we end the proof of Theorem. □

### Remark

It is not difficult to notice that the estimation (37) with \(M_{2}\) from (38) guarantees the stability of the solution, in the sense of the Hölder with regard to the initial values.