Consider a cross-section D of a prismatic cylinder and the boundary of the section, ∂D, assumed to be piecewise continuously differentiable. We choose the system of Cartesian rectangular axis so that its origin is in the center of the cylinder base and the positive \(x_{3}\)-axis is directed along the cylinder. If we denote by L the length of the cylinder, then the lateral boundary of the cylinder is \(\mathcal{S}=\partial D\times[0,L]\). The contents of the prismatic cylinder is a micropolar thermoelastic body which is homogeneous and anisotropic.
The cylinder is free of load on the lateral boundary surface, that is, we have a zero body force, couple force and heat supply and zero displacement, microrotations, and thermal displacements. But over the base of cylinder are specified the displacements, microrotations, and thermal displacement, all of which are assumed to be harmonic in time. Therefore, besides the system of equations (9) we can adjoin the following lateral boundary conditions:
$$\begin{aligned} u_{i}(x,t)=0, \qquad \varphi_{i}(x,t)=0,\qquad \tau(x,t)=0, \qquad (x,t)\in\mathcal{S}\times(0,\infty), \end{aligned}$$
(10)
and the base boundary conditions
$$ \begin{aligned} &u_{i}(x_{1},x_{2}, 0,t)={\tilde{u}}_{i}(x_{1},x_{2})e^{\iota \omega t}, \quad \varphi_{i}(x_{1},x_{2}, 0,t)={\tilde{\varphi}}_{i}(x_{1},x_{2})e^{\iota \omega t}, \\ &\tau(x_{1},x_{2}, 0,t)={\tilde{t}}(x_{1},x_{2})e^{\iota \omega t}, \quad (x_{1},x_{2})\in D(0), t>0, \end{aligned} $$
(11)
where \({\tilde{u}}_{i}(x_{1},x_{2})\), \({\tilde{\varphi}}_{i}(x_{1},x_{2})\), and \({\tilde{t}}(x_{1},x_{2})\) are prescribed smooth functions, ι is the complex unit, and ω is a prescribed positive constant.
Loads from (11) induce inside the cylinder some vibrations harmonic in time, having the form
$$ \begin{aligned} &u_{i}(x_{1},x_{2}, x_{3},t)=U_{i}(x_{1},x_{2},x_{3})e^{\iota \omega t}, \qquad \varphi_{i}(x_{1},x_{2}, x_{3},t)= \Phi_{i}(x_{1},x_{2},x_{3})e^{\iota \omega t}, \\ &\tau(x_{1},x_{2}, x_{3},t)=T(x_{1},x_{2},x_{3})e^{\iota \omega t}, \quad (x_{1},x_{2}, x_{3},t)\in B\times(0,\infty). \end{aligned} $$
(12)
The amplitude \((U_{i},\Phi_{i}, T )\) of the vibrations satisfies the following system of differential equations:
$$ \begin{aligned} & \bigl[A_{ijmn} (U_{n, m}+ \varepsilon_{mnk}\Phi_{k} ) +B_{ijmn} \Phi_{n, m}- \iota \omega D_{ij} T \bigr]_{, j}+\varrho \omega^{2} U_{i}=0, \\ & \bigl[B_{mnij} (U_{n, m}+\varepsilon_{mnk} \Phi_{k} )+ C_{ijmn}\Phi_{n, m}- \iota \omega E_{ij} T \bigr]_{,j} \\ &\quad{}+\varepsilon_{ijk} \bigl[A_{jkmn} ( U_{n, m}+ \varepsilon_{mnk}\Phi_{k} ) + B_{jkmn} \Phi_{n, m}- \iota \omega D_{jk} T \bigr]+ I_{ij} \omega^{2} \Phi_{j} =0, \\ & \biggl(\frac{1}{\theta_{0}}K_{ij} T_{, j} \biggr)_{, i}- \iota \omega D_{ij} (U_{j, i}+ \varepsilon_{jik}\Phi_{k} )- \iota \omega E_{ij} \Phi_{j, i}+\frac{c}{\theta_{0}}\omega^{2} T=0. \end{aligned} $$
(13)
The lateral boundary conditions get the form
$$\begin{aligned} U_{i}(x)=0, \qquad \Phi_{i}(x)=0,\qquad T(x)=0, \quad x\in\mathcal{S}, \end{aligned}$$
(14)
and the base boundary conditions become
$$ \begin{aligned} &U_{i}(x_{1},x_{2}, 0)={\tilde{U}}_{i}(x_{1},x_{2}),\qquad \Phi_{i}(x_{1},x_{2}, 0)={\tilde{\Phi}}_{i}(x_{1},x_{2}), \\ &T (x_{1},x_{2}, 0)={\tilde{T}}(x_{1},x_{2}), \quad (x_{1},x_{2})\in D(0). \end{aligned} $$
(15)
In the case of a finite cylinder we will have to prescribe a boundary condition on the superior base of the cylinder, \(D(L)\). For a forced oscillation the spatial behavior of the amplitude was studied by Chirita [19] and Ciarletta [21], in the case of a rhombic thermoelastic material, provided the exciting frequency is less than a certain critical frequency. The main goal of our study is to estimate how the amplitude evolves with respect to the axial distance to the excited and.
In the following we want to prove some estimates on a solution of the system of equations (13), with the lateral boundary conditions (14) and the base boundary conditions (15). We will use the notation \(\mathcal{U}_{j, i}=U_{j, i}+\varepsilon_{jik}\Phi_{k}\).
In the following theorem we will state and prove four auxiliary identities on which will be based the main result.
Theorem 1
Let
\((U_{i},\Phi_{i}, T )\)
be a solution of the boundary value problem consisting of equations (13)-(15). Then the following equalities are satisfied:
$$\begin{aligned} &2 \int_{D(x_{3})} \bigl\{ A_{ijmn}\mathcal{U}_{j, i} \bar{\mathcal{U}}_{n, m}+ C_{ijmn}\Phi_{n, m}{\bar{\Phi}}_{j, i} \\ &\qquad{}+ B_{ijmn} [\mathcal{U}_{j, i}{\bar{\Phi}}_{n, m} + \bar{\mathcal{U}}_{j, i}\Phi_{n, m} ] - \varrho\omega^{2}U_{i}{\bar{U}}_{i} - I_{ij}\omega^{2}\Phi_{i}{\bar{\Phi}}_{j} \bigr\} \,dA \\ &\qquad{}+ \int_{D(x_{3})} \bigl\{ \iota \omega D_{ij} ( {\bar{T}} \mathcal{U}_{j, i} - T\bar{\mathcal{U}}_{j, i} ) + \iota \omega E_{ij} ( {\bar{T}} \Phi_{i, j} - T {\bar{\Phi}}_{i, j} ) \bigr\} \,dA \\ &\quad= \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn}\mathcal{U}_{n, m}+B_{3jmn} \Phi_{n, m}- \iota \omega D_{3j}T ]{\bar{U}}_{j} \bigr\} \,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn}\bar{\mathcal{U}}_{n, m}+B_{3jmn}{ \bar{\Phi}}_{n, m}+ \iota \omega D_{3j}{\bar{T}} ]U_{j} \bigr\} \,dA \\ &\qquad{}+ \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\mathcal{U}_{n, m}+C_{3jmn} \Phi_{n, m}- \iota \omega E_{3j}T ] {\bar{\Phi}}_{j} \bigr\} \,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\bar{\mathcal{U}}_{n, m}+C_{3jmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{3j}{\bar{T}} ] \Phi_{j} \bigr\} \,dA, \end{aligned}$$
(16)
$$\begin{aligned} & \int_{D(x_{3})} \bigl[ \iota \omega D_{ij} ( {\bar{T}} \mathcal{U}_{j, i} + T\bar{\mathcal{U}}_{j, i} ) + \iota \omega E_{ij} ( {\bar{T}} \Phi_{i, j} + T {\bar{\Phi}}_{i, j} ) \bigr]\,dA \\ &\quad= \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn}\bar{\mathcal{U}}_{n, m}+B_{3jmn}{ \bar{\Phi}}_{n, m}+ \iota \omega D_{3j}{\bar{T}} ]U_{j} \bigr\} \,dA \\ &\qquad{}- \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn}\mathcal{U}_{n, m}+B_{3jmn} \Phi_{n, m}- \iota \omega D_{3j}T ]{\bar{U}}_{j} \bigr\} \,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\bar{\mathcal{U}}_{n, m}+C_{3jmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{3j}{\bar{T}} ] \Phi_{j} \bigr\} \,dA \\ &\qquad{}- \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\mathcal{U}_{n, m}+C_{3jmn} \Phi_{n, m}- \iota \omega E_{3j}T ] {\bar{\Phi}}_{j} \bigr\} \,dA, \end{aligned}$$
(17)
$$\begin{aligned} &2 \int_{D(x_{3})} \frac{1}{\theta_{0}} \bigl(K_{ij} T_{, i}{\bar{T}}_{, j}- c \omega^{2} T{\bar{T}} \bigr)\,dA+ \int_{D(x_{3})} \iota \omega E_{ij} (\Phi_{j, i} { \bar{T}}- {\bar{\Phi}}_{j, i} T )\,dA \\ &\qquad{}+ \int_{D(x_{3})} \iota \omega D_{ij} (\mathcal{U}_{j, i}{ \bar{T}}- \bar{\mathcal{U}}_{j, i} T )\,dA = \frac{d}{d x_{3}} \int_{D(x_{3})} \frac{1}{\theta_{0}}K_{33} ({\bar{T}}T_{, 3}+ T {\bar{T}}_{, 3} )\,dA, \end{aligned}$$
(18)
$$\begin{aligned} & \int_{D(x_{3})} \iota \omega D_{ij} (\mathcal{U}_{j, i}{ \bar{T}}+ \mathcal{U}_{j, i} T )\,dA \\ &\quad{}+ \int_{D(x_{3})} \iota \omega E_{ij} ({\bar{\Phi}}_{j, i} T + \Phi_{j, i}{\bar{T}} )\,dA = \frac{d}{d x_{3}} \int_{D(x_{3})} \frac{1}{\theta_{0}}K_{3j} ({\bar{T}}T_{, j} - T{\bar{T}}_{, j} )\,dA, \end{aligned}$$
(19)
where
z̄
is the notation for the complex conjugate of
z.
Proof
Considering equations (13)1 and (13)2 we can prove the following equality:
$$\begin{aligned} & \bigl\{ [A_{ijmn}\mathcal{U}_{n, m}+B_{ijmn} \Phi_{n, m}- \iota \omega D_{ij}T ]_{, j}+ \varrho \omega^{2}U_{i} \bigr\} {\bar{U}}_{i} \\ &\quad{}+ \bigl\{ [A_{ijmn}\bar{\mathcal{U}}_{n, m}+B_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega D_{ij}{\bar{T}} ]_{, j}+ \varrho\omega^{2}{\bar{U}}_{i} \bigr\} U_{i} \\ &\quad{}+ [B_{mnij}\mathcal{U}_{n, m}+C_{ijmn} \Phi_{n, m}- \iota \omega E_{ij}T ]_{, j} {\bar{\Phi}}_{i} \\ &\quad{}+\varepsilon_{ijk} [A_{jkmn}\mathcal{U}_{n, m}+B_{jkmn} \Phi_{n, m}- \iota \omega D_{jk}T ] {\bar{\Phi}}_{i}+ I_{ij}\omega^{2}\Phi_{i}{\bar{\Phi}}_{j} \\ &\quad{}+ [B_{mnij}\bar{\mathcal{U}}_{n, m}+C_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{ij}{\bar{T}} ]_{, j} \Phi_{i} \\ &\quad{}+\varepsilon_{ijk} [A_{jkmn}\bar{\mathcal{U}}_{n, m}+B_{jkmn}{ \bar{\Phi}}_{n, m}+\iota \omega D_{jk}{\bar{T}} ] \Phi_{i}+ I_{ij}\omega^{2}\Phi_{i}{\bar{\Phi}}_{j}=0. \end{aligned}$$
(20)
With some calculations, equality (20) can be written in the form
$$\begin{aligned} &2 \bigl\{ A_{ijmn}\mathcal{U}_{j, i}\bar{ \mathcal{U}}_{n, m}+ C_{ijmn}\Phi_{n, m}{\bar{\Phi}}_{n, m} \\ &\qquad{}+ B_{ijmn} [\mathcal{U}_{j, i}{\bar{\Phi}}_{n, m}+ \bar{\mathcal{U}}_{j, i}\Phi_{n, m} ]- \varrho\omega^{2}U_{i}{\bar{U}}_{i}- I_{ij} \omega^{2}\Phi_{i}{\bar{\Phi}}_{j} \bigr\} \\ &\qquad{}+ \iota \omega D_{ij} ( {\bar{T}}\mathcal{U}_{j, i}-T \bar{\mathcal{U}}_{j, i} )+ \iota \omega E_{ij} ( {\bar{T}} \Phi_{i, j}-T {\bar{\Phi}}_{i, j} ) \\ &\quad= \bigl\{ [A_{ijmn}\mathcal{U}_{n, m}+B_{ijmn} \Phi_{n, m}- \iota \omega D_{ij}T ]{\bar{U}}_{i} \bigr\} _{, j} \\ &\qquad{}+ \bigl\{ [A_{ijmn}\bar{\mathcal{U}}_{n, m}+B_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega D_{ij}{\bar{T}} ]U_{i} \bigr\} _{, j} \\ &\qquad{}+ \bigl\{ [B_{mnij}\mathcal{U}_{n, m}+C_{ijmn} \Phi_{n, m}- \iota \omega E_{ij}T ] {\bar{\Phi}}_{i} \bigr\} _{, j} \\ &\qquad{}+ \bigl\{ [B_{mnij}\bar{\mathcal{U}}_{n, m}+C_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{ij}{\bar{T}} ] \Phi_{i} \bigr\} _{, j}. \end{aligned}$$
(21)
Integrate equality (21) over \(D(x_{3})\), apply the divergence theorem, and use the lateral conditions (14); we get the equality (16).
Now, if we again consider equations (13)1 and (13)2 then it is easy to prove the following equality:
$$\begin{aligned} & \bigl\{ [A_{ijmn} \mathcal{U}_{n, m}+B_{ijmn} \Phi_{n, m}- \iota \omega D_{ij}T ]_{, j}+ \varrho \omega^{2}U_{i} \bigr\} {\bar{U}}_{i} \\ &\quad{}- \bigl\{ [A_{ijmn}\bar{\mathcal{U}}_{n, m}+B_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega D_{ij}{\bar{T}} ]_{, j}+ \varrho\omega^{2}{\bar{U}}_{i} \bigr\} U_{i} \\ &\quad{}+ [B_{mnij} \mathcal{U}_{n, m}+C_{ijmn} \Phi_{n, m}- \iota \omega E_{ij}T ]_{, j} {\bar{\Phi}}_{i} \\ &\quad{}+\varepsilon_{ijk} [A_{jkmn}\mathcal{U}_{n, m}+B_{jkmn} \Phi_{n, m}- \iota \omega D_{jk}T ] {\bar{\Phi}}_{i}+ I_{ij}\omega^{2}\Phi_{i}{\bar{\Phi}}_{j} \\ &\quad{}- [B_{mnij}\bar{\mathcal{U}}_{n, m}+C_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{ij}{\bar{T}} ]_{, j} \Phi_{i} \\ &\quad{}-\varepsilon_{ijk} [A_{jkmn}\bar{\mathcal{U}}_{n, m}+B_{jkmn}{ \bar{\Phi}}_{n, m}+\iota \omega D_{jk}{\bar{T}} ] \Phi_{i}- I_{ij}\omega^{2}\Phi_{i}{\bar{\Phi}}_{j}=0. \end{aligned}$$
(22)
With some calculations, equality (22) can be written in the form
$$ \begin{aligned}[b] &\iota \omega D_{ij} ( {\bar{T}}\mathcal{U}_{j, i}+T \bar{\mathcal{U}}_{j, i} )+ \iota \omega E_{ij} ( {\bar{T}} \Phi_{i, j}+T {\bar{\Phi}}_{i, j} ) \\ &\quad=+ \bigl\{ [A_{ijmn}\bar{\mathcal{U}}_{n, m}+B_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega D_{ij}{\bar{T}} ]U_{i} \bigr\} _{, j} \\ &\qquad{}- \bigl\{ [A_{ijmn}\mathcal{U}_{n, m}+B_{ijmn} \Phi_{n, m}- \iota \omega D_{ij}T ]{\bar{U}}_{i} \bigr\} _{, j} \\ &\qquad{}+ \bigl\{ [B_{mnij}\bar{\mathcal{U}}_{n, m}+C_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{ij}{\bar{T}} ] \Phi_{i} \bigr\} _{, j} \\ &\qquad{}- \bigl\{ [B_{mnij}\mathcal{U}_{n, m}+C_{ijmn} \Phi_{n, m}- \iota \omega E_{ij}T ] {\bar{\Phi}}_{i} \bigr\} _{, j}. \end{aligned} $$
(23)
Integrate equality (23) over \(D(x_{3})\), apply the divergence theorem; if we use the lateral conditions (14) we get the equality (17).
With the help of equation (13)3, we can deduce immediately the equality
$$\begin{aligned} &{\bar{T}} \biggl[ \biggl(\frac{1}{\theta_{0}}K_{ij} T_{, j} \biggr)_{, i}- \iota \omega D_{ij} \mathcal{U}_{j, i}- \iota \omega E_{ij}\Phi_{j, i}+ \frac{c}{\theta_{0}}\omega^{2} T \biggr] \\ &\quad{}+T \biggl[ \biggl(\frac{1}{\theta_{0}}K_{ij} {\bar{T}}_{, j} \biggr)_{, i}+\iota \omega D_{ij} \mathcal{U}_{j, i}+ \iota \omega E_{ij}{\bar{\Phi}}_{j, i}+\frac{c}{\theta_{0}}\omega^{2} {\bar{T}} \biggr]=0. \end{aligned}$$
(24)
With some calculations, equality (24) can be written in the form
$$\begin{aligned} &\frac{2}{\theta_{0}} \bigl(K_{ij} T_{, i}{\bar{T}}_{, j}- c \omega^{2} T{\bar{T}} \bigr)+ \iota \omega D_{ij} (\mathcal{U}_{j, i}{\bar{T}}- \mathcal{U}_{j, i} T )+\iota \omega E_{ij} (\Phi_{j, i} {\bar{T}}- {\bar{\Phi}}_{j, i} T ) \\ &\quad= \biggl[\frac{1}{\theta_{0}}K_{ij} ({\bar{T}}T_{, j}+ T {\bar{T}}_{, j} ) \biggr]_{, i}. \end{aligned}$$
(25)
Integrate equality (25) over \(D(x_{3})\), apply the divergence theorem, and if we use the lateral conditions (14) we get the equality (18).
Finally, we use again equation (13)3 thus we will obtain, in a trivial way, the equality
$$\begin{aligned} &{\bar{T}} \biggl[ \biggl(\frac{1}{\theta_{0}}K_{ij} T_{, j} \biggr)_{, i}- \iota \omega D_{ij} \mathcal{U}_{j, i}- \iota \omega E_{ij}\Phi_{j, i}+ \frac{c}{\theta_{0}}\omega^{2} T \biggr] \\ &\quad{}-T \biggl[ \biggl(\frac{1}{\theta_{0}}K_{ij} {\bar{T}}_{, j} \biggr)_{, i}+\iota \omega D_{ij} \mathcal{U}_{j, i}+ \iota \omega E_{ij}{\bar{\Phi}}_{j, i}+\frac{c}{\theta_{0}}\omega^{2} {\bar{T}} \biggr]=0. \end{aligned}$$
(26)
With some calculations, equality (26) can be written in the form
$$\begin{aligned} \iota \omega D_{ij} (\mathcal{U}_{j, i}{\bar{T}}+ \mathcal{U}_{j, i} T ) + \iota \omega E_{ij} ({\bar{\Phi}}_{j, i} T+\Phi_{j, i}{\bar{T}} )= \biggl[ \frac{1}{\theta_{0}}K_{ij} ({\bar{T}}T_{, j}-T{\bar{T}}_{, j} ) \biggr]_{, i}. \end{aligned}$$
(27)
Integrate equality (27) over \(D(x_{3})\), apply the divergence theorem; if we use the lateral conditions (14) we get the equality (19) and the proof of Theorem 1 is completed. □
The next theorem is also dedicated to a proof of two auxiliary identities on which will be based the main result.
Theorem 2
Let
\((U_{i},\Phi_{i}, T )\)
be a solution of the boundary value problem consisting of equations (13)-(15). Then we have the identities
$$\begin{aligned} & \int_{D(x_{3})} [ A_{ijmn} \mathcal{U}_{n, m}\bar{ \mathcal{U}}_{j, i}+ C_{ijmn}\Phi_{i,j}{\bar{\Phi}}_{n, m} ]\,dA \\ &\qquad{} + \int_{D(x_{3})} \bigl\{ B_{ijmn} [ \mathcal{U}_{n, m} {\bar{\Phi}}_{i,j} + \bar{\mathcal{U}}_{n, m} \Phi_{i,j} ] - 3\omega^{2} ( \varrho U_{i}{\bar{U}}_{i} + I_{ij} \Phi _{i}{\bar{\Phi}}_{j} ) \bigr\} \,dA \\ &\qquad{}-2\iota\omega \int_{D(x_{3})} \bigl\{ D_{ij} ( T \bar{ \mathcal{U}}_{j, i} - {\bar{T}} \mathcal{U}_{j, i} ) + E_{ij} ( T{\bar{\Phi}}_{j,i}-{\bar{T}} \Phi_{j,i} ) \bigr\} \,dA \\ &\qquad{}-\iota\omega \int_{D(x_{3})} \bigl\{ D_{ij} x_{p} ( T_{,p} \bar{\mathcal{U}}_{j, i} - {\bar{T}}_{,p} \mathcal{U}_{j, i} ) + E_{ij} x_{p} ( T_{,p}{\bar{\Phi}}_{j,i} - {\bar{T}}_{,p} \Phi_{j,i} ) \bigr\} \,dA \\ &\quad=- \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn} \mathcal{U}_{n, m} +B_{3jmn} \Phi_{n, m}- \iota \omega D_{3j} T ]x_{p}{\bar{U}}_{j, p} \bigr\} \,dA \\ &\qquad{}-\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn} \bar{\mathcal{U}}_{n, m} +B_{3jmn} {\bar{\Phi}}_{n, m}+\iota \omega D_{3j} {\bar{T}} ]x_{p} U_{j, p} \bigr\} \,dA \\ &\qquad{}-\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn} \mathcal{U}_{n, m}+ C_{3jmn}\Phi_{n, m}- \iota \omega E_{3j} T ]x_{p} {\bar{\Phi}}_{j, p} \bigr\} \,dA \\ &\qquad{}-\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn} \bar{\mathcal{U}}_{n, m}+ C_{3jmn}{\bar{\Phi}}_{n, m}+ \iota \omega E_{3j} {\bar{T}} ]x_{p} \Phi_{j, p} \bigr\} \,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} x_{3} [ A_{ijmn} \bar{ \mathcal{U}}_{n, m}\bar{\mathcal{U}}_{j, i}+ C_{ijmn} \Phi_{i,j}{\bar{\Phi}}_{n, m} ]\,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} x_{3} \bigl\{ B_{ijmn} [ \mathcal{U}_{n, m} {\bar{\Phi}}_{i,j} + \bar{\mathcal{U}}_{n, m} \Phi_{i,j} ] - \varrho\omega^{2} U_{i}{\bar{U}}_{i} \bigr\} \,dA \\ &\qquad{}- \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ \iota \omega x_{3} D_{ij} ( T \bar{\mathcal{U}}_{j, i} - {\bar{T}} \mathcal{U}_{j, i} ) \bigr\} \,dA \\ &\qquad{}-\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ \iota \omega x_{3} E_{ij} ( T{\bar{\Phi}}_{j,i}-{\bar{T}} \Phi_{j,i} )-x_{3} I_{ij} \omega ^{2}\Phi_{i}{\bar{\Phi}}_{j} \bigr\} \,dA \\ &\qquad{} + \int_{\partial D(x_{3})} x_{p} n_{p} \biggl( A_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial U_{i}}{\partial n} \frac{\partial{\bar{U}}_{m}}{\partial n} + B_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial U_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{m}}{\partial n} \\ &\qquad{}+ C_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\Phi_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{m}}{\partial n} \biggr)\,ds, \end{aligned}$$
(28)
$$\begin{aligned} & \int_{D(x_{3})} \frac{1}{\theta_{0}} \bigl(K_{ij} T_{, i}{\bar{T}}_{, j}- 3c\omega^{2} T {\bar{T}} \bigr)\,dA \\ &\qquad{}+ \int_{D(x_{3})} \iota \omega E_{ij} ({\bar{\Phi}}_{j, i}T_{, p}-\Phi_{j, i}{\bar{T}}_{, p} )\,dA \\ &\qquad{}+ \int_{D(x_{3})} \iota \omega D_{ij} x_{p} ( \mathcal{U}_{j, i}T_{, p}- \mathcal{U}_{j, i}{\bar{T}}_{, p} )\,dA \\ &\qquad{}+ \int_{\partial D(x_{3})} \frac{1}{\theta_{0}} x_{p} n_{p} K_{\alpha\beta}n_{\alpha}n_{\beta}\frac {\partial T}{\partial n} \frac{\partial{\bar{T}}}{\partial n}\,ds \\ &\quad=- \frac{d}{dx_{3}} \int_{D(x_{3})} \frac{1}{\theta_{0}} \bigl[x_{\alpha} K_{3\beta} ({\bar{T}}_{, \alpha }T_{, \beta}+T_{, \alpha}{\bar{T}}_{, \beta} ) +x_{\alpha}K_{33} (T_{, 3}{\bar{T}}_{, \alpha}+{\bar{T}}_{3} T_{, \alpha} ) \bigr]\,dA \\ &\qquad{}- \frac{d}{dx_{3}} \int_{D(x_{3})} \frac{x_{3}}{\theta_{0}} \bigl(K_{33} T_{, 3} {\bar{T}}_{3}-K_{\alpha\beta} T_{, \alpha}{\bar{T}}_{, \beta}+ c\omega^{2} T {\bar{T}} \bigr)\,dA. \end{aligned}$$
(29)
Proof
Considering equations (13)1 and (13)2 it is easy to prove the following equality:
$$\begin{aligned} & \bigl\{ [A_{ijmn} \mathcal{U}_{n, m} +B_{ijmn} \Phi_{n, m}- \iota \omega D_{ij} T ]_{, j}+\varrho \omega^{2} U_{i} \bigr\} x_{p}{\bar{U}}_{i, p} \\ &\quad{}+ [B_{mnij} \mathcal{U}_{n, m}+ C_{ijmn} \Phi_{n, m}- \iota \omega E_{ij} T ]_{,j}x_{p}{ \bar{\Phi}}_{i, p} \\ &\quad{} + \varepsilon_{ijk} [A_{jkmn} \mathcal{U}_{n, m} + B_{jkmn} \Phi_{n, m} - \iota \omega D_{jk} T ]x_{p} {\bar{\Phi}}_{i, p} + I_{ij} \omega^{2} x_{p} {\bar{\Phi}}_{i, p}\Phi_{j} \\ &\quad{}+ \bigl\{ [A_{ijmn} \bar{\mathcal{U}}_{n, m} +B_{ijmn} {\bar{\Phi}}_{n, m}+ \iota \omega D_{ij} {\bar{T}} ]_{, j}+\varrho\omega^{2} {\bar{U}}_{i} \bigr\} x_{p} U_{i, p} \\ &\quad{}+ [B_{mnij} \bar{\mathcal{U}}_{n, m}+ C_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{ij} {\bar{T}} ]_{,j}x_{p} \Phi_{i, p} \\ &\quad{} + \varepsilon_{ijk} [A_{jkmn} \bar{ \mathcal{U}}_{n, m} + B_{jkmn} {\bar{\Phi}}_{n, m} + \iota \omega D_{jk} {\bar{T}} ] x_{p} \Phi_{i, p} + I_{ij} \omega^{2} x_{p} \Phi_{i, p}{\bar{\Phi}}_{j} = 0. \end{aligned}$$
(30)
With simple calculations, equality (30) can be written in the form
$$\begin{aligned} & \bigl\{ [A_{ijmn} \mathcal{U}_{n, m} +B_{ijmn} \Phi_{n, m}- \iota \omega D_{ij} T ]x_{p}{\bar{U}}_{i, p} \bigr\} _{, j} \\ &\quad{}- [A_{ijmn} \mathcal{U}_{n, m} + B_{ijmn} \Phi_{n, m} - \iota \omega D_{ij} T ]x_{p}{\bar{U}}_{i, pj} + \varrho\omega^{2}x_{p} U_{i} { \bar{U}}_{i, p} \\ &\quad{}+ \bigl\{ [B_{mnij} \mathcal{U}_{n, m}+ C_{ijmn} \Phi_{n, m}- \iota \omega E_{ij} T ]x_{p} {\bar{\Phi}}_{i, p} \bigr\} _{,j} \\ &\quad{}- [B_{mnij} \mathcal{U}_{n, m}+ C_{ijmn} \Phi_{n, m}- \iota \omega E_{ij} T ]x_{p} {\bar{\Phi}}_{i, pj} \\ &\quad{} + \varepsilon_{ijk} [A_{jkmn} \mathcal{U}_{n, m} + B_{jkmn} \Phi_{n, m} - \iota \omega D_{jk} T ]x_{p} {\bar{\Phi}}_{i, p} + I_{ij} \omega^{2} x_{p} {\bar{\Phi}}_{i, p}\Phi_{j} \\ &\quad{}+ \bigl\{ [A_{ijmn} \bar{\mathcal{U}}_{n, m} +B_{ijmn} {\bar{\Phi}}_{n, m}+\iota \omega D_{ij} {\bar{T}} ]x_{p} U_{i, p} \bigr\} _{, j} \\ &\quad{} - [A_{ijmn} \bar{\mathcal{U}}_{n, m} + B_{ijmn} {\bar{\Phi}}_{n, m} + \iota \omega D_{ij} {\bar{T}} ]x_{p} U_{i, pj} + \varrho\omega^{2}x_{p} {\bar{U}}_{i} U_{i, p} \\ &\quad{}+ \bigl\{ [B_{mnij} \bar{\mathcal{U}}_{n, m}+ C_{ijmn}{\bar{\Phi}}_{n, m}+ \iota \omega E_{ij} {\bar{T}} ]x_{p} \Phi_{i, p} \bigr\} _{,j} \\ &\quad{}- [B_{mnij} \bar{\mathcal{U}}_{n, m}+ C_{ijmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{ij} {\bar{T}} ]x_{p} \Phi_{i, pj} \\ &\quad{} + \varepsilon_{ijk} [A_{jkmn} \bar{ \mathcal{U}}_{n, m} + B_{jkmn} {\bar{\Phi}}_{n, m} + \iota \omega D_{jk} {\bar{T}} ]x_{p} \Phi_{i, p} + I_{ij} \omega^{2} x_{p} \Phi_{i, p}{\bar{\Phi}}_{j} = 0. \end{aligned}$$
(31)
This equality leads to
$$\begin{aligned} &A_{ijmn} \mathcal{U}_{n, m}\bar{ \mathcal{U}}_{j, i}+ C_{ijmn}\Phi_{i,j}{\bar{\Phi}}_{n, m} \\ &\qquad{}+B_{ijmn} ( \mathcal{U}_{n, m} {\bar{\Phi}}_{i,j}+ \bar{\mathcal{U}}_{n, m} \Phi_{i,j} ) -3 \omega^{2} (\varrho U_{i}{\bar{U}}_{i}+ I_{ij} \Phi_{i}{\bar{\Phi}}_{j} ) \\ &\qquad{}-2\iota \omega D_{ij} ( T \bar{\mathcal{U}}_{j, i}-{ \bar{T}} \mathcal{U}_{j, i} ) -2\iota \omega E_{ij} ( T{\bar{\Phi}}_{j,i}-{\bar{T}} \Phi_{j,i} ) \\ &\qquad{}- \iota\omega D_{ij} x_{p} ( T_{,p} \bar{\mathcal{U}}_{j, i} - {\bar{T}}_{,p} \mathcal{U}_{j, i} )- \iota \omega E_{ij} x_{p} ( T_{,p}{\bar{\Phi}}_{j,i}-{\bar{T}}_{,p} \Phi_{j,i} ) \\ &\quad=- \bigl\{ [A_{ijmn} \mathcal{U}_{n, m} +B_{ijmn} \Phi_{n, m}- \iota \omega D_{ij} T ]x_{p}{\bar{U}}_{i, p} \bigr\} _{, j} \\ &\qquad{}- \bigl\{ [A_{ijmn} \bar{\mathcal{U}}_{n, m} +B_{ijmn} {\bar{\Phi}}_{n, m}+\iota \omega D_{ij} {\bar{T}} ]x_{p} U_{i, p} \bigr\} _{, j} \\ &\qquad{}- \bigl\{ [B_{mnij} \mathcal{U}_{n, m}+ C_{ijmn}\Phi_{n, m}- \iota \omega E_{ij} T ]x_{p} {\bar{\Phi}}_{i, p} \bigr\} _{,j} \\ &\qquad{}- \bigl\{ [B_{mnij} \bar{\mathcal{U}}_{n, m}+ C_{ijmn}{\bar{\Phi}}_{n, m}+ \iota \omega E_{ij} {\bar{T}} ]x_{p} \Phi_{i, p} \bigr\} _{,j} \\ &\qquad{}+ [x_{k} A_{ijmn} \mathcal{U}_{n, m}\bar{ \mathcal{U}}_{j, i}+x_{p} C_{ijmn}\Phi_{i,j}{ \bar{\Phi}}_{n, m} ]_{, p} \\ &\qquad{}+ \bigl\{ x_{p} B_{ijmn} [ \mathcal{U}_{n, m} {\bar{\Phi}}_{i,j}+ \bar{\mathcal{U}}_{n, m} \Phi_{i,j} ] -x_{p}\varrho\omega^{2} U_{i}{\bar{U}}_{i} \bigr\} _{, p} \\ &\qquad{}- \bigl\{ \iota \omega x_{p} D_{ij} ( T \bar{ \mathcal{U}}_{j, i}-{\bar{T}} \mathcal{U}_{j, i} ) \bigr\} _{, p} \\ &\qquad{}- \bigl\{ \iota \omega x_{p} E_{ij} ( T{\bar{\Phi}}_{j,i}-{\bar{T}} \Phi_{j,i} )-x_{p} I_{ij} \omega ^{2}\Phi_{i}{\bar{\Phi}}_{j} \bigr\} _{, p}. \end{aligned}$$
(32)
We integrate equality (32) and use the lateral boundary condition (14); then we are led to the equality
$$\begin{aligned} & \int_{D(x_{3})} [ A_{ijmn} \bar{\mathcal{U}}_{n, m} \bar{\mathcal{U}}_{j, i}+ C_{ijmn}\Phi_{i,j}{\bar{\Phi}}_{n, m} ]\,dA \\ &\qquad{}+ \int_{D(x_{3})} \bigl\{ B_{ijmn} [ \mathcal{U}_{n, m} {\bar{\Phi}}_{i,j}+ \bar{\mathcal{U}}_{n, m} \Phi_{i,j} ] -3\omega^{2} (\varrho U_{i}{\bar{U}}_{i}+ I_{ij} \Phi_{i}{\bar{\Phi}}_{j} ) \bigr\} \,dA \\ &\qquad{}-2\iota\omega \int_{D(x_{3})} \bigl\{ D_{ij} ( T \bar{ \mathcal{U}}_{j, i}-{\bar{T}} \mathcal{U}_{j, i} ) + E_{ij} ( T{\bar{\Phi}}_{j,i}-{\bar{T}} \Phi_{j,i} ) \bigr\} \,dA \\ &\qquad{}- \iota\omega \int_{D(x_{3})} \bigl\{ D_{ij} x_{p} ( T_{,p} \bar{\mathcal{U}}_{j, i}-{\bar{T}}_{,p} \mathcal{U}_{j, i} )+ E_{ij} x_{p} ( T_{,p}{\bar{\Phi}}_{j,i}-{\bar{T}}_{,p} \Phi_{j,i} ) \bigr\} \,dA \\ &\quad=- \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn} \mathcal{U}_{n, m} +B_{3jmn} \Phi_{n, m}- \iota \omega D_{3j} T ]x_{p}{\bar{U}}_{j, p} \bigr\} \,dA \\ &\qquad{}-\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn} \bar{\mathcal{U}}_{n, m} +B_{3jmn} {\bar{\Phi}}_{n, m}+\iota \omega D_{3j} {\bar{T}} ]x_{p} U_{j, p} \bigr\} \,dA \\ &\qquad{}-\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\mathcal{U}_{n, m}+ C_{3jmn}\Phi_{n, m}- \iota \omega E_{3j} T ]x_{p} {\bar{\Phi}}_{j, p} \bigr\} \,dA \\ &\qquad{}-\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\bar{\mathcal{U}}_{n, m}+ C_{3jmn}{\bar{\Phi}}_{n, m}+ \iota \omega E_{3j} {\bar{T}} ]x_{p} \Phi_{j, p} \bigr\} \,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} x_{3} [ A_{ijmn} \bar{ \mathcal{U}}_{n, m}\bar{\mathcal{U}}_{j, i}+ C_{ijmn} \Phi_{i,j}{\bar{\Phi}}_{n, m} ]\,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} x_{3} \bigl\{ B_{ijmn} [ \mathcal{U}_{n, m} {\bar{\Phi}}_{i,j}+ \bar{\mathcal{U}}_{n, m} \Phi_{i,j} ] -\varrho\omega^{2} U_{i}{\bar{U}}_{i} \bigr\} \,dA \\ &\qquad{}- \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ \iota \omega x_{3} D_{ij} ( T \bar{\mathcal{U}}_{j, i}-{\bar{T}} \mathcal{U}_{j, i} ) \bigr\} \,dA \\ &\qquad{}-\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ \iota \omega x_{3} E_{ij} ( T{\bar{\Phi}}_{j,i}-{\bar{T}} \Phi_{j,i} )-x_{3} I_{ij} \omega ^{2}\Phi_{i}{\bar{\Phi}}_{j} \bigr\} \,dA \\ &\qquad{}- \int_{\partial D(x_{3})} [ x_{p} \bar{\mathcal{U}}_{s, p} A_{psmn} \bar{\mathcal{U}}_{n, m}+ x_{p} \mathcal{U}_{s, p} A_{psmn} \bar{\mathcal{U}}_{s, p} ] n_{p} \,ds \\ &\qquad{} - \int_{\partial D(x_{3})} [ x_{p}{\bar{\Phi}}_{s,p}B_{psmn} \mathcal{U}_{n, m}+ x_{p} \Phi_{s,p}B_{psmn} \bar{\mathcal{U}}_{n, m} ] n_{p} \,ds \\ &\qquad{} - \int_{\partial D(x_{3})} [ x_{p}{\bar{\Phi}}_{s,p}C_{psmn} \Phi_{n,m}+ x_{p} \Phi_{s,p}C_{psmn} {\bar{\Phi}}_{n,m} ] n_{p} \,ds \\ &\qquad{}+ \int_{\partial D(x_{3})} x_{p} n_{p} \bigl\{ A_{ijmn} \bar{\mathcal{U}}_{n, m}\mathcal{U}_{j, i}+C_{ijmn} \Phi_{j, i}{\bar{\Phi}}_{n, m} \\ &\qquad{}+ B_{ijmn} [ \mathcal{U}_{j, i}{\bar{\Phi}}_{n,m}+ \bar{\mathcal{U}}_{j, i}\Phi_{n,m} ] \bigr\} \,ds. \end{aligned}$$
(33)
If we take into account the lateral boundary condition (14) we conclude that
$$\begin{aligned} U_{i, 3}=0 \quad \mbox{on } \partial D(x_{3}). \end{aligned}$$
(34)
On the curve ∂D we have
$$U_{i, \alpha}=n_{\alpha}\frac{\partial U_{i}}{\partial n}+ \tau_{\alpha} \frac{\partial U_{i}}{\partial\tau}, $$
where \(\tau_{\alpha}\) are components of the unit vector tangent to ∂D and \(\partial/ \partial\tau\) is the tangential derivative. According to the lateral boundary condition (14) we deduce \(\partial U_{i}/\partial\tau=0 \) on the curve ∂D and hence we obtain
$$\begin{aligned} U_{i, \alpha}=n_{\alpha} \frac{\partial U_{i}}{\partial n} \quad\mbox{on the curve } \partial D. \end{aligned}$$
(35)
Using equations (34) and (35), the last integral in (33) becomes
$$\begin{aligned} & \int_{\partial D(x_{3})} x_{p} n_{p} ( A_{ijmn} \mathcal{U}_{j,i} \bar{\mathcal{U}}_{n,m}+B_{ijmn} \mathcal{U}_{j,i}{\bar{\Phi}}_{n,m}+ C_{ijmn} \Phi_{j,i}{\bar{\Phi}}_{n,m} )\,ds \\ &\quad = \int_{\partial D(x_{3})} x_{p} n_{p} \biggl( A_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\mathcal{U}_{i}}{\partial n} \frac{\partial\bar{\mathcal{U}}_{m}}{\partial n} + B_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\mathcal{U}_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{m}}{\partial n} \\ &\qquad{}+ C_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\Phi_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{m}}{\partial n} \biggr)\,ds. \end{aligned}$$
(36)
For the other integrals in (33) we obtain
$$ \begin{aligned} & \int_{\partial D(x_{3})} [ x_{p} \bar{\mathcal{U}}_{s,p} A_{psmn} \mathcal{U}_{n,m} +x_{p} \mathcal{U}_{s,p} A_{psmn} \bar{\mathcal{U}}_{n,m} ]n_{p} \,ds \\ &\quad=2 \int_{\partial D(x_{3})} x_{p} n_{p} A_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\mathcal{U}_{i}}{\partial n} \frac{\partial\bar{\mathcal{U}}_{m}}{\partial n}\,ds, \\ & \int_{\partial D(x_{3})} [ x_{p} \bar{\mathcal{U}}_{s,p} B_{psmn} \Phi_{n,m} +x_{p} \mathcal{U}_{s,p} C_{psmn} {\bar{\Phi}}_{n,m} ]n_{p} \,ds \\ &\quad=2 \int_{\partial D(x_{3})} x_{p} n_{p} B_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\mathcal{U}_{i}}{\partial n} \frac{\partial{\bar{\Phi}}_{m}}{\partial n}\,ds, \\ & \int_{\partial D(x_{3})} [ x_{p} {\bar{\Phi}}_{s,p} C_{psmn} \Phi_{n,m} +x_{p} \Phi_{s,p} C_{psmn} {\bar{\Phi}}_{n,m} ]n_{p} \,ds \\ &\quad=2 \int_{\partial D(x_{3})} x_{p} n_{p} C_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\Phi_{i}}{\partial n} \frac{\partial{\bar{\Phi}}_{m}}{\partial n}\,ds. \end{aligned} $$
(37)
If we substitute the results of equations (36) and (37) in the equality (33), we obtain the first relation of Theorem 2, namely equation (28).
To prove equation (29) we start from the following equality, which is evident:
$$\begin{aligned} &x_{p} {\bar{T}}_{p} \biggl[ \biggl( \frac{1}{\theta_{0}}K_{ij} T_{, j} \biggr)_{, i}- \iota \omega D_{ij}\mathcal{U}_{j, i}- \iota \omega E_{ij} \Phi_{j, i}+\frac{c}{\theta_{0}}\omega^{2} T \biggr] \\ &\quad{}+ x_{p} T_{p} \biggl[ \biggl(\frac{1}{\theta_{0}}K_{ij} {\bar{T}}_{, j} \biggr)_{, i}+\iota \omega D_{ij} \mathcal{U}_{j, i}- \iota \omega E_{ij}{\bar{\Phi}}_{j, i}+\frac{c}{\theta_{0}}\omega^{2} {\bar{T}} \biggr]=0. \end{aligned}$$
(38)
After some direct calculations, equality (38) acquires the form
$$\begin{aligned} &\iota \omega D_{ij} x_{p} ( \mathcal{U}_{j, i}T_{, p}- \mathcal{U}_{j, i}{\bar{T}}_{, p} )+ \iota \omega E_{ij} x_{p} ({\bar{\Phi}}_{j, i}T_{, p}-\Phi_{j, i}{\bar{T}}_{, p} ) \\ &\quad=- x_{p} \biggl(\frac{c}{\theta_{0}}\omega^{2} T {\bar{T}} \biggr)_{, p}+ \frac{2}{\theta_{0}} K_{ij} T_{, i}{ \bar{T}}_{, j} \\ &\qquad{}- \biggl[\frac{1}{\theta_{0}}x_{p} K_{ij} ({\bar{T}}_{, p}T_{, j}+T_{, p}{\bar{T}}_{, j} ) \biggr]_{, i}+ x_{p} \biggl( \frac{1}{\theta_{0}}K_{ij} T_{, i}{\bar{T}}_{, j} \biggr)_{, p}. \end{aligned}$$
(39)
This equality can be rewritten as follows:
$$\begin{aligned} &\frac{1}{\theta_{0}} K_{ij} T_{, i}{\bar{T}}_{, j}- \frac{3c}{\theta_{0}}\omega^{2} T {\bar{T}}+ \iota \omega E_{ij} ({\bar{\Phi}}_{j, i}T_{, p}- \Phi_{j, i}{\bar{T}}_{, p} ) + \iota \omega D_{ij} x_{p} (\mathcal{U}_{j, i}T_{, p}- \mathcal{U}_{j, i}{\bar{T}}_{, p} ) \\ &\quad=- \biggl(\frac{c}{\theta_{0}}\omega^{2} T {\bar{T}} \biggr)_{, p}- \biggl[\frac{1}{\theta_{0}}x_{p} K_{ij} ({\bar{T}}_{, p}T_{, j}+T_{, p}{\bar{T}}_{, j} ) \biggr]_{, i} + \biggl( \frac{x_{p}}{\theta_{0}}K_{ij} T_{, i}{\bar{T}}_{, j} \biggr)_{, p}. \end{aligned}$$
(40)
Now we integrate the equality (4) on \(D(x_{3})\) and, after using the lateral boundary condition (14), we are led to
$$\begin{aligned} & \int_{D(x_{3})} \frac{1}{\theta_{0}} \bigl(K_{ij} T_{, i}{\bar{T}}_{, j}- 3c\omega^{2} T {\bar{T}} \bigr)\,dA+ \int_{D(x_{3})} \iota \omega E_{ij} ({\bar{\Phi}}_{j, i}T_{, p}-\Phi_{j, i}{\bar{T}}_{, p} )\,dA \\ &\qquad{}+ \int_{D(x_{3})} \iota \omega D_{ij} x_{p} ( \mathcal{U}_{j, i}T_{, p}- \mathcal{U}_{j, i}{\bar{T}}_{, p} )\,dA \\ &\quad=- \frac{d}{dx_{3}} \int_{D(x_{3})} \biggl[\frac{1}{\theta_{0}}x_{p} K_{3j} ({\bar{T}}_{, p}T_{, j}+T_{, p}{\bar{T}}_{, j} ) -\frac{x_{3}}{\theta_{0}}K_{ij} T_{, i}{\bar{T}}_{, j}+ \frac{x_{3}}{\theta_{0}}c\omega^{2} T {\bar{T}} \biggr]\,dA \\ &\qquad{}+ \int_{\partial D(x_{3})} \frac{1}{\theta_{0}} \bigl[x_{p} n_{p}K_{ij}T_{, i}{\bar{T}}_{, j}- x_{p} K_{pj} ({\bar{T}}_{, p}T_{, j}+T_{, p}{ \bar{T}}_{, j} )n_{p} \bigr]\,ds. \end{aligned}$$
(41)
As we have already shown in the proof of equality (28), the lateral boundary condition implies
$$T_{, 3}=0,\qquad T_{, \alpha}=n_{\alpha}\frac{\partial T}{\partial n}, $$
on the curve \(\partial D(x_{3})\).
With these arguments, the equality (41) implies equation (29), therefore the proof of Theorem 2 is completed. □
The conservation laws which will be proved in the following theorem will be used to derive a priori estimates for a solution of our mixed problem.
Theorem 3
Let
\((U_{i},\Phi_{i}, T )\)
be a solution of the boundary value problem consisting of equations (13)-(15). Then the following two conservation laws are satisfied:
$$\begin{aligned} &\frac{d}{d x_{3}} \int_{D(x_{3})}\omega^{2} \biggl( \varrho U_{j}{ \bar{U}}_{j}+I_{ij} \Phi_{i}{\bar{\Phi}}_{j}+\frac{c}{\theta_{0}} T {\bar{T}} \biggr)\,dA \\ &\qquad{}+ \frac{d}{d x_{3}} \int_{D(x_{3})} \frac{c}{\theta_{0}} \biggl(\frac{}{} K_{33} T_{, 3} {\bar{T}}_{, 3}- K_{\alpha\beta} T_{, \alpha} {\bar{T}}_{, \beta} \biggr)\,dA \\ &\qquad{} + \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl[ A_{i3m3} \mathcal{U}_{i,3} \bar{\mathcal{U}}_{m,3} + B_{i3m3} (\mathcal{U}_{i,3}{ \bar{\Phi}}_{m,3} + \bar{\mathcal{U}}_{i,3} \Phi_{m,3} ) + C_{i3m3} \Phi_{i,3}{\bar{\Phi}}_{m,3} \bigr]\,dA \\ &\qquad{} - \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl[ A_{i\alpha m\beta} \mathcal{U}_{i,\alpha} \bar{\mathcal{U}}_{m,\beta} + B_{i\alpha m3} (\mathcal{U}_{i,\alpha}{ \bar{\Phi}}_{m,\beta} + \bar{\mathcal{U}}_{i,\alpha} \Phi_{m,\beta} ) \\ &\qquad{}+ C_{i\alpha m\beta} \Phi_{i,\alpha}{\bar{\Phi}}_{m, \beta} \bigr]\,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl[ \iota\omega D_{i\alpha} (T\bar{ \mathcal{U}}_{i, \alpha}-{\bar{T}} \mathcal{U}_{i, \alpha} )+ \iota\omega E_{i\alpha} (T{\bar{\Phi}}_{i, \alpha}-{\bar{T}} \Phi _{i, \alpha} ) \bigr]\,dA=0, \end{aligned}$$
(42)
$$\begin{aligned} &\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn}\bar{\mathcal{U}}_{n, m}+B_{3jmn}{ \bar{\Phi}}_{n, m}+ \iota \omega D_{3j}{\bar{T}} ]U_{j} \bigr\} \,dA \\ &\qquad{}- \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn}\mathcal{U}_{n, m}+B_{3jmn} \Phi_{n, m}- \iota \omega D_{3j}T ]{\bar{U}}_{j} \bigr\} \,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\bar{\mathcal{U}}_{n, m}+C_{3jmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{3j}{\bar{T}} ] \Phi_{j} \bigr\} \,dA \\ &\qquad{}- \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\mathcal{U}_{n, m}+C_{3jmn} \Phi_{n, m}- \iota \omega E_{3j}T ] {\bar{\Phi}}_{j} \bigr\} \,dA \\ &\quad= \frac{d}{d x_{3}} \int_{D(x_{3})} \biggl[ \frac{1}{\theta_{0}}K_{3j} ({\bar{T}}T_{, j}-T{\bar{T}}_{, j} ) \biggr]\,dA. \end{aligned}$$
(43)
Proof
To prove equation (42) we start by using equations (13)1 and (13)2; with the help of these we obtain the following equality:
$$\begin{aligned} & \bigl\{ [A_{ijmn} \mathcal{U}_{n,m}+B_{ijmn} \Phi_{n,m}-\iota\omega D_{ij} T ]_{, i}+\varrho \omega^{2} U_{j} \bigr\} {\bar{U}}_{j, 3} \\ &\quad{}+ \bigl\{ [B_{ijmn} \mathcal{U}_{n,m}+C_{ijmn} \Phi_{n,m}-\iota\omega E_{ij} T ]_{, i} \\ &\quad{} + \varepsilon_{jik} [ A_{ikmn} \mathcal{U}_{n,m}+B_{ikmn} \Phi_{n,m}-\iota\omega D_{ik} T ]+ I_{ij}\omega^{2} \Phi_{i} \bigr\} {\bar{\Phi}}_{j, 3} \\ &\quad{}+ \bigl\{ [A_{ijmn} \bar{\mathcal{U}}_{n,m}+B_{ijmn} {\bar{\Phi}}_{n,m}+\iota\omega D_{ij} {\bar{T}} ]_{, i}+\varrho\omega^{2} {\bar{U}}_{j} \bigr\} U_{j, 3} \\ &\quad{}+ \bigl\{ [B_{ijmn} \bar{\mathcal{U}}_{n,m}+C_{ijmn}{ \bar{\Phi}}_{n,m}+\iota\omega E_{ij} {\bar{T}} ]_{, i} \\ &\quad{} + \varepsilon_{jik} [ A_{ikmn} \bar{ \mathcal{U}}_{n,m}+B_{ikmn} {\bar{\Phi}}_{n,m}+\iota\omega D_{ik} {\bar{T}} ]+ I_{ij}\omega^{2} {\bar{\Phi}}_{i} \bigr\} \Phi_{j, 3}=0. \end{aligned}$$
(44)
Performing direct calculations on equality (44) we are led to
$$\begin{aligned} &\frac{d}{d x_{3}} \bigl[\varrho\omega^{2} U_{j}{\bar{U}}_{j}+I_{ij}\omega^{2} \Phi _{i}{\bar{\Phi}}_{j}+ A_{i3m3} \mathcal{U}_{i,3} \bar{\mathcal{U}}_{m, 3}+ B_{i3m3} (\mathcal{U}_{i,3}{ \bar{\Phi}}_{m, 3}+ \bar{\mathcal{U}}_{i,3} \Phi_{m, 3} ) \\ &\quad{}+C_{i3m3} \Phi_{i,3}{\bar{\Phi}}_{m, 3}-A_{i\alpha m\beta} \mathcal{U}_{i,\alpha}\bar{\mathcal{U}}_{m, \beta}- B_{i\alpha m3} ( \mathcal{U}_{i,\alpha}{\bar{\Phi}}_{m, \beta}+ \bar{\mathcal{U}}_{i,\alpha} \Phi_{m, \beta} ) \\ &\quad{} -C_{i\alpha m\beta} \Phi_{i,\alpha}{\bar{\Phi}}_{m, \beta}+ \iota\omega D_{i\alpha} (T\bar{\mathcal{U}}_{i, \alpha}-{\bar{T}} \mathcal{U}_{i, \alpha} )+ \iota\omega E_{i\alpha} (T{\bar{\Phi}}_{i, \alpha}-{\bar{T}} \Phi _{i, \alpha} ) \bigr] \\ &\quad{}+ \bigl[ A_{i\alpha m3} \mathcal{U}_{m,3}\bar{ \mathcal{U}}_{i, \alpha} + B_{i\alpha m3} (\mathcal{U}_{m, 3}{\bar{\Phi}}_{i, \alpha} + \bar{\mathcal{U}}_{m,3} \Phi_{i, \alpha} ) + C_{i\alpha m3} \Phi_{m, 3}{\bar{\Phi}}_{i, \alpha} \bigr]_{, \alpha} \\ &\quad{}+ \bigl[\iota\omega D_{i\alpha} ({\bar{T}} \mathcal{U}_{i, 3}- T \bar{\mathcal{U}}_{i, 3} ) \bigr]_{, \alpha}+ \bigl[\iota\omega E_{i\alpha} ({\bar{T}} \Phi_{i, 3}- T {\bar{\Phi}}_{i, 3} ) \bigr]_{, \alpha} \\ &\quad{}+\iota\omega D_{ij} ({\bar{T}}_{, 3} \mathcal{U}_{i, j}- T_{, 3} \bar{\mathcal{U}}_{i, j} )+ \iota\omega E_{ij} ({\bar{T}}_{, 3} \Phi_{i, j}- T_{, 3} {\bar{\Phi}}_{i, j} )=0. \end{aligned}$$
(45)
Now integrate equality (45) and use the lateral boundary condition (14); we get
$$\begin{aligned} &\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl[\varrho\omega^{2} U_{j} \bar{\mathcal{U}}_{j} + I_{ij}\omega^{2} \Phi_{i}{\bar{\Phi}}_{j} + A_{i3m3} U_{i,3} \bar{\mathcal{U}}_{m, 3} + B_{i3m3} (\mathcal{U}_{i,3}{ \bar{\Phi}}_{m, 3} + \bar{\mathcal{U}}_{i,3} \Phi_{m, 3} ) \\ &\quad{}+ C_{i3m3} \Phi_{i,3}{\bar{\Phi}}_{m, 3}-A_{i\alpha m\beta} \mathcal{U}_{i,\alpha}\bar{\mathcal{U}}_{m, \beta}- B_{i\alpha m3} ( \mathcal{U}_{i,\alpha}{\bar{\Phi}}_{m, \beta}+ \bar{\mathcal{U}}_{i,\alpha} \Phi_{m, \beta} ) \\ &\quad{}-C_{i\alpha m\beta} \Phi_{i,\alpha}{\bar{\Phi}}_{m, \beta}+ \iota\omega D_{i\alpha} (T\bar{\mathcal{U}}_{i, \alpha}-{\bar{T}} \mathcal{U}_{i, \alpha} )+ \iota\omega E_{i\alpha} (T{\bar{\Phi}}_{i, \alpha}-{\bar{T}} \Phi _{i, \alpha} ) \bigr]\,dA \\ &\quad{}+ \int_{D(x_{3})} \bigl[ \iota\omega D_{ij} ({\bar{T}}_{, 3} \mathcal{U}_{i, j}- T_{, 3} \bar{ \mathcal{U}}_{i, j} )+ \iota\omega E_{ij} ({\bar{T}}_{, 3} \Phi_{i, j}- T_{, 3} {\bar{\Phi}}_{i, j} ) \bigr]\,dA=0. \end{aligned}$$
(46)
Using equation (13)3, it is clear that
$$\begin{aligned} &{\bar{T}}_{, 3} \biggl[ \frac{1}{\theta_{0}}K_{ij} T_{, ij}- \iota\omega (D_{ij}\mathcal{U}_{i, j}+E_{ij} \Phi_{i, j} )+ \frac{c}{\theta_{0}}\omega^{2} T \biggr] \\ &\quad{}+T_{, 3} \biggl[ \frac{1}{\theta_{0}}K_{ij} {\bar{T}}_{, ij}+ \iota\omega (D_{ij}\bar{\mathcal{U}}_{i, j}+E_{ij}{ \bar{\Phi}}_{i, j} )+ \frac{c}{\theta_{0}}\omega^{2} {\bar{T}} \biggr]=0. \end{aligned}$$
(47)
After doing some calculations, we can write equation (47) in the form
$$\begin{aligned} &\frac{d}{d x_{3}} \biggl( \frac{c}{\theta_{0}}\omega^{2} T {\bar{T}}+ \frac{1}{\theta_{0}} K_{33} T_{, 3} {\bar{T}}_{, 3}- \frac{1}{\theta_{0}}K_{\alpha\beta} T_{, \alpha} {\bar{T}}_{, \beta } \biggr) \\ &\quad{}+ \biggl( \frac{2}{\theta_{0}} K_{\alpha3} T_{, 3} {\bar{T}}_{, 3} \biggr)_{, \alpha}+ \iota\omega D_{ij} (T_{, 3} \bar{\mathcal{U}}_{i, j}- {\bar{T}}_{, 3} \mathcal{U}_{i, j} ) \\ &\quad{}+ \iota\omega E_{ij} (T_{, 3} {\bar{\Phi}}_{i, j}- {\bar{T}}_{, 3} \Phi_{i, j} )=0. \end{aligned}$$
(48)
Now we integrate (48) on \(D(x_{3})\) and use the lateral boundary condition (14); we arrive at the equality
$$\begin{aligned} &\frac{d}{d x_{3}} \int_{D(x_{3})} \biggl( \frac{c}{\theta_{0}}\omega^{2} T {\bar{T}}+ \frac{1}{\theta_{0}} K_{33} T_{, 3} {\bar{T}}_{, 3}- \frac{1}{\theta_{0}} K_{\alpha\beta} T_{, \alpha} {\bar{T}}_{, \beta } \biggr)\,dA \\ &\quad{}+ \int_{D(x_{3})} \bigl[ \iota\omega D_{ij} (T_{, 3} \bar{\mathcal{U}}_{i, j}- {\bar{T}}_{, 3} \mathcal{U}_{i, j} )+ \iota\omega E_{ij} (T_{, 3} {\bar{\Phi}}_{i, j}- { \bar{T}}_{, 3} \Phi_{i, j} ) \bigr]\,dA=0. \end{aligned}$$
(49)
By using equations (49) and (46) we obtain the equality (42). The conservation law (43) is obtained immediately equaling the right-side members of equality (17) and (19). This concludes the proof of Theorem 3. □
Combining equalities (16)-(19) of Theorem 1 with equalities (28)-(29) of Theorem 2 and those of Theorem 3, namely (42)-(43), we obtain various measures associated with the amplitude \((U_{i},\Phi_{i}, T )\). With the help of these measures, we will obtain suitable spatial estimates to describe the spatial behavior of the respective amplitude.
The next result is a first estimate which describes the spatial behavior of the solution.
Theorem 4
Let
\((U_{i},\Phi_{i}, T )\)
be a solution of the boundary value problem consisting of equations (13)-(15). Then the following equality holds:
$$\begin{aligned} & \int_{D(x_{3})} \biggl[ A_{ijmn}\mathcal{U}_{j, i} \bar{\mathcal{U}}_{n, m}+ B_{ijmn} (\mathcal{U}_{j, i}{ \bar{\Phi}}_{n, m}+\bar{\mathcal{U}}_{j, i}\Phi_{n, m} )+ C_{ijmn}\Phi_{j, i}{\bar{\Phi}}_{n, m} \\ &\qquad{} - \omega^{2} \biggl( \varrho U_{i} {\bar{U}}_{i}+I_{ij}\Phi_{i}{\bar{\Phi}}_{j}+ \frac{c}{\theta_{0}}T \bar{T} \biggr)+ \frac{c}{\theta_{0}}K_{ij}T_{, i} \bar{T}_{, j} \biggr]\,dA \\ &\qquad{}+ \int_{D(x_{3})} \bigl[ \iota\omega D_{ij} ({\bar{T}} \mathcal{U}_{j, i}- T \bar{\mathcal{U}}_{j, i} )+ \iota\omega E_{ij} ({\bar{T}}\Phi_{j, i}- T {\bar{\Phi}}_{j, i} ) \bigr]\,dA \\ &\quad=\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn}\mathcal{U}_{n, m}+B_{3jmn} \Phi_{n, m}- \iota \omega D_{3j}T ]{\bar{U}}_{j} \bigr\} \,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [A_{3jmn}\bar{\mathcal{U}}_{n, m}+B_{3jmn}{ \bar{\Phi}}_{n, m}+ \iota \omega D_{3j}{\bar{T}} ]U_{j} \bigr\} \,dA \\ &\qquad{}+ \frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\mathcal{U}_{n, m}+C_{3jmn} \Phi_{n, m}- \iota \omega E_{3j}T ] {\bar{\Phi}}_{j} \bigr\} \,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl\{ [B_{3jmn}\bar{\mathcal{U}}_{n, m}+C_{3jmn}{ \bar{\Phi}}_{n, m}+ \iota \omega E_{3j}{\bar{T}} ] \Phi_{j} \bigr\} \,dA \\ &\qquad{}+ \frac{d}{d x_{3}} \int_{D(x_{3})} \frac{1}{\theta_{0}}K_{33} (T \bar{T}_{, 3}+ \bar{T}T_{, 3} )\,dA. \end{aligned}$$
(50)
Proof
By combining equations (16) and (18) we obtain immediately the above desired identity (50). □
Another a priori estimate will be proved in the next theorem.
Theorem 5
If
\((U_{i},\Phi_{i}, T )\)
is a solution of the boundary value problem consisting of equations (13)-(15), then we have
$$\begin{aligned} & \int_{D(x_{3})} \biggl[ A_{ijmn}\mathcal{U}_{j, i} \bar{\mathcal{U}}_{n, m}+ B_{ijmn} (\mathcal{U}_{j, i}{ \bar{\Phi}}_{n, m}+{\bar{U}}_{j, i}\Phi _{n, m} )+ C_{ijmn}\Phi_{j, i}{\bar{\Phi}}_{n, m} \\ &\qquad{} + \frac{1}{\theta_{0}}K_{ij}T_{, i}{\bar{T}}_{, j}+\omega^{2} \biggl(\varrho U_{i}{\bar{U}}_{i}+I_{ij}\Phi_{i}{\bar{\Phi}}_{j}+ \frac{1}{\theta_{0}}T{\bar{T}} \biggr) \biggr]\,dA \\ &\qquad{}- \int_{\partial D(x_{3})} x_{p} n_{p} \biggl( A_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\mathcal{U}_{i}}{\partial n} \frac{\partial\bar{\mathcal{U}}_{m}}{\partial n} + B_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\mathcal{U}_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{m}}{\partial n} \\ &\qquad{}+ C_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\Phi_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{m}}{\partial n} \biggr)\,ds - \int_{\partial D(x_{3})} \frac{1}{\theta_{0}} x_{p} n_{p} K_{\alpha\beta}n_{\alpha}n_{\beta}\frac {\partial T}{\partial n} \frac{\partial{\bar{T}}}{\partial n}\,ds \\ &\quad=\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl[ ( A_{3jmn}\mathcal{U}_{n, m}+B_{3jmn} \Phi_{n, m}-\iota\omega D_{3j}T ) (\bar{\mathcal{U}}_{j}+x_{p} \bar{\mathcal{U}}_{j, p} ) \\ &\qquad{} + ( A_{3jmn}\bar{\mathcal{U}}_{n, m}+B_{3jmn}{ \bar{\Phi}}_{n, m}+\iota\omega D_{3j}{\bar{T}} ) ( \mathcal{U}_{j}+x_{p} \mathcal{U}_{j, p} ) \bigr]\,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl[ ( B_{3jmn}\mathcal{U}_{n, m}+C_{3jmn} \Phi_{n, m}-\iota\omega E_{3j}T ) ({\bar{\Phi}}_{j}+x_{p}{ \bar{\Phi}}_{j, p} ) \\ &\qquad{} + ( B_{3jmn}\bar{\mathcal{U}}_{n, m}+C_{3jmn}{ \bar{\Phi}}_{n, m}+\iota\omega E_{3j}{\bar{T}} ) ( \Phi_{j}+x_{p} \Phi_{j, p} ) \bigr]\,dA \\ &\qquad{}+ \frac{d}{d x_{3}} \int_{D(x_{3})} \frac{1}{\theta_{0}}K_{33} (T \bar{T}_{, 3}+ \bar{T}T_{, 3} )\,dA \\ &\qquad{}+ \frac{d}{d x_{3}} \int_{D(x_{3})} \frac{x_{\alpha}}{\theta_{0}} \bigl[K_{3\alpha} ({\bar{T}}_{, \alpha }T_{, \beta}+T_{, \alpha}{\bar{T}}_{, \beta} )+ K_{33} ({\bar{T}}_{, \alpha}T_{, 3}+T_{, \alpha}{\bar{T}}_{, 3} ) \bigr]\,dA \\ &\qquad{}+ \frac{d}{d x_{3}} \int_{D(x_{3})}\biggl\{ x_{3} \bigl[ A_{i3m3} \mathcal{U}_{i, 3}\bar{\mathcal{U}}_{m, 3}+ B_{i3m3} ( \mathcal{U}_{i, 3}{\bar{\Phi}}_{m, 3}+\bar{\mathcal{U}}_{i, 3} \Phi_{m, 3} )+ C_{i3m3}\Phi_{i, 3}{\bar{\Phi}}_{m, 3} \bigr] \\ &\qquad{}+ x_{3} \bigl[ A_{i\alpha m\beta}\mathcal{U}_{i, \alpha} \bar{\mathcal{U}}_{m, \beta}+ B_{i\alpha m\beta} (\mathcal{U}_{i, \alpha}{ \bar{\Phi}}_{m, \beta }+\bar{\mathcal{U}}_{i, \alpha}\Phi_{m, \beta} )+ C_{i\alpha m\beta}\Phi_{i, \alpha}{\bar{\Phi}}_{m, \beta} \bigr] \\ &\qquad{}+ x_{3} \iota\omega \bigl[ D_{i\alpha} (T \bar{ \mathcal{U}}_{i, \alpha}- {\bar{T}} \mathcal{U}_{i, \alpha} )+ E_{i\alpha} (T {\bar{\Phi}}_{i, \alpha}- {\bar{T}} \Phi_{i, \alpha} ) \bigr] \\ &\qquad{} + \frac{x_{3}}{\theta_{0}} (K_{33}T_{, 3}{\bar{T}}_{, 3}- K_{\alpha\beta}T_{, \alpha}{\bar{T}}_{, \beta} )+ x_{3}\omega^{2} \biggl(\varrho U_{i}{\bar{U}}_{i}+I_{ij}\Phi_{i}{\bar{\Phi}}_{j}+ \frac{c}{\theta _{0}}T{\bar{T}} \biggr) \biggr\} \,dA. \end{aligned}$$
(51)
Proof
We arrive at the equality (51) if we combine the results from equalities (28) and (29) of Theorem 2 with equation (50) of Theorem 4.
The result of the spatial behavior will be based on equality (51). For the result to be rigorous, we specify assumptions which are really common in continuum mechanics. Thus, we assume that the tensors of the micropolar thermoelasticity satisfy the strong ellipticity condition,
$$ \textstyle\begin{array}{@{}l@{\quad}l} A_{ijmn} x_{i} x_{m} y_{j} y_{n} >0,\\ B_{ijmn}x_{i} x_{m} y_{j} y_{n} >0, & \mbox{for all non-zero vectors } (x_{1},x_{2},x_{3} ), (y_{1},y_{2},y_{3} ),\\ C_{ijmn}x_{i} x_{m} y_{j} y_{n} >0. \end{array} $$
(52)
Also, the specific heat c and the conductivity tensor \(K_{ij}\) satisfy the conditions
$$\begin{aligned} c>0, \quad K_{ij}x_{i} x_{j} >0, \quad \mbox{for all non-zero vector } (x_{1},x_{2},x_{3} ). \end{aligned}$$
(53)
It is clear that from (52) that we can deduce
$$ \textstyle\begin{array}{@{}l@{\quad}l} A_{i3m3} x_{i} x_{m} >0,\\ B_{i3m3} x_{i} x_{m} >0, & \mbox{for all non-zero vector } (x_{1},x_{2},x_{3} ),\\ C_{i3m3} x_{i} x_{m} >0. \end{array} $$
(54)
Since the curve ∂D was presumed regular, we deduce that there is \(h_{0}>0\) such that \(x_{p} n_{p}\ge h_{0}>0\). Then we have the inequalities
$$\begin{aligned} 0 \le& \int_{\partial D(x_{3})} x_{p} n_{p} \biggl( A_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial U_{i}}{\partial n} \frac{\partial{\bar{U}}_{m}}{\partial n} + 2 B_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial U_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{m}}{\partial n} \\ &{}+ C_{i\alpha m\beta}n_{\alpha} n_{\beta} \frac{\partial\Phi_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{m}}{\partial n} \biggr)\,ds \\ \le& MC \int_{\partial D(x_{3})} \biggl( \frac{\partial U_{i}}{\partial n}\frac{\partial{\bar{U}}_{i}}{\partial n}+ \frac{\partial\Phi_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{i}}{\partial n} \biggr)\,ds, \end{aligned}$$
(55)
where we have used the notations
$$\begin{aligned}& C= (A_{i\alpha m\beta}A_{i\alpha m\beta}+2B_{i\alpha m\beta }B_{i\alpha m\beta}+ C_{i\alpha m\beta}C_{i\alpha m\beta} )^{1/2}, \end{aligned}$$
(56)
$$\begin{aligned}& M=\sup_{(x_{1},x_{2})\in\partial D}\sqrt{ \bigl(x_{1}^{2}+x_{2}^{2} \bigr)}. \end{aligned}$$
(57)
Also, for the conductivity tensor \(K_{ij}\) we have
$$\begin{aligned} 0\le \int_{\partial D(x_{3})} \frac{1}{\theta_{0}} x_{p} n_{p} K_{\alpha\beta}n_{\alpha}n_{\beta}\frac {\partial T}{\partial n} \frac{\partial{\bar{T}}}{\partial n}\,ds \le \frac{M K}{\theta_{0}} \int_{\partial D(x_{3})} \frac{\partial T}{\partial n} \frac{\partial{\bar{T}}}{\partial n}\,ds, \end{aligned}$$
(58)
where M is defined in (57) and
$$\begin{aligned} K= (K_{\alpha\beta}K_{\alpha\beta} )^{1/2}. \end{aligned}$$
(59)
Now we introduce the quantities \(m_{0}\), \(m_{1}\), \(\omega_{0}^{*}\), and \(\omega _{1}^{*}\) by
$$\begin{aligned}& m_{0}=\max_{x_{3}\in[0,L]} \frac{ \int_{\partial D(x_{3})} ( \frac{\partial U_{i}}{\partial n}\frac{\partial{\bar{U}}_{i}}{\partial n}+ \frac{\partial\Phi_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{i}}{\partial n} )\,ds}{ \int_{D(x_{3})} ( U_{i} {\bar{U}}_{i}+\Phi_{i} {\bar{\Phi}}_{i} )\,ds},\qquad \omega_{0}^{*}=\frac {1}{\varrho}M C m_{0}, \end{aligned}$$
(60)
$$\begin{aligned}& m_{1}=\max_{x_{3}\in[0,L]} \frac{ \int_{\partial D(x_{3})} \frac{\partial T}{\partial n}\frac{\partial{\bar{T}}}{\partial n}\,ds}{ \int_{D(x_{3})} T {\bar{T}}\,ds}, \qquad \omega_{0}^{*}=\frac{1}{c}M K m_{1}. \end{aligned}$$
(61)
We can assume that
$$\begin{aligned}& \omega>\omega^{*}=\max \bigl\{ \omega_{0}^{*}, \omega_{1}^{*} \bigr\} , \end{aligned}$$
(62)
$$\begin{aligned}& m_{0}\le m_{0}^{*},\qquad m_{1}\le m_{1}^{*}, \end{aligned}$$
(63)
where
$$\begin{aligned}& m_{0}^{*}=\max \frac{ \int_{\partial D(x_{3})} ( \frac{\partial U_{i}}{\partial n}\frac{\partial{\bar{U}}_{i}}{\partial n}+ \frac{\partial\Phi_{i}}{\partial n}\frac{\partial{\bar{\Phi}}_{i}}{\partial n} )\,ds}{ \int_{D(x_{3})} ( U_{i} {\bar{U}}_{i}+\Phi_{i} {\bar{\Phi}}_{i} )\,ds}, \end{aligned}$$
(64)
$$\begin{aligned}& m_{1}^{*}=\max_{T\in H_{0}^{1}(D)}\frac{ \int_{\partial D(x_{3})} \frac{\partial T}{\partial n}\frac{\partial{\bar{T}}}{\partial n}\,ds}{ \int_{D(x_{3})}T {\bar{T}}\,ds}. \end{aligned}$$
(65)
Here the maximum from \(m_{0}^{*}\) is calculated for \(U_{i}\in H_{0}^{1}(D)\), \(\Phi _{i}\in H_{0}^{1}(D)\), where \(H_{0}^{1}(D)\) is the usual Sobolev space. In this way we obtain an explicit critical value for the frequency of the vibration, namely
$$\omega^{*}=\max \biggl\{ \frac{1}{\varrho}M C m_{0}^{*}, \frac{1}{c}M K m_{1}^{*} \biggr\} . $$
Combining the results from equations (51), (55), (58), and (62) we obtain the following estimate of the spatial behavior of the amplitude \((U_{i},\Phi_{i}, T )\):
$$\begin{aligned} &\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl[ ( A_{3jmn}\mathcal{U}_{n, m}+B_{3jmn} \Phi_{n, m}-\iota\omega D_{3j}T ) (\bar{\mathcal{U}}_{j}+x_{p}{ \bar{U}}_{j, p} ) \\ &\qquad{}+ ( A_{3jmn}\bar{\mathcal{U}}_{n, m}+B_{3jmn}{ \bar{\Phi}}_{n, m}+\iota\omega D_{3j}{\bar{T}} ) ( \mathcal{U}_{j}+x_{p} \mathcal{U}_{j, p} ) \bigr]\,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \bigl[ ( B_{3jmn}\mathcal{U}_{n, m}+C_{3jmn} \Phi_{n, m}-\iota\omega E_{3j}T ) ({\bar{\Phi}}_{j}+x_{p}{ \bar{\Phi}}_{j, p} ) \\ &\qquad{} + ( B_{3jmn}\bar{\mathcal{U}}_{n, m}+C_{3jmn}{ \bar{\Phi}}_{n, m}+\iota\omega E_{3j}{\bar{T}} ) ( \Phi_{j}+x_{p} \Phi_{j, p} ) \bigr]\,dA \\ &\qquad{}+\frac{d}{d x_{3}} \int_{D(x_{3})} \frac{x_{\alpha}}{\theta_{0}} \bigl[K_{3\alpha} ({\bar{T}}_{, \alpha }T_{, \beta}+T_{, \alpha}{\bar{T}}_{, \beta} )+ K_{33} ({\bar{T}}_{, \alpha}T_{, 3}+T_{, \alpha}{\bar{T}}_{, 3} ) \bigr]\,dA \\ &\qquad{}+ \frac{d}{d x_{3}} \int_{D(x_{3})} \frac{1}{\theta_{0}} \biggl[K_{33} (T \bar{T}_{, 3}+ \bar{T}T_{, 3} ) +x_{3} \omega^{2} \biggl(\varrho U_{i}{\bar{U}}_{i}+I_{ij} \Phi_{i}{\bar{\Phi}}_{j}+ \frac{1}{\theta_{0}}T{\bar{T}} \biggr) \biggr]\,dA \\ &\qquad{} + \frac{d}{d x_{3}} \int_{D(x_{3})}\biggl\{ x_{3} \bigl[ A_{i3m3} \mathcal{U}_{i, 3}\bar{\mathcal{U}}_{m, 3} + B_{i3m3} ( \mathcal{U}_{i, 3}{\bar{\Phi}}_{m, 3} + \bar{\mathcal{U}}_{i, 3} \Phi_{m, 3} ) + C_{i3m3}\Phi_{i, 3}{\bar{\Phi}}_{m, 3} \bigr] \\ &\qquad{}- x_{3} \bigl[ A_{i\alpha m\beta}\mathcal{U}_{i, \alpha} \bar{\mathcal{U}}_{m, \beta}+ B_{i\alpha m\beta} (\mathcal{U}_{i, \alpha}{ \bar{\Phi}}_{m, \beta }+\bar{\mathcal{U}}_{i, \alpha}\Phi_{m, \beta} )+ C_{i\alpha m\beta}\Phi_{i, \alpha}{\bar{\Phi}}_{m, \beta} \bigr] \\ &\qquad{}- x_{3} \iota\omega \bigl[ D_{i\alpha} (T \bar{ \mathcal{U}}_{i, \alpha}- {\bar{T}} \mathcal{U}_{i, \alpha} )+ E_{i\alpha} (T {\bar{\Phi}}_{i, \alpha}- {\bar{T}} \Phi_{i, \alpha} ) \bigr] \\ &\qquad{} + \frac{x_{3}}{\theta_{0}} (K_{33}T_{, 3}{\bar{T}}_{, 3}- K_{\alpha\beta}T_{, \alpha}{\bar{T}}_{, \beta} )+ x_{3}\omega^{2} \biggl(\varrho U_{i}{\bar{U}}_{i}+I_{ij}\Phi_{i}{\bar{\Phi}}_{j}+ \frac{c}{\theta _{0}}T{\bar{T}} \biggr) \biggr\} \,dA \\ &\quad\ge \int_{D(x_{3})} \biggl[ A_{ijmn}\mathcal{U}_{j, i} \bar{\mathcal{U}}_{n, m}+ B_{ijmn} (\mathcal{U}_{j, i}{ \bar{\Phi}}_{n, m}+\bar{\mathcal{U}}_{j, i}\Phi_{n, m} ) \\ &\qquad{} +C_{ijmn}\Phi_{j, i}{\bar{\Phi}}_{n, m}+ \frac{1}{\theta_{0}}K_{ij}T_{, i}{\bar{T}}_{, j} \biggr]\,dA. \end{aligned}$$
(66)
With this the proof of Theorem 5 is complete. □
Conclusion
It is appropriate to note that the differential inequality (66) is different from those used for a deduction of the estimates of Saint-Venant type.
To deduce these estimates we used only the strong ellipticity assumptions for the thermoelastic coefficients.
Therefore, these results can be applied to a large scale of materials.