At the beginning of this section we will define the notion of domain of influence.
After that we will formulate and demonstrate an inequality that relates to the domain of influence. We have to say that this inequality is a generalization of similar inequality proposed in the papers [21–23]. This section and, in fact, our study ends with the demonstration of the theorem that establishes the domain of influence or, in other words, the relaxed Saint–Venant principle for the thermoelastic dipolar bodies with pores.
It is clear that the above mentioned results will be obtained if some conditions are met. The next assumptions on the properties of the material will help us in this endeavor:
-
(i)
\(\varrho >0\), \(I_{ij}>0\), \(\kappa >0\), \(T_{0}>0\), \(a>0\);
-
(ii)
$$\begin{aligned}& A_{ijmn}\xi_{ij}\xi_{mn}+2G_{ijmn} \xi_{ij}\zeta_{mn}+ B_{ijmn}\zeta _{ij} \zeta_{mn} \\& \quad\quad{} + 2F_{mnrij}\xi_{ij}\nu_{mnr}+ 2D_{ijmnr} \zeta_{ij}\nu_{mnr}+ C_{ijkmnr} \nu_{ijk} \nu_{mnr} \\& \quad\quad{}+2a_{ij}\xi_{ij}\omega +2b_{ij} \zeta_{ij}\omega + 2c_{ijk}\nu_{ijk} \omega +2D_{ijk}\xi_{ij}\omega_{k} \\& \quad\quad{}+2E_{ijk}\zeta_{ij}\omega_{k}+ 2F_{ijkm} \nu_{ijk}\omega_{m}+2d_{i} \omega_{i} \omega + \xi \omega^{2}+g_{ij} \omega_{i} \omega_{j} \\& \quad \ge \alpha \bigl(\xi_{ij}\xi_{ij}+\zeta_{ij} \zeta_{ij}+ \nu_{ijk}\nu_{ijk}+ \omega_{i} \omega_{i}+\omega^{2} \bigr), \end{aligned}$$
for all \(\xi_{ij}=\xi_{ji}\), \(\zeta_{ij}\), \(\nu_{ijk}\), \(\omega_{i}\), ω;
-
(iii)
\(k_{ij}x_{i}x_{j}\ge \gamma x_{i}x_{i} \) for all \(x_{i}\).
In these inequalities, α and γ are conveniently chosen positive constants.
We must note that these hypotheses are usual restrictions imposed in the mechanics of solids. For instance, hypothesis (iii) is a considerable strengthening of inequality (13) which, in turn, is a consequence of the inequality of entropy production.
Let us consider \(V_{\alpha }(x)\), a smooth non-decreasing function, defined by
$$ V_{\alpha }(x)= \textstyle\begin{cases} 0,& \text{if }x\in (-\infty ,0], \\ 1, & \text{if }x\in [\alpha ,\infty ), \end{cases} $$
for sufficiently small \(\alpha >0\).
Function \(V_{\alpha }\) is inspired by known Heaviside step function.
Using the function \(V_{\alpha }\), for some fixed positive \(\mathcal{R} _{1}\) and t, and for \(d= \vert {\mathbf{x}}-\mathbf{x}_{0} \vert \), we define another useful function G as follows:
$$\begin{aligned} W(x,s)=V_{\alpha } \biggl( \frac{\mathcal{R}-d}{v}+ t-s \biggr) , \quad\quad W:B \times [0,t]\to R, \end{aligned}$$
(14)
where v is a positive constant having the dimension of velocity to be determined later and \(\mathbf{x}_{0}\) is an arbitrary fixed point in B.
It is not difficult to find that the function \(W(x,s)\) is a smooth function on the cylinder \(B\times [0,t]\), and it vanishes outside the set
$$\begin{aligned} \Sigma = \bigcup_{s\in [0,t]} {\mathcal{S}} \bigl[{ \mathbf{x}}_{0},\mathcal{R}+v(t-s) \bigr]. \end{aligned}$$
Here \(\mathcal{S}(\mathbf{x}_{0},r)\) is a sphere defined by
$$\begin{aligned} {\mathcal{S}}(\mathbf{x}_{0},r)= \bigl\{ \mathbf{x} \in R^{3}: \vert {\mathbf{x}}-\mathbf{x} _{0} \vert < r \bigr\} . \end{aligned}$$
(15)
The following inequality is a necessary step to obtain our main result.
Proposition 1
If the system of equations (12) admits a solution
\((u_{i}, \varphi_{ij},\phi ,\theta )\)
which satisfies the initial data (9) and the conditions to limit (10), then we have the following inequality:
$$\begin{aligned}& \bigl[ \varrho \dot{u}_{i}\dot{u}_{i}+ I_{kr} \dot{\varphi }_{jr} \dot{\varphi }_{jk}+ \varrho \kappa \dot{ \phi }^{2}+a\theta^{2}+A_{ijmn} \varepsilon_{ij} \varepsilon_{mn} \\& \quad\quad{} +2G_{ijmn}\varepsilon_{ij}\gamma_{mn} +B_{ijmn}\gamma_{ij}\gamma_{mn}+ 2F_{mnrij} \varepsilon_{ij}\chi_{mnr} \\& \quad\quad{} +2D_{ijmnr}\gamma_{ij}\chi_{mnr}+ C_{ijkmnr} \chi_{ijk}\chi_{mnr}+2a _{ij} \varepsilon_{ij} \phi \\& \quad\quad{}+2b_{ij}\gamma_{ij}\phi +2c_{ijk} \chi_{ijk}\phi + 2d_{ijk}\varepsilon _{ij} \phi_{,k}+2e_{ijk}\gamma_{ij}\phi_{,k} \\& \quad\quad{}+ 2f_{ijkm}\chi_{ijk}\phi_{,k}+ 2d_{i}\phi \phi_{,i}+g_{ij} \phi_{,i} \phi_{,j}+\xi \phi^{2} \bigr] (x,s) \\& \quad \ge \bigl[ \varrho \dot{u}_{i}\dot{u}_{i}+ I_{kr}\dot{\varphi }_{jr} \dot{\varphi }_{jk}+ \varrho \kappa \dot{\phi }^{2}+a\theta^{2} \\& \quad\quad{}+\varepsilon_{ij}\varepsilon_{ij}+ \gamma_{ij} \gamma_{ij}+\chi _{ijk} \chi_{ijk}+ \phi^{2}+ \phi_{,i}\phi_{,i} \bigr] (x,s). \end{aligned}$$
(16)
Proof
The inequality is immediately obtained as a consequence of hypotheses (i) and (ii). □
The following inequality uses inequality (16) and is the basis for obtaining the most important result of our paper.
Theorem 1
If the system of equations (12) admits a solution
\(( u _{i},\varphi_{ij},\phi ,\theta ) \)
which satisfies the initial data (9) and the data at the limit (10), then for any
\(t>0\), \(\mathcal{R}>0\), and
\(\mathbf{x}_{0}\in B\), the following inequality holds:
$$\begin{aligned}& \int_{\Gamma [{\mathbf{x}}_{0},\mathcal{R}]}P(\mathbf{x},t) \,dV+\frac{1}{T_{0}} \int_{0}^{t} \int_{\Gamma [{\mathbf{x}}_{0},\mathcal{R}+v(t-s)]}k_{ij}\theta _{,i} \theta_{,j}\,dV \\& \quad \le \int_{\Gamma [{\mathbf{x}}_{0},\mathcal{R}+vt]}P(\mathbf{x},0)\,dV \\& \quad\quad{} + \int_{0}^{t} \int_{\Gamma [{\mathbf{x}}_{0}, \mathcal{R}+v(t-s)]} \varrho \biggl[F _{i}\dot{u}_{i}+M_{jk} \dot{\varphi }_{jk}+L\dot{\phi }+ \frac{1}{T _{0}} r \theta \biggr] \,dV\,ds \\& \quad\quad{}+ \int_{0}^{t} \int_{\partial \Gamma [{\mathbf{x}}_{0}, \mathcal{R}+v(t-s)]} \biggl[ \bar{t}_{i}\dot{u}_{i}+ \bar{\mu }_{jk}\dot{\varphi }_{jk}+\bar{h} \dot{\phi }+ \frac{1}{T_{0}}\bar{q}\theta \biggr]\,dA\,ds, \end{aligned}$$
(17)
where
\(\Gamma (\mathbf{x}_{0},r)= \{\mathbf{x} \in B : \vert { \mathbf{x}}-\mathbf{x} _{0} \vert < r \}\), \(\partial \Gamma ( \mathbf{x}_{0},r)= \{\mathbf{x} \in \partial B : \vert { \mathbf{x}}-\mathbf{x}_{0} \vert =r \}\).
Proof
First, we must specify that the function \(P(x,t)\) used in (17) is the potential energy and has the expression
$$\begin{aligned} P(x,s)&=\frac{1}{2} \bigl[ \varrho \dot{u}_{i} \dot{u}_{i}+ I_{kr} \dot{\varphi }_{jr}\dot{\varphi }_{jk}+ \varrho \kappa \dot{\phi } ^{2}+a \theta^{2}+C_{ijmn}\varepsilon_{ij} \varepsilon_{mn} \\ & \quad {} +2G_{ijmn}\varepsilon_{ij}\gamma_{mn} +B_{ijmn}\gamma_{ij}\gamma_{mn}+ 2F_{mnrij} \varepsilon_{ij}\chi_{mnr} \\ & \quad {} +2D_{ijmnr}\gamma_{ij}\chi_{mnr}+ A_{ijkmnr} \chi_{ijk}\chi_{mnr}+ 2a _{ij}\varepsilon_{ij} \phi \\ & \quad {} +2b_{ij}\gamma_{ij}\phi +2c_{ijk} \chi_{ijk}\phi 2d_{ijk}\varepsilon _{ij} \phi_{,k}+2e_{ijk}\gamma_{ij}\phi_{,k} \\ & \quad {} +2f_{ijkm}\chi_{ijk}\phi_{,k} +2d_{i}\phi \phi_{,i}+g_{ij}\phi _{,i}\phi_{,j}+\xi \phi^{2} \bigr] (x,s). \end{aligned}$$
(18)
Also, the kinetic energy is \(K(x,s)\) which is a function defined by
$$\begin{aligned} \begin{aligned}[b] K(x,s)&=\frac{1}{2} \bigl[ \varrho \dot{u}_{i} \dot{u}_{i}+ I_{kr} \dot{\varphi }_{jr}\dot{\varphi }_{jk}+\varrho \kappa \dot{\phi }^{2}+a \theta^{2} \\ &\quad{} +\varepsilon_{ij}\varepsilon_{ij}+ \gamma_{ij} \gamma_{ij}+\chi _{ijk}\chi_{ijk}+ \phi^{2}+ \phi_{,i}\phi_{,i} \bigr] (x,s). \end{aligned} \end{aligned}$$
(19)
If we multiply both members of equation (12)1 by \(W\dot{u}_{i}\), we are led to the relation
$$\begin{aligned} \begin{aligned}[b] \frac{1}{2}W\frac{d}{dt}(\varrho \dot{u}_{i} \dot{u}_{i})&= \varrho WF _{i}\dot{u}_{i}+(Wt_{ij} \dot{u}_{i})_{,j}-W_{,j}t_{ij} \dot{u}_{i} \\ &\quad{} -W(A_{ijmn}\varepsilon_{mn}+ B_{ijmn} \gamma_{mn}+ B_{ij}\phi +D_{ijk} \phi_{,k}- \beta_{ij}\theta )\dot{u}_{i,j}. \end{aligned} \end{aligned}$$
(20)
By analogy, we multiply both members of equation (12)2 by \(W\dot{\varphi }_{jk}\), so that we obtain
$$\begin{aligned} \begin{aligned}[b] \frac{1}{2}W\frac{d}{dt}(I_{kr}\dot{\varphi }_{jr}\dot{\varphi }_{jk})&= \varrho GM_{jk}\dot{ \varphi }_{jk}+(W\mu_{jk}\dot{\varphi }_{jk})_{,j}- W_{,j}\mu_{jk}\dot{\varphi }_{jk} \\ &\quad{} -W(B_{mnij}\varepsilon_{mn}+ C_{ijmn} \gamma_{mn}+C_{ij}\phi +E_{ijk} \phi_{,k}- \alpha_{ij}\theta )\dot{\varphi }_{jk,j} \\ &\quad{}+\varepsilon_{ijk}(A_{jkmn}\varepsilon_{mn}+ B_{jkmn}\gamma_{mn}+B _{jk}\phi +D_{jkm} \phi_{,m}- \beta_{jk}\theta )\dot{\varphi }_{jk}. \end{aligned} \end{aligned}$$
(21)
Now, we multiply both members of equation (12)3 by Wϕ̇, so we deduce that
$$\begin{aligned} \begin{aligned}[b] \frac{1}{2}W\frac{d}{dt} \bigl(\varrho \kappa \dot{ \phi }^{2} \bigr)&= \varrho WL \dot{\phi }+(Wh_{i}\dot{\phi })_{,i}-W_{,i}h_{i}\dot{\phi } \\ &\quad{}-W(A_{ij}\phi_{,j}\dot{\phi }_{,i}+D_{mni} \varepsilon_{mn}\dot{\phi } _{,i}+ E_{mni} \gamma_{mn}\dot{\phi }_{,i}+ d_{i}\phi \dot{\phi }_{,i}-a _{i}\theta \dot{\phi }_{,i}) \\ &\quad{}-W(B_{ij}\varepsilon_{ij}\dot{\phi }+ C_{ij} \gamma_{ij}\dot{\phi }+ \xi \phi \dot{\phi }+ d_{i} \phi_{,i}\dot{\phi }-m\theta \dot{\phi }). \end{aligned} \end{aligned}$$
(22)
Finally, if we multiply both members of equation (12)4 by Wθ, we are led to
$$\begin{aligned} \begin{aligned}[b] \frac{1}{2}W\frac{d}{dt} \bigl(a\theta^{2} \bigr)&=\frac{1}{T_{0}}W r\theta + \frac{1}{ \varrho T_{0}} \bigl[ (W\theta q_{i})_{,i}-W_{,i}\theta q_{i} \bigr] \\ &\quad{} -\frac{1}{\varrho T_{0}}Wk_{ij}\theta_{,i}\theta_{,j}- W(\beta_{ij} \theta \dot{\varepsilon }_{ij}+ \alpha_{ij}\theta \dot{\gamma }_{ij}+ m\theta \dot{\phi }+ a_{i}\theta \dot{\phi }_{,i}). \end{aligned} \end{aligned}$$
(23)
Summing up equations (20), (21), (22), and (23) term by term results in
$$\begin{aligned}& \frac{1}{2}W\frac{d}{dt} \bigl(\varrho \dot{u}_{i}\dot{u}_{i}+ I_{kr} \dot{\varphi }_{jr}\dot{\varphi }_{jk}+ \varrho \kappa \dot{\phi } ^{2}+a\theta^{2} \bigr) \\& \quad = \varrho WF_{i} \dot{u}_{i}+\varrho WM_{jk} \dot{\varphi }_{jk} \\& \quad \quad {} +\varrho WL\dot{\phi }+\frac{1}{T_{0}}Wr\theta + W \biggl(t_{ij} \dot{u}_{i}+\mu_{jk} \dot{\varphi }_{jk}+ h_{j}\dot{\phi }+\frac{1}{\varrho T_{0}}\theta q _{j} \biggr)_{,j} \\& \quad \quad {} -W \bigl[A_{ijmn}\varepsilon_{mn}\dot{\varepsilon }_{ij} +B_{ijmn}( \varepsilon_{mn}\dot{\gamma }_{ij}+ \dot{\varepsilon }_{mn}\gamma_{ij}) +C_{ijmn}\gamma_{mn}\dot{\gamma }_{ij} \\& \quad \quad {} +B_{ij}(\dot{\varepsilon }_{ij}\phi +\varepsilon_{ij} \dot{\phi })+ C _{ij}(\dot{\gamma }_{ij}\phi + \gamma_{ij}\dot{\phi }) +D_{ijk}( \varepsilon_{ij} \dot{\phi }_{,k}+\dot{\varepsilon }_{ij}\phi_{,k}) \\& \quad \quad {} +E_{ijk}(\gamma_{ij}\dot{\phi }_{,k}+\dot{\gamma }_{ij}\phi_{,k})+ d _{i}(\phi \dot{\phi }_{,i}+\dot{\phi }\phi_{,i})+ A_{ij} \phi_{,i} \dot{\phi }_{,j}+\xi \phi \dot{\phi } \bigr] \\& \quad \quad {} -W_{,j}t_{ij}\dot{u}_{i}-W_{,j} \mu_{jk}\dot{\varphi }_{jk}- W_{,i}h _{i} \dot{\phi }-\frac{1}{\varrho T_{0}} W_{,i}q_{i}\theta - \frac{1}{ \varrho T_{0}}Wk_{ij}\theta_{,i}\theta_{,j}. \end{aligned}$$
(24)
Clearly, relation (24) can be rewritten in the following form:
$$\begin{aligned}& \frac{1}{2}W\frac{d}{dt} \bigl(\varrho \dot{u}_{i}\dot{u}_{i}+ I_{kr} \dot{\varphi }_{jr}\dot{\varphi }_{jk}+ \varrho \kappa \dot{\phi } ^{2}+a\theta^{2}+ A_{ijmn}\varepsilon_{mn} \varepsilon_{ij}+ 2B_{ijmn} \gamma_{mn} \varepsilon_{ij} \\& \quad \quad {} +C_{ijmn}\gamma_{mn}\gamma_{ij}+ 2B_{ij} \varepsilon_{ij}\phi + 2C_{ij} \gamma_{ij}\phi + 2D_{ijk}\varepsilon_{ij}\phi_{,k} \\& \quad \quad {} +2E_{ijk}\gamma_{ij}\phi_{,k}+ 2d_{i} \phi \phi_{,i}+ A_{ij}\phi_{,i} \phi_{,j}+ \xi \phi^{2} \bigr) \\& \quad =\varrho W \biggl(F_{i}\dot{u}_{i}+M_{jk}\dot{ \varphi }_{jk}+ \varrho L \dot{\phi }+\frac{1}{T_{0}}r\theta \biggr) \\& \quad \quad {} +W \biggl(t_{ij}\dot{u}_{i}+\mu_{jk}\dot{\varphi }_{jk}+ h_{j}\dot{\phi }+\frac{1}{ \varrho T_{0}}\theta q_{j} \biggr)_{,j} \\& \quad \quad {} -W_{,j}t_{ij}\dot{u}_{i}-W_{,j} \mu_{jk}\dot{\varphi }_{jk}- W_{,i}h _{i} \dot{\phi }-W_{,i}\frac{1}{\varrho T_{0}}\theta q_{i}- \frac{1}{ \varrho T_{0}}k_{ij}\theta_{,i}\theta_{,j}, \end{aligned}$$
(25)
or, equivalently,
$$\begin{aligned} \frac{1}{2}W\dot{U}+\frac{1}{\varrho T_{0}}k_{ij} \theta_{,i}\theta _{,j}&= W \biggl(\varrho F_{i} \dot{u}_{i}+\varrho M_{jk}\dot{\varphi }_{jk}+ \varrho L\dot{\phi }+\frac{1}{T_{0}}\varrho r\theta \biggr) \\ &\quad{} +W \biggl(t_{ij}\dot{u}_{i}+\mu_{jk}\dot{\varphi }_{jk}+h_{j}\dot{\phi }+ \frac{1}{ \varrho T_{0}}\theta q_{j} \biggr)_{,j} \\ &\quad{} -W_{,j} \biggl(t_{ij}\dot{u}_{i}+ \mu_{jk}\dot{\varphi }_{jk}+h_{j} \dot{\phi }+ \frac{1}{\varrho T_{0}}\theta q_{j} \biggr). \end{aligned}$$
(26)
We now integrate on cylinder \(B\times [0,t]\) both members of equality (26), and afterwards we use the divergence theorem and the boundary conditions (10), so we get
$$\begin{aligned}& \int_{B}WU(x,t)\,dV+\frac{1}{\varrho T_{0}} \int_{0}^{t} \int_{B}Wk_{ij} \theta_{,i} \theta_{,j}\,dV\,ds \\& \quad = \int_{B}WU(x,0)\,dV \\& \quad \quad {} + \int_{0}^{t} \int_{\partial B} W \biggl( \bar{t}_{i}\dot{u}_{i}+ \bar{ \mu }_{jk}\dot{\varphi }_{jk}+\bar{h}\dot{\phi }+ \frac{1}{\varrho T _{0}}\bar{q}\theta \biggr) \,dV\,ds \\& \quad \quad {} + \int_{0}^{t} \int_{B}\varrho W \biggl( F_{i} \dot{u}_{i}+M_{jk} \dot{\varphi }_{jk}+ L\dot{\phi }+\frac{1}{T_{0}}r\theta \biggr) \,dV\,ds \\& \quad \quad {} + \int_{0}^{t} \int_{B}\dot{W}U(x,s)\,dV\,ds- \int_{0}^{t} \int_{B}W_{,j} \biggl( t_{ij} \dot{u}_{i}+\mu_{jk}\dot{\varphi }_{jk}+ h_{j} \dot{\phi }+\frac{1}{\varrho T_{0}}q_{j}\theta \biggr) \,dV \,ds. \end{aligned}$$
(27)
Considering definition (14) of the function W, it is not difficult to find that
$$\begin{aligned}& \biggl\vert -W_{,j}t_{ij} \dot{u}_{i}-W_{,j}\mu_{jk}\dot{\varphi }_{jk}- W_{,i}h_{i}\dot{\phi }-\frac{1}{\varrho T_{0}} W_{,i}q_{i}\theta \biggr\vert \\& \quad = \biggl\vert \frac{1}{v}V_{\alpha }^{\prime} \frac{x_{j}}{\mathbf{r}}t_{ij} \dot{u}_{i}+ \frac{1}{c}V_{\alpha }^{\prime} \frac{x_{j}}{\mathbf{r}}\mu_{jk} \dot{\varphi }_{jk}+ \frac{1}{c}V_{\alpha }^{\prime}\frac{x_{i}}{\mathbf{r}}h _{i}\dot{\phi }+ \frac{1}{c\varrho T_{0}}V_{\alpha }^{\prime} \frac{x_{i}}{ \mathbf{r}}q_{i}\theta \biggr\vert \\& \quad = \biggl\vert \frac{1}{v}V_{\alpha }^{\prime} \frac{1}{\mathbf{r}} \biggl[ ( A _{ijmn}\varepsilon_{mn}x_{j}+B_{ijmn} \gamma_{mn}x_{j}+ B_{ij}\phi x _{j}+D_{ijk} \phi_{,k}x_{j}- \beta_{ij}\theta x_{j} ) \dot{u}_{i} \\& \quad \quad {}+ ( B_{mnij}\varepsilon_{mn}x_{j}+C_{ijmn} \gamma_{mn}x_{j}+ C_{ij} \phi x_{j}+E_{ijk} \phi_{,k}x_{j}-\beta_{ij}\theta x_{j} ) \dot{\varphi }_{jk} \\& \quad \quad{} + ( D_{mni}\varepsilon_{mn}x_{i}+E_{mni} \gamma_{mn}x _{i}+ A_{ij}\phi_{,j}x_{i}+d_{i} \phi x_{i}- a_{i}\theta x_{i} ) \dot{\phi }+ \frac{1}{\varrho T_{0}}k_{ij}\theta_{,j}\theta x_{i} \biggr] \biggr\vert , \end{aligned}$$
(28)
where
$$ V_{\alpha }^{\prime}=\frac{dV_{\alpha }}{d\mathbf{r}}. $$
The elementary arithmetic–geometric mean inequality
$$\begin{aligned} ab \le \frac{1}{2} \biggl(\frac{a^{2}}{p^{2}}+b^{2}p^{2} \biggr) \end{aligned}$$
(29)
is now used to the last terms of relation (28). Thus, we can choose some suitable parameters p and can find v such that
$$\begin{aligned} \biggl\vert -W_{,j}t_{ij} \dot{u}_{i}-W_{,j}\mu_{jk}\dot{\varphi }_{jk}- W_{,i}h_{i}\dot{\phi }- \frac{1}{T_{0}}W_{,i}q_{i}\theta \biggr\vert \le V_{\alpha }^{\prime} K(x,s), \end{aligned}$$
(30)
and also
$$\begin{aligned} \begin{aligned}[b] & \int_{0}^{t} \int_{B}\dot{W}U(x,s)\,dV\,ds- \int_{0}^{t} \int_{B} \biggl( W _{,j}t_{ij} \dot{u}_{i}+W_{,j}\mu_{jk}\dot{\varphi }_{jk}+ W_{,i} h _{i}\dot{\phi }+ \frac{1}{T_{0}}W_{,i} q_{i}\theta \biggr) \,dV\,ds \\ &\quad \le \int_{0}^{t} \int_{B}V_{\alpha }^{\prime}(x,s) \bigl[K(x,s)-U(x,s) \bigr]\,dV\,ds \le 0. \end{aligned} \end{aligned}$$
(31)
Considering inequality (31), from equation (27) we are led to
$$\begin{aligned}& \int_{B}WU(x,t)\,dV+\frac{1}{T_{0}} \int_{0}^{t} \int_{B}Wk_{ij} \theta _{,i} \theta_{,j}\,dV\,ds \\& \quad \le \int_{B}WU(x,0)\,dV \\& \quad \quad {} + \int_{0}^{t} \int_{B}\varrho W \biggl( F_{i} \dot{u}_{i}+M_{jk} \dot{\varphi }_{jk}+ L\dot{\phi }+\frac{1}{\varrho^{2} T_{0}}r\theta \biggr) \,dV\,ds \\& \quad \quad {} + \int_{0}^{t} \int_{\partial B} W \biggl( \bar{t}_{i}\dot{u}_{i}+ \bar{ \mu }_{jk}\dot{\varphi }_{jk}+\bar{h}\dot{\phi }+ \frac{1}{\varrho T _{0}}\bar{q}\theta \biggr) \,dV\,ds. \end{aligned}$$
(32)
We will pass to the limit as \(\alpha \rightarrow 0\) into relation (32), so that we deduce that W tends boundedly to the characteristic function of the set Σ (before (15)) and, as a consequence, we obtain inequality (17) and the proof of Theorem 1 is complete. □
We will use previous estimates from Proposition 1 and Theorem 1 to demonstrate the basic result of the present study, namely an extension of the relaxed Saint–Venant principle or, in other words, a generalized theorem of the domain of influence.
We will denote by \(B(t)\) the set of all points ∈B̄ having the following properties:
-
(1)
if \(x \in B\) then \(u_{i}^{0} \ne 0\) or \(u_{i} ^{1} \ne 0\) or \(\varphi_{jk}^{0} \ne 0\) or \(\varphi_{jk}^{1} \ne 0\) or \(\phi^{0} \ne 0 \) or \(\phi^{1} \ne 0\) or \(\theta^{0} \ne 0\) or \(\exists \tau \in [0,t]\) such that \(F_{i}(x, \tau ) \ne 0\) or \(M_{i}(x,\tau ) \ne 0\) or \(L(x,\tau ) \ne 0\) or \(r(x,\tau ) \ne 0\);
-
(2)
if \(x \in \partial B_{1}\) then \(\exists \tau \in [0,t]\) such that \(\bar{u}_{i}(x,\tau ) \ne 0\);
-
(3)
if \(x \in \partial B_{1}^{c}\) then \(\exists \tau \in [0,t]\) such that \(\bar{t}_{i}(x,\tau ) \ne 0\);
-
(4)
if \(x \in \partial B_{2}\) then \(\exists \tau \in [0,t]\) such that \(\bar{\varphi }_{jk}(x,\tau )\ne 0\);
-
(5)
if \(x \in \partial B_{2}^{c}\) then \(\exists \tau \in [0,t]\) such that \(\bar{\mu }_{jk}(x,\tau ) \ne 0\);
-
(6)
if \(x \in \partial B_{3}\) then \(\exists \tau \in [0,t]\) such that \(\bar{\phi }(x,\tau ) \ne 0\);
-
(7)
if \(x \in \partial B_{3}^{c}\) then \(\exists \tau \in [0,t]\) such that \(\bar{h}(x,\tau ) \ne 0\);
-
(8)
if \(x \in \partial B_{4}\) then \(\exists \tau \in [0,t]\) such that \(\bar{\theta }(x,\tau ) \ne 0\);
-
(9)
if \(x \in \partial B_{4}^{c}\) then \(\exists \tau \in [0,t]\) such that \(\bar{q}(x,\tau ) \ne 0\).
For the data of our mixed problem, we define the domain of influence at instant t as follows:
$$\begin{aligned} {\mathcal{D}}_{t}= \bigl\{ \mathbf{x}_{0}\in \bar{B} : B(t)\cap \bar{\Gamma }(\mathbf{x} _{0},vt) \ne \Phi \bigr\} . \end{aligned}$$
(33)
Here we denote by Φ the empty set.
Theorem 2
If the system of equations (12) admits a solution
\(( u _{i},\varphi_{ij},\phi ,\theta ) \)
which satisfies the initial data (9) and the data at the limit (10), then for any
\(t>0\)
we obtain
$$\begin{aligned} u_{i}=0, \quad\quad \varphi_{ij}=0, \quad\quad \phi =0, \quad\quad \theta =0 \quad \textit{on }\{\bar{B} \setminus {\mathcal{D}}_{t}\} \times [0,t]. \end{aligned}$$
Proof
We will arbitrarily fix \(\mathbf{x}_{0}\in \bar{B}\setminus {\mathcal{D}}_{t}\) and \(\tau \in [0,t]\), and use inequality (17) taking \(t=\tau \) and \(\mathcal{R}=v(\tau -s)\). In this way, we are led to
$$\begin{aligned}& \int_{\Gamma [{\mathbf{x}}_{0}, v(\tau -s)]}U(\mathbf{x},\tau )\,dV+\frac{1}{T _{0}} \int_{0}^{\tau } \int_{\Gamma [{\mathbf{x}}_{0}, v(\tau -s)]}k_{ij} \theta_{,i} \theta_{,j}\,dV\,d\alpha \\& \quad \le \int_{\Gamma [{\mathbf{x}}_{0}, v\tau ]}U(\mathbf{x},0)\,dV+ \int_{0}^{ \tau } \int_{\Gamma [{\mathbf{x}}_{0}, v(\tau -s)]}\varrho \biggl(F_{i}\dot{u} _{i}+M_{jk}\dot{\varphi }_{jk}+ L\dot{\phi }+ \frac{1}{T_{0}} r \theta \biggr)\,dV\,d\alpha \\& \quad\quad{} + \int_{0}^{\tau } \int_{\partial \Gamma [{\mathbf{x}}_{0}, v(\tau -s)]} \varrho \biggl(\bar{t}_{i} \dot{u}_{i}+\bar{\mu }_{jk}\dot{\varphi }_{jk}+ \bar{h} \dot{\phi }+\frac{1}{T_{0}}\bar{q}\theta \biggr)\,dA\,d\alpha . \end{aligned}$$
(34)
Taking into account that \(\mathbf{x}_{0}\in \bar{B}\setminus {\mathcal{D}} _{t}\), we deduce that \(\mathbf{x}\in \Gamma [{\mathbf{x}}_{0}, v\tau ]\) which involves that \(\mathbf{x} \notin B(t)\). So, we deduce
$$\begin{aligned} \int_{\Gamma [{\mathbf{x}}_{0}, v\tau ]}U(\mathbf{x},0)\,dV=0. \end{aligned}$$
(35)
Now, we take into account that \(\Gamma [{\mathbf{x}}_{0}, v(\tau -s)] \subseteq \Gamma [{\mathbf{x}}_{0}, v\tau ]\) in order to obtain the following two equalities:
$$\begin{aligned}& \int_{0}^{\tau } \int_{\Gamma [{\mathbf{x}}_{0}, v(\tau -s)]}\varrho \biggl(F _{i}\dot{u}_{i}+M_{jk} \dot{\varphi }_{jk}+ L\dot{\phi }+\frac{1}{T _{0}} r \theta \biggr) \,dV\,d \alpha =0, \end{aligned}$$
(36)
$$\begin{aligned}& \int_{0}^{\tau } \int_{\Gamma [{\mathbf{x}}_{0}, v(\tau -s)]} \biggl(\bar{t}_{i} \dot{u}_{i}+ \bar{\mu }_{jk}\dot{\varphi }_{jk}+ \bar{h} \dot{\phi }+ \frac{1}{T _{0}} \bar{q} \theta \biggr)\,dV\,d\alpha =0. \end{aligned}$$
(37)
Considering assumption (iii) and relations (32)–(34), we deduce that
$$\begin{aligned} \int_{\Gamma [{\mathbf{x}}_{0}, v(\tau -s)]}U(\mathbf{x},\tau )\,dV \le 0. \end{aligned}$$
(38)
From (38), by using inequality (16), we deduce
$$\begin{aligned} \int_{\Gamma [{\mathbf{x}}_{0}, v(\tau -s)]}K(\mathbf{x},\tau )\,dV \le 0. \end{aligned}$$
(39)
By using this inequality and considering the definition of function K, we are led to the following null values:
$$\begin{aligned} \dot{u}_{i}(\mathbf{x}_{0},\tau )=0, \quad\quad \dot{\varphi }_{jk}(\mathbf{x}_{0}, \tau )=0 , \quad\quad \phi (\mathbf{x}_{0}, \tau )=0, \quad\quad \theta (\mathbf{x}_{0},\tau )=0 \end{aligned}$$
for any \((\mathbf{x}_{0},\tau )\in \{\bar{B}\setminus {\mathcal{D}}_{t}\} \times [0,t]\).
But \(u_{i}(\mathbf{x}_{0},0)=0\), \(\varphi_{jk}(\mathbf{x}_{0},0)=0\) for any \(\mathbf{x}_{0}\in \bar{B} \setminus {\mathcal{D}}_{t}\), so we deduce that
$$\begin{aligned} u_{i}(\mathbf{x}_{0},\tau )=0, \quad\quad \varphi_{jk}( \mathbf{x}_{0},\tau )=0, \quad\quad \phi (\mathbf{x}_{0},\tau )=0, \quad\quad \theta (\mathbf{x}_{0},\tau )=0 \end{aligned}$$
for any \((\mathbf{x}_{0},\tau )\in \{\bar{B}\setminus {\mathcal{D}}_{t}\} \times [0,t]\), which ends the proof of Theorem 2. □