On mixed problem in thermoelasticity of type III for Cosserat media

Abstract

In our present study, we approach a linear theory for the thermoelasticity of type III for Cosserat media. At the beginning we introduce the equations and conditions, specific for a mixed problem in this context, namely the motion and energy equations, the initial condition and boundary relations. After that, we establish two results regarding the solution unique to the problem and two results on the continuous dependence of solutions, for the same mixed problem. Both problems that we address in our paper (uniqueness and continuous dependence) are not based on the material symmetries of the medium. Moreover, our results concern the theory of type III thermoelasticity for Cosserat media in their most general form, namely anisotropic.

Keywords:

• Cosserat bodies
• thermoelasticity
• type III thermoelasticity
• uniqueness
• continuous dependence

1. Introduction

In the present study, we first formulate the mixed problem with initial and boundary conditions in the context of the theory of Cosserat thermoelastic bodies and to obtain two types of properties of the solutions of this problem, namely, the uniqueness of solution and the continuous dependence of solutions with respect o the data. The thermodynamic theory of Cosserat media sparked the attention of a large number of researchers, since its appearance, one of the reasons was that by means of this theory it is predicting a finite speed for signals of heat. In fact, this is one of the reasons for which most of the new thermoelasticity theories were created. A specific thing that characterizes this theory introduced by F. and E. Cosserat in [1] is the consideration of a mechanic of continuous solids which starts from the following principle: all points of the media possess a number of six degrees of freedom, as it has a rigid solid. Ever since the foundation of this theory and especially in the last two decades, a great number of studies have appeared in which the advantages of the new theory in comparison with classical thermoelasticity are highlighted. The practical importance of this theory is also noted. The well-known books of Nowacki [2] and Eringen [3] refer to a large number of specialists who have dedicated their studies to the theory of Cosserat media and highlight the contributions of all of them to the development of the theory. We have selected the papers [4–20], from this long list of specialists and their works. Specialists appreciate that a natural fibre composite, like in the case of human bones or animal bones, has a bending and torsional deformation which can be described more faithfully in the context of the Cosserat elasticity, compared to the results in classical elasticity.

The paper [21] is devoted to analysing the long time behaviour of solutions of the system of thermoelasticity of type III in a bounded domain. The authors have proven in [22] the exponential stability in one space dimension for different boundary conditions. An elastic beam equation described by a fourth-order fractional difference equation is proposed in [23] with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. In [24] the authors study a thermoelastic problem involving binary mixtures, for type III thermal theory. The study [25] aims at investigating the stability results in the sense of Hyers and Ulam with the application of MittagLeffler function hybrid fractional order difference equation of the second type. In [26], it is considered a newly defined partial ($\mathrm{\Phi },\chi$)-fractional integral and derivative to study a new class of partial fractional differential equations with impulses. The expansion method is applied in [27] to construct many families of exact solutions of nonlinear evolution equations via the nonlinear diffusive predator-prey system and the Bogoyavlenskii equations. The paper [28] studies the novel generalized-expansion technique to two nonlinear evolution equations. In paper [29] it is employed the space-time fractional nonlinear Bogoyavlenskii equation and Schrodinger equation The paper [30] manifests kink wave answers, mixed singular optical solitons, the mixed dark–bright lump answer, the mixed dark–bright periodic wave answer, and periodic wave answers to the conformable fractional ZK model. Some reciprocal relations and a minimum variational principle in the thermodynamics of Cosserat bodies are established in [31].

We must say that there are results similar to those in our study, but they are obtained in the context of classical thermoelasticity. Also, in certain case, the results are deduced by using the Laplace transform. In other cases, in order to obtain such kind of results, the authors reformulated the problem with initial data so they included the initial data into the equations of motion, respectively the energy equation. In our present work, we did not use any of these two listed procedures. We present now the structure of our paper. At the beginning of Section 2 we consider the basic relations and conditions which are specific to a mixed problem from the thermodynamics theory for Cosserat bodies. So, we stated the motion equations, the equation of energy and formulated the boundary relations and initial conditions. Also, in this section, we presented the regularity conditions satisfied by all functions with which we are dealing to be able to deduce the desired results. The basic results of our paper are presented and demonstrated in Section 3. More precisely, here we have proven two reciprocity results, an uniqueness theorem and detail a variational principle. This principle is an extension of some analogous principles from the classical theory of thermoelasticity. See [32–38].

We should emphasize the following aspects about the novelty and contribution of the results of our manuscript. As can be seen from the specialized literature, the results of existence, of uniqueness and of continuous dependence are usually established by using the logarithmic convexity method or by using Lagrange's identity. Both of these methods are based on the material symmetries of the medium, which does not happen in the case of our study. Also, our material is considered in its most general form, that is, inhomogeneous and isotropic. Moreover, according to our documentation, the results present in our study that concern the theory of type III thermoelasticity for Cosserat media have not been addressed so far.

2. The problem formulation

We shall take into account an anisotropic and inhomogeneous Cosserat material which is located in a regular region D of the space $R^3$, that is the three-dimensional Euclidean vector. The boundary of the domain D is denoted by $\mathrm{\partial }D$ and we assume that it is a closed and piece-wise smooth surface. During the study, some scalar, vectorial and tensorial functions are used which depend on time variable $t\in [0,\mathrm{\infty })$ and on points $x=(x_m)$ of the domain D. A superimposed dot is for partial derivative with regards to time t, while a subscript m after a comma is to designate partial derivative regarding the corresponding variable $x_m$. In the case there is no possibility of confusion, it is possible to avoid writing the dependence on the time variable or space variable of a function.

The behaviour of a Cosserat thermoelastic material will be described by using the following constitutive variables:

• the vector of displacement having the elements $v_m$;

• the vector of microrotation having the elements ${\varphi }_m$;

• the temperature variation denoted by T;

• the thermal displacement ϑ, defined by $\dot{\vartheta }=T$;

By considering the internal variables $v_m$ and ${\varphi }_m$, we can obtain the tensors of strain, having the components ${\mathrm{e}}_{mn}$ and ${\epsilon }_{mn}$, and defined with the help of the following geometric equations: (1) $\begin{array}{rl}{\mathrm{e}}_{mn}& =v_{n,m}+{ϵ}_{mnk}{\varphi }_k,\\ {\epsilon }_{mn}& ={\varphi }_{n,m},\\ {\alpha }_k& ={\vartheta }_{,}k.\end{array}$(1) With the independent constitutive variables, above defined, and the strain tensor, we can introduce the free energy of Helmholtz by means of the following formula: (2) $\begin{array}{rl}\varrho H& =\frac{1}{2}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}+\frac{1}{2}{C}_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+\\ & +\frac{1}{2}{k}_{mn}{\alpha }_m{\alpha }_n+a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+b_{mnk}{\epsilon }_{mn}{\alpha }_k-\\ & -\frac{1}{2}a{\dot{\vartheta }}^2-{\beta }_{mn}{\mathrm{e}}_{mn}\dot{\vartheta }-{\gamma }_{mn}{\epsilon }_{mn}\dot{\vartheta }+{\delta }_m\dot{\vartheta }{\alpha }_m,\end{array}$(2) and the corresponding internal energy, defined by: (3) $\begin{array}{rl}\varrho e& =\frac{1}{2}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}+\frac{1}{2}{C}_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+\\ & +\frac{1}{2}{k}_{mn}{\alpha }_m{\alpha }_n+a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k\\ & +b_{mnk}{\epsilon }_{mn}{\alpha }_k+\frac{1}{2}a{\dot{\vartheta }}^2.\end{array}$(3) Being in the context of a linear theory, it is normal to consider both the Helmholtz free energy and the internal energy to be quadratic forms.

For the internal rate of supply of heat per unit mass we will use the relation: (4) $\begin{array}{r}T\zeta ={\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n,\end{array}$(4) and, as a consequence, the tensor of the heat conductivity ${\mu }_{mn}$ satisfies the inequality of dissipation: (5) $\begin{array}{r}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\ge 0.\end{array}$(5) With the help of a procedure used in Green and Lindsay [39] and Eringen [40], we can deduce the main differential equations, specific to the linear theory of the thermoelastic Cosserat media (see [40]):

• the equations of motion: (6) $\begin{array}{r}{\tau }_{mn,n}+\varrho (f_m-{\ddot{v}}_m)=0,\\ {\sigma }_{mn,n}+{ϵ}_{mjk}{\tau }_{jk}+\varrho g_m-I_{mn}{\ddot{\varphi }}_n=0;\end{array}$(6)

• the energy equation: (7) $\begin{array}{r}\varrho (\dot{\eta }-r)-q_{m,m}=0;\end{array}$(7)

• the constitutive relations: (8) $\begin{array}{rl}{\tau }_{mn}& =A_{klmn}{\mathrm{e}}_{kl}+B_{klmn}{\epsilon }_{kl}+a_{mnk}{\alpha }_k-{\beta }_{mn}\dot{\vartheta },\\ {\sigma }_{mn}& =B_{mnkl}{\mathrm{e}}_{kl}+C_{klmn}{\epsilon }_{kl}+b_{mnk}{\alpha }_k-{\gamma }_{mn}\dot{\vartheta },\\ q_m& =a_{mnk}{\mathrm{e}}_{nk}+b_{mnk}{\epsilon }_{nk}+K_{mn}{\alpha }_n+{\delta }_m\dot{\vartheta }+{\mu }_{mn}{\dot{\alpha }}_n,\\ \varrho S& ={\beta }_{mn}{\mathrm{e}}_{mn}+{\gamma }_{mn}{\epsilon }_{mn}-{\delta }_m{\alpha }_m+a\dot{\vartheta }.\end{array}$(8)

We must specify that all previous equations take place for $(t,x)\in [0,\mathrm{\infty })\times D$.

Also, in above Equations (2(2) $\begin{array}{rl}\varrho H& =\frac{1}{2}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}+\frac{1}{2}{C}_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+\\ & +\frac{1}{2}{k}_{mn}{\alpha }_m{\alpha }_n+a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+b_{mnk}{\epsilon }_{mn}{\alpha }_k-\\ & -\frac{1}{2}a{\dot{\vartheta }}^2-{\beta }_{mn}{\mathrm{e}}_{mn}\dot{\vartheta }-{\gamma }_{mn}{\epsilon }_{mn}\dot{\vartheta }+{\delta }_m\dot{\vartheta }{\alpha }_m,\end{array}$(2) )–(8(8) $\begin{array}{rl}{\tau }_{mn}& =A_{klmn}{\mathrm{e}}_{kl}+B_{klmn}{\epsilon }_{kl}+a_{mnk}{\alpha }_k-{\beta }_{mn}\dot{\vartheta },\\ {\sigma }_{mn}& =B_{mnkl}{\mathrm{e}}_{kl}+C_{klmn}{\epsilon }_{kl}+b_{mnk}{\alpha }_k-{\gamma }_{mn}\dot{\vartheta },\\ q_m& =a_{mnk}{\mathrm{e}}_{nk}+b_{mnk}{\epsilon }_{nk}+K_{mn}{\alpha }_n+{\delta }_m\dot{\vartheta }+{\mu }_{mn}{\dot{\alpha }}_n,\\ \varrho S& ={\beta }_{mn}{\mathrm{e}}_{mn}+{\gamma }_{mn}{\epsilon }_{mn}-{\delta }_m{\alpha }_m+a\dot{\vartheta }.\end{array}$(8) ) we used the notations with the next significance:

$v_m$ – the component elements of the displacement, ${\varphi }_m$ – the component elements of microrotation, $f_m$ – body force, $g_m$ – body couple force, ${\tau }_{mn}$ – the component elements of the stress tensor, ${\sigma }_{mn}$ – the component elements of microrotation stress tensor, $q_m$ – the component elements of the vector of conduction of the heat, S – the specific entropy, T – the variation of the temperature, $I_{mn}$ – the component elements of tensor of inertia and ${ϵ}_{mjk}$ – Ricci tensor (or the alternating symbol).

The tensors used in (8(8) $\begin{array}{rl}{\tau }_{mn}& =A_{klmn}{\mathrm{e}}_{kl}+B_{klmn}{\epsilon }_{kl}+a_{mnk}{\alpha }_k-{\beta }_{mn}\dot{\vartheta },\\ {\sigma }_{mn}& =B_{mnkl}{\mathrm{e}}_{kl}+C_{klmn}{\epsilon }_{kl}+b_{mnk}{\alpha }_k-{\gamma }_{mn}\dot{\vartheta },\\ q_m& =a_{mnk}{\mathrm{e}}_{nk}+b_{mnk}{\epsilon }_{nk}+K_{mn}{\alpha }_n+{\delta }_m\dot{\vartheta }+{\mu }_{mn}{\dot{\alpha }}_n,\\ \varrho S& ={\beta }_{mn}{\mathrm{e}}_{mn}+{\gamma }_{mn}{\epsilon }_{mn}-{\delta }_m{\alpha }_m+a\dot{\vartheta }.\end{array}$(8) ) satisfy the next symmetry conditions: (9) $\begin{array}{r}{A}_{klmn}=A_{mnkl},C_{klmn}=C_{mnkl},k_{mn}=k_{nm},I_{mn}=I_{nm}.\end{array}$(9) If we assume an additional hypothesis that the solid, in its initial state, possesses a centre of symmetry in all points and otherwise it is anisotropic, we obtain that: (10) $\begin{array}{r}{a}_{mnk}=0,b_{mnk}=0,{\delta }_m=0,\end{array}$(10) so that the constitutive equations (8(8) $\begin{array}{rl}{\tau }_{mn}& =A_{klmn}{\mathrm{e}}_{kl}+B_{klmn}{\epsilon }_{kl}+a_{mnk}{\alpha }_k-{\beta }_{mn}\dot{\vartheta },\\ {\sigma }_{mn}& =B_{mnkl}{\mathrm{e}}_{kl}+C_{klmn}{\epsilon }_{kl}+b_{mnk}{\alpha }_k-{\gamma }_{mn}\dot{\vartheta },\\ q_m& =a_{mnk}{\mathrm{e}}_{nk}+b_{mnk}{\epsilon }_{nk}+K_{mn}{\alpha }_n+{\delta }_m\dot{\vartheta }+{\mu }_{mn}{\dot{\alpha }}_n,\\ \varrho S& ={\beta }_{mn}{\mathrm{e}}_{mn}+{\gamma }_{mn}{\epsilon }_{mn}-{\delta }_m{\alpha }_m+a\dot{\vartheta }.\end{array}$(8) ) receive the following simpler form: (11) $\begin{array}{rl}{\tau }_{mn}& =A_{klmn}{\mathrm{e}}_{kl}+B_{klmn}{\epsilon }_{kl}-{\beta }_{mn}\dot{\vartheta },\\ {\sigma }_{mn}& =B_{mnkl}{\mathrm{e}}_{kl}+C_{klmn}{\epsilon }_{kl}-{\gamma }_{mn}\dot{\vartheta },\\ q_m& =K_{mn}{\alpha }_n+{\mu }_{mn}{\dot{\alpha }}_n,\\ \varrho S& ={\beta }_{mn}{\mathrm{e}}_{mn}+{\gamma }_{mn}{\epsilon }_{mn}+a\dot{\vartheta }.\end{array}$(11) In order to formulate a mixed problem with initial conditions and boundary relations in the present context, we add the next initial conditions: (12) $\begin{array}{rl}{v}_m(0)& =v_m^0,{\dot{v}}_m(0)=v_m^1,\\ {\varphi }_m(0)& ={\varphi }_m^0,{\dot{\varphi }}_m(0)={\varphi }_m^1,\\ \vartheta (0)& =0,\dot{\vartheta }(0)=T^0,\end{array}$(12) in which I did not write the dependence of the functions on variable x.

Also, for a mixed problem, some boundary conditions are needed. But for this we have to introduce the flux of heat q and also the normal tractions on the surface $t_k$ and $m_k$: (13) $\begin{array}{r}q=q_k{n}_k,t_k={\tau }_{kl}{n}_l,m_k={\sigma }_{kl}{n}_l,\end{array}$(13) which take place in any point of the cylinder $[0,\mathrm{\infty })\times \mathrm{\partial }D$.

Here $n_l$ designate the components of the vector of the normal for the surface $\mathrm{\partial }D$, with orientation outward.

Also, we need the following division of the set $\mathrm{\partial }D$: $\begin{array}{rl}\mathrm{\partial }D& ={\mathrm{\Sigma }}_1\cup {\mathrm{\Sigma }}_1^c={\mathrm{\Sigma }}_2\cup {\mathrm{\Sigma }}_2^c={\mathrm{\Sigma }}_3\cup {\mathrm{\Sigma }}_3^c,\\ {\mathrm{\Sigma }}_1& \cap {\mathrm{\Sigma }}_1^c={\mathrm{\Sigma }}_2\cap {\mathrm{\Sigma }}_2^c={\mathrm{\Sigma }}_3\cap {\mathrm{\Sigma }}_3^c=\mathrm{\varnothing },\end{array}$where the sets ${\mathrm{\Sigma }}_1^c$, ${\mathrm{\Sigma }}_2^c$ and ${\mathrm{\Sigma }}_3^c$ are the complements of the sets ${\mathrm{\Sigma }}_1$, ${\mathrm{\Sigma }}_2$ and ${\mathrm{\Sigma }}_3$, respectively, with respect to the surface $\mathrm{\partial }D$.

So, we can add the next boundary conditions: (14) $\begin{array}{rl}{v}_m& ={\tilde{v}}_{m}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_1,t_m={\tilde{t}}_{m}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_1^c,\\ {\varphi }_k& ={\tilde{\varphi }}_{k}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_2,m_k={\tilde{m}}_{k}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_2^c,\\ \vartheta & =\tilde{\vartheta }in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_3,q=\tilde{q}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_3^c.\end{array}$(14) Considering the basic equations (6(6) $\begin{array}{r}{\tau }_{mn,n}+\varrho (f_m-{\ddot{v}}_m)=0,\\ {\sigma }_{mn,n}+{ϵ}_{mjk}{\tau }_{jk}+\varrho g_m-I_{mn}{\ddot{\varphi }}_n=0;\end{array}$(6) ) and (7(7) $\begin{array}{r}\varrho (\dot{\eta }-r)-q_{m,m}=0;\end{array}$(7) ), in which we substitute the constitutive relations (8(8) $\begin{array}{rl}{\tau }_{mn}& =A_{klmn}{\mathrm{e}}_{kl}+B_{klmn}{\epsilon }_{kl}+a_{mnk}{\alpha }_k-{\beta }_{mn}\dot{\vartheta },\\ {\sigma }_{mn}& =B_{mnkl}{\mathrm{e}}_{kl}+C_{klmn}{\epsilon }_{kl}+b_{mnk}{\alpha }_k-{\gamma }_{mn}\dot{\vartheta },\\ q_m& =a_{mnk}{\mathrm{e}}_{nk}+b_{mnk}{\epsilon }_{nk}+K_{mn}{\alpha }_n+{\delta }_m\dot{\vartheta }+{\mu }_{mn}{\dot{\alpha }}_n,\\ \varrho S& ={\beta }_{mn}{\mathrm{e}}_{mn}+{\gamma }_{mn}{\epsilon }_{mn}-{\delta }_m{\alpha }_m+a\dot{\vartheta }.\end{array}$(8) ), we get the next differential equations: (15) $\begin{array}{rl}\varrho ({\ddot{v}}_m-f_m)& =C_{klmn}{\mathrm{e}}_{kl,n}+B_{klmn}{\epsilon }_{kl,n}\\ & +a_{mnk}{\alpha }_{k,n}-{\beta }_{mn}{\dot{\vartheta }}_{,n},\\ I_{mn}{\ddot{\varphi }}_n-\varrho g_m& =B_{mnkl}{\mathrm{e}}_{kl,n}+A_{klmn}{\epsilon }_{kl,n}\\ & +b_{mnk}{\alpha }_{k,n}-{\gamma }_{mn}{\dot{\vartheta }}_{,n},\\ & +{ϵ}_{mjk}(A_{jkln}{\mathrm{e}}_{ln}+B_{jkln}{\epsilon }_{ln}\\ & +a_{jkn}{\alpha }_n-{\beta }_{jk}\dot{\vartheta }),\\ a\ddot{\vartheta }-\varrho r& =-{\beta }_{mn}\dot{}{\mathrm{e}}_{mn}-{\gamma }_{mn}{\dot{\epsilon }}_{mn}+a_{mnk}{\mathrm{e}}_{mn,k}\\ & +b_{mnk}{\epsilon }_{mn,k}+\\ & +K_{mn}{\vartheta }_{,mn}+{\delta }_m{\dot{\vartheta }}_{,m}+{\mu }_{mn}{\dot{T}}_{,mn},\end{array}$(15) which take places in any $(t,x)$ from the cylinder $[0,\mathrm{\infty })\times D$.

In the case of a centrosymmetric body, that is, if the constitutive relations (8(8) $\begin{array}{rl}{\tau }_{mn}& =A_{klmn}{\mathrm{e}}_{kl}+B_{klmn}{\epsilon }_{kl}+a_{mnk}{\alpha }_k-{\beta }_{mn}\dot{\vartheta },\\ {\sigma }_{mn}& =B_{mnkl}{\mathrm{e}}_{kl}+C_{klmn}{\epsilon }_{kl}+b_{mnk}{\alpha }_k-{\gamma }_{mn}\dot{\vartheta },\\ q_m& =a_{mnk}{\mathrm{e}}_{nk}+b_{mnk}{\epsilon }_{nk}+K_{mn}{\alpha }_n+{\delta }_m\dot{\vartheta }+{\mu }_{mn}{\dot{\alpha }}_n,\\ \varrho S& ={\beta }_{mn}{\mathrm{e}}_{mn}+{\gamma }_{mn}{\epsilon }_{mn}-{\delta }_m{\alpha }_m+a\dot{\vartheta }.\end{array}$(8) ) are replaced by relations (11(11) $\begin{array}{rl}{\tau }_{mn}& =A_{klmn}{\mathrm{e}}_{kl}+B_{klmn}{\epsilon }_{kl}-{\beta }_{mn}\dot{\vartheta },\\ {\sigma }_{mn}& =B_{mnkl}{\mathrm{e}}_{kl}+C_{klmn}{\epsilon }_{kl}-{\gamma }_{mn}\dot{\vartheta },\\ q_m& =K_{mn}{\alpha }_n+{\mu }_{mn}{\dot{\alpha }}_n,\\ \varrho S& ={\beta }_{mn}{\mathrm{e}}_{mn}+{\gamma }_{mn}{\epsilon }_{mn}+a\dot{\vartheta }.\end{array}$(11) ), the system (15(15) $\begin{array}{rl}\varrho ({\ddot{v}}_m-f_m)& =C_{klmn}{\mathrm{e}}_{kl,n}+B_{klmn}{\epsilon }_{kl,n}\\ & +a_{mnk}{\alpha }_{k,n}-{\beta }_{mn}{\dot{\vartheta }}_{,n},\\ I_{mn}{\ddot{\varphi }}_n-\varrho g_m& =B_{mnkl}{\mathrm{e}}_{kl,n}+A_{klmn}{\epsilon }_{kl,n}\\ & +b_{mnk}{\alpha }_{k,n}-{\gamma }_{mn}{\dot{\vartheta }}_{,n},\\ & +{ϵ}_{mjk}(A_{jkln}{\mathrm{e}}_{ln}+B_{jkln}{\epsilon }_{ln}\\ & +a_{jkn}{\alpha }_n-{\beta }_{jk}\dot{\vartheta }),\\ a\ddot{\vartheta }-\varrho r& =-{\beta }_{mn}\dot{}{\mathrm{e}}_{mn}-{\gamma }_{mn}{\dot{\epsilon }}_{mn}+a_{mnk}{\mathrm{e}}_{mn,k}\\ & +b_{mnk}{\epsilon }_{mn,k}+\\ & +K_{mn}{\vartheta }_{,mn}+{\delta }_m{\dot{\vartheta }}_{,m}+{\mu }_{mn}{\dot{T}}_{,mn},\end{array}$(15) ) receives the following simpler form: (16) $\begin{array}{rl}\varrho ({\ddot{v}}_m-f_m)& =C_{klmn}{\mathrm{e}}_{kl,n}+B_{klmn}{\epsilon }_{kl,n}-{\beta }_{mn}{\dot{\vartheta }}_{,n},\\ I_{mn}{\ddot{\varphi }}_n-\varrho g_m& =B_{mnkl}{\mathrm{e}}_{kl,n}+A_{klmn}{\epsilon }_{kl,n}-{\gamma }_{mn}{\dot{\vartheta }}_{,n},\\ & +{ϵ}_{mjk}(A_{jkln}{\mathrm{e}}_{ln}+B_{jkln}{\epsilon }_{ln}-{\beta }_{jk}\dot{\vartheta }),\\ a\ddot{\vartheta }-\varrho r& =-{\beta }_{mn}\dot{}{\mathrm{e}}_{mn}-{\gamma }_{mn}{\dot{\epsilon }}_{mn}\\ & +K_{mn}{\vartheta }_{,mn}+{\mu }_{mn}{\dot{T}}_{,mn}.\end{array}$(16) In what follows we consider the mixed problem (with initial and boundary values) in the thermoelasticity of type III for Cosserat bodies, denote by $\mathcal{P}$, which is constituted by the system of equations (16(16) $\begin{array}{rl}\varrho ({\ddot{v}}_m-f_m)& =C_{klmn}{\mathrm{e}}_{kl,n}+B_{klmn}{\epsilon }_{kl,n}-{\beta }_{mn}{\dot{\vartheta }}_{,n},\\ I_{mn}{\ddot{\varphi }}_n-\varrho g_m& =B_{mnkl}{\mathrm{e}}_{kl,n}+A_{klmn}{\epsilon }_{kl,n}-{\gamma }_{mn}{\dot{\vartheta }}_{,n},\\ & +{ϵ}_{mjk}(A_{jkln}{\mathrm{e}}_{ln}+B_{jkln}{\epsilon }_{ln}-{\beta }_{jk}\dot{\vartheta }),\\ a\ddot{\vartheta }-\varrho r& =-{\beta }_{mn}\dot{}{\mathrm{e}}_{mn}-{\gamma }_{mn}{\dot{\epsilon }}_{mn}\\ & +K_{mn}{\vartheta }_{,mn}+{\mu }_{mn}{\dot{T}}_{,mn}.\end{array}$(16) ), the initial values (12(12) $\begin{array}{rl}{v}_m(0)& =v_m^0,{\dot{v}}_m(0)=v_m^1,\\ {\varphi }_m(0)& ={\varphi }_m^0,{\dot{\varphi }}_m(0)={\varphi }_m^1,\\ \vartheta (0)& =0,\dot{\vartheta }(0)=T^0,\end{array}$(12) ) and the conditions to the limit (14(14) $\begin{array}{rl}{v}_m& ={\tilde{v}}_{m}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_1,t_m={\tilde{t}}_{m}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_1^c,\\ {\varphi }_k& ={\tilde{\varphi }}_{k}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_2,m_k={\tilde{m}}_{k}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_2^c,\\ \vartheta & =\tilde{\vartheta }in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_3,q=\tilde{q}in[0,\mathrm{\infty })\times {\mathrm{\Sigma }}_3^c.\end{array}$(14) ).

It is necessary to emphasize that our subsequent results concern only the solutions to problem $\mathcal{P}$.

We now want to systematize the conditions of regularity that are necessary to obtain the results we proposed. It will be noted that these conditions are not restrictive, they are usually encountered in the mechanics of continuous media.

We will use three sets of hypotheses:

1. (17) $\begin{array}{rl}a& >0,\varrho >0,\\ A_{mnkl}{x}_{mn}{x}_{kl}& \ge c_1{x}_{mn}{x}_{mn},C_{mnkl}{x}_{mn}{x}_{kl}\\ & \ge c_2{x}_{mn}{x}_{mn},\mathrm{\forall }{x}_{mn}=x_{nm},\\ {\mu }_{mn}{y}_m{y}_n& \ge c_3{y}_m,\mathrm{\forall }{y}_m.\end{array}$(17)

2. (18) $\begin{array}{rl}\varrho & >0,\\ A_{mnkl}{x}_{mn}{x}_{kl}& \ge c_1{x}_{mn}{x}_{mn},C_{mnkl}{x}_{mn}{x}_{kl}\\ & \ge c_2{x}_{mn}{x}_{mn},\mathrm{\forall }{x}_{mn}=x_{nm},\\ {\mu }_{mn}{y}_m{y}_n& \ge c_3{y}_m,\mathrm{\forall }{y}_m.\end{array}$(18) Assumptions (i) and (ii) ensure that the elasticity tensors $A_{mnkl}$ and $C_{mnkl}$ are positive definite and the heat conductivity tensor ${\mu }_{mn}$ is also positive definite.

3. (19) $\begin{array}{rl}& \varrho >0,\\ & A_{mnkl}{x}_{mn}{x}_{kl}+2B_{mnkl}{x}_{mn}{y}_{kl}+C_{mnkl}{y}_{mn}{y}_{kl}+\\ & +k_{mn}{z}_m{z}_n+2a_{mnk}{x}_{mn}{z}_k\\ & +2b_{mnk}{y}_{mn}{z}_k+a{\xi }^2\ge 0,\mathrm{\forall }{x}_{mn},y_{mn},z_m,\xi ,\\ & {\mu }_{mn}{z}_m{z}_n\ge 0,\mathrm{\forall }{z}_m.\end{array}$(19)

In (iii) it is observed that the specific internal energy e (defined in (3)) is positive and also the tensor of the heat conductivity ${\mu }_{mn}$ is positive. Also, in (17(17) $\begin{array}{rl}a& >0,\varrho >0,\\ A_{mnkl}{x}_{mn}{x}_{kl}& \ge c_1{x}_{mn}{x}_{mn},C_{mnkl}{x}_{mn}{x}_{kl}\\ & \ge c_2{x}_{mn}{x}_{mn},\mathrm{\forall }{x}_{mn}=x_{nm},\\ {\mu }_{mn}{y}_m{y}_n& \ge c_3{y}_m,\mathrm{\forall }{y}_m.\end{array}$(17) ) and (18(18) $\begin{array}{rl}\varrho & >0,\\ A_{mnkl}{x}_{mn}{x}_{kl}& \ge c_1{x}_{mn}{x}_{mn},C_{mnkl}{x}_{mn}{x}_{kl}\\ & \ge c_2{x}_{mn}{x}_{mn},\mathrm{\forall }{x}_{mn}=x_{nm},\\ {\mu }_{mn}{y}_m{y}_n& \ge c_3{y}_m,\mathrm{\forall }{y}_m.\end{array}$(18) ) the positive constants $c_1$, $c_2$ and $c_3$ are defined by the minimum eigenvalues of the elasticity tensors $A_{mnkl}$ and $C_{mnkl}$ and the tensor of the heat conductivity ${\mu }_{mn}$, respectively.

3. Results

Let us denote by E a new form of energy: (20) $\begin{array}{rl}E(t)& ={\int }_D[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]\\ & +{\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(20) Our first result provides a law of conservation, with respect to the energy E.

Proposition 3.1

If $(v_m,{\varphi }_m,\vartheta )$ is a solution to the mixed problem $\mathcal{P}$, then we have the identity: (21) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }t}[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]+{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n=\\ & =\varrho (f_m{\dot{v}}_m+g_m{\dot{\varphi }}_m+r\dot{\vartheta })\\ & +{({\tau }_{mn}{\dot{v}}_n+{\sigma }_{mn}{\dot{\varphi }}_n+q_m\dot{\vartheta })}_{,m}.\end{array}$(21) Also, the following law of conservation takes place: (22) $\begin{array}{rl}E(t)& =E(0)+{\int }_0^t{\int }_D\varrho (f_m{\dot{v}}_m+g_m{\dot{\varphi }}_m+r\dot{\vartheta })\mathrm{d}V\mathrm{d}s+\\ & +{\int }_0^t{\int }_{\mathrm{\partial }D}({\tau }_{nk}{\dot{v}}_n+{\sigma }_{nk}{\dot{\varphi }}_n+q_k\dot{\vartheta })n_k\mathrm{d}A\mathrm{d}s.\end{array}$(22)

Proof.

The identity (21(21) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }t}[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]+{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n=\\ & =\varrho (f_m{\dot{v}}_m+g_m{\dot{\varphi }}_m+r\dot{\vartheta })\\ & +{({\tau }_{mn}{\dot{v}}_n+{\sigma }_{mn}{\dot{\varphi }}_n+q_m\dot{\vartheta })}_{,m}.\end{array}$(21) ) is easily obtained, considering the basic differential equations (16(16) $\begin{array}{rl}\varrho ({\ddot{v}}_m-f_m)& =C_{klmn}{\mathrm{e}}_{kl,n}+B_{klmn}{\epsilon }_{kl,n}-{\beta }_{mn}{\dot{\vartheta }}_{,n},\\ I_{mn}{\ddot{\varphi }}_n-\varrho g_m& =B_{mnkl}{\mathrm{e}}_{kl,n}+A_{klmn}{\epsilon }_{kl,n}-{\gamma }_{mn}{\dot{\vartheta }}_{,n},\\ & +{ϵ}_{mjk}(A_{jkln}{\mathrm{e}}_{ln}+B_{jkln}{\epsilon }_{ln}-{\beta }_{jk}\dot{\vartheta }),\\ a\ddot{\vartheta }-\varrho r& =-{\beta }_{mn}\dot{}{\mathrm{e}}_{mn}-{\gamma }_{mn}{\dot{\epsilon }}_{mn}\\ & +K_{mn}{\vartheta }_{,mn}+{\mu }_{mn}{\dot{T}}_{,mn}.\end{array}$(16) ) and the constitutive relations (11(11) $\begin{array}{rl}{\tau }_{mn}& =A_{klmn}{\mathrm{e}}_{kl}+B_{klmn}{\epsilon }_{kl}-{\beta }_{mn}\dot{\vartheta },\\ {\sigma }_{mn}& =B_{mnkl}{\mathrm{e}}_{kl}+C_{klmn}{\epsilon }_{kl}-{\gamma }_{mn}\dot{\vartheta },\\ q_m& =K_{mn}{\alpha }_n+{\mu }_{mn}{\dot{\alpha }}_n,\\ \varrho S& ={\beta }_{mn}{\mathrm{e}}_{mn}+{\gamma }_{mn}{\epsilon }_{mn}+a\dot{\vartheta }.\end{array}$(11) ). Taking into account the identity (21(21) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }t}[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]+{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n=\\ & =\varrho (f_m{\dot{v}}_m+g_m{\dot{\varphi }}_m+r\dot{\vartheta })\\ & +{({\tau }_{mn}{\dot{v}}_n+{\sigma }_{mn}{\dot{\varphi }}_n+q_m\dot{\vartheta })}_{,m}.\end{array}$(21) ) and using the initial values (12(12) $\begin{array}{rl}{v}_m(0)& =v_m^0,{\dot{v}}_m(0)=v_m^1,\\ {\varphi }_m(0)& ={\varphi }_m^0,{\dot{\varphi }}_m(0)={\varphi }_m^1,\\ \vartheta (0)& =0,\dot{\vartheta }(0)=T^0,\end{array}$(12) ) we are led to the conservation law (22(22) $\begin{array}{rl}E(t)& =E(0)+{\int }_0^t{\int }_D\varrho (f_m{\dot{v}}_m+g_m{\dot{\varphi }}_m+r\dot{\vartheta })\mathrm{d}V\mathrm{d}s+\\ & +{\int }_0^t{\int }_{\mathrm{\partial }D}({\tau }_{nk}{\dot{v}}_n+{\sigma }_{nk}{\dot{\varphi }}_n+q_k\dot{\vartheta })n_k\mathrm{d}A\mathrm{d}s.\end{array}$(22) ).

Our next result provides another form of the positivity of the tensor of the heat conductivity ${\mu }_{mn}$.

Proposition 3.2

Assume that the thermal displacement gradient vector $\mathit{\alpha }=({\alpha }_k)$ is a continuous differentiable function regarding the variable t and she is subject to the initial condition: (23) $\begin{array}{r}{\alpha }_k(0,x)=0,\mathrm{\forall }x\in \overline{D}.\end{array}$(23) Moreover, we suppose that the heat conductivity tensor ${\mu }_{mn}$ is positive definite in the sense of (17(17) $\begin{array}{rl}a& >0,\varrho >0,\\ A_{mnkl}{x}_{mn}{x}_{kl}& \ge c_1{x}_{mn}{x}_{mn},C_{mnkl}{x}_{mn}{x}_{kl}\\ & \ge c_2{x}_{mn}{x}_{mn},\mathrm{\forall }{x}_{mn}=x_{nm},\\ {\mu }_{mn}{y}_m{y}_n& \ge c_3{y}_m,\mathrm{\forall }{y}_m.\end{array}$(17) ) or (18(18) $\begin{array}{rl}\varrho & >0,\\ A_{mnkl}{x}_{mn}{x}_{kl}& \ge c_1{x}_{mn}{x}_{mn},C_{mnkl}{x}_{mn}{x}_{kl}\\ & \ge c_2{x}_{mn}{x}_{mn},\mathrm{\forall }{x}_{mn}=x_{nm},\\ {\mu }_{mn}{y}_m{y}_n& \ge c_3{y}_m,\mathrm{\forall }{y}_m.\end{array}$(18) ).

Then we have the following estimation: (24) $\begin{array}{rl}& {\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\\ & \ge c_4{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,\mathrm{\forall }t\in [0,\mathrm{\infty }),\end{array}$(24) where the constant $c_4>0$ is arbitrary.

Proof.

Considering the initial relation (23(23) $\begin{array}{r}{\alpha }_k(0,x)=0,\mathrm{\forall }x\in \overline{D}.\end{array}$(23) ) and taking into account that the function $({\alpha }_k)$ is continuous differentiable regarding the variable t, we have ensured the existence of a $t_1\ge 0$ for which we have: (25) $\begin{array}{r}{\alpha }_k(t,x)=0,\mathrm{\forall }(t,x)\in [0,t_1]\times \overline{D}.\end{array}$(25) As a consequence, the inequality is satisfied on the interval $[0,t_1]$. It remains to prove the correctness of inequality (24(24) $\begin{array}{rl}& {\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\\ & \ge c_4{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,\mathrm{\forall }t\in [0,\mathrm{\infty }),\end{array}$(24) ) for $t\in (t_1,t_2)$, with $t_2>t_1$, which we will do by reduction to the absurd.

Let us assume that there is $t_2>0$ such that (26) $\begin{array}{rl}& {\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\\ & <c_4{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,\mathrm{\forall }t\in (t_1,t_2).\end{array}$(26) In this situation we deduce: (27) $\begin{array}{r}{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V>0,\mathrm{\forall }t\in (t_1,t_2).\end{array}$(27) But, by using (25(25) $\begin{array}{r}{\alpha }_k(t,x)=0,\mathrm{\forall }(t,x)\in [0,t_1]\times \overline{D}.\end{array}$(25) ) and the Cauchy–Schwarz inequality, it results (28) $\begin{array}{rl}& {\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V={\int }_{D(t_1)}{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V\\ & +{\int }_{t_1}^t{\int }_{D(s)}2{\mu }_{mn}{\dot{\alpha }}_m{\alpha }_n\mathrm{d}V\mathrm{d}s\le \\ & \le 2\left({\int }_{t_1}^t{\int }_{D(s)}2{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\right)\\ & \times \left({\int }_{t_1}^t{\int }_{D(s)}2{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V\mathrm{d}s\right),\end{array}$(28) from where, by using the inequality (26(26) $\begin{array}{rl}& {\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\\ & <c_4{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,\mathrm{\forall }t\in (t_1,t_2).\end{array}$(26) ), we deduce: (29) $\begin{array}{rl}& {\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V\\ & \le 4c_4{\int }_{t_1}^t{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }t\in (t_1,t_2).\end{array}$(29) Above, the notation $D(s)$ is to specify the fact that the expression under integral is calculated at the moment s.

If we use the notation: (30) $\begin{array}{r}u(t)={\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V>0,\end{array}$(30) then the inequality (29(29) $\begin{array}{rl}& {\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V\\ & \le 4c_4{\int }_{t_1}^t{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }t\in (t_1,t_2).\end{array}$(29) ) is a Gronwall inequality of the form: (31) $\begin{array}{r}u(t)\le 4c_4{\int }_{D(t)}u(s)\mathrm{d}V>0,\mathrm{\forall }t\in (t_1,t_2),\end{array}$(31) or, in a simpler form: (32) $\begin{array}{r}u(t)\le v^2(t),\mathrm{\forall }t\in (t_1,t_2),\end{array}$(32) where we used the notation: (33) $\begin{array}{r}{v}^2(t)=4c_4{\int }_{D(t)}u(s)\mathrm{d}V.\end{array}$(33) If we derive in (33(33) $\begin{array}{r}{v}^2(t)=4c_4{\int }_{D(t)}u(s)\mathrm{d}V.\end{array}$(33) ) and take into account (32(32) $\begin{array}{r}u(t)\le v^2(t),\mathrm{\forall }t\in (t_1,t_2),\end{array}$(32) ), we get: (34) $\begin{array}{r}\dot{u}(t)\le 2c_{4}u(t),\end{array}$(34) whence, if we integrate, it results $u(t)=0$ so that considering (30(30) $\begin{array}{r}u(t)={\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V>0,\end{array}$(30) ) we obtain: ${\alpha }_k(x,t)=0,\text{for any}(x,t)\in \overline{D}\times (t_1,t_2),$which contradicts the assumption in (27(27) $\begin{array}{r}{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V>0,\mathrm{\forall }t\in (t_1,t_2).\end{array}$(27) ), so that the hypothesis (26(26) $\begin{array}{rl}& {\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\\ & <c_4{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,\mathrm{\forall }t\in (t_1,t_2).\end{array}$(26) ) can't be true.

The proof of Proposition 3.2 is complete.

To simplify the writing, we will use the notation: (35) $\begin{array}{rl}\mathcal{F}(t)& ={\int }_{D(t)}(A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\\ & +2B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}+C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+\\ & +2a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+2b_{mnk}{\epsilon }_{mn}{\alpha }_k\\ & +k_{mn}{\alpha }_m{\alpha }_n)\mathrm{d}V+\\ & +2{\int }_0^t{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,t\in [0,\mathrm{\infty }),\end{array}$(35) and in the following sentence, we will prove the positivity of the function $\mathcal{F}(t)$.

Proposition 3.3

If the elasticity tensors $A_{mnkl}$ and $C_{mnkl}$ and the tensor of the heat conductivity ${\mu }_{mn}$ are positive definite, then for any solution $(v_m,{\varphi }_m,\vartheta )$ of the problem $\mathcal{P}$ we have the condition: (36) $\begin{array}{r}\mathcal{F}(t)\ge 0.\end{array}$(36)

Proof.

We will use the Cauchy–Schwarz inequality and an arithmetic-geometric mean inequality of the form: $ab\le \frac{1}{2}(\frac{a^2}{p^2}+b^2{p}^2),$where p is a non-null parameter.

So, we have: (37) $\begin{array}{rl}& |a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+b_{mnk}{\epsilon }_{mn}{\alpha }_k|\le \\ & \le {\left(a_{mnk}{a}_{mnk}\right)}^{1/2}{\left({\mathrm{e}}_{rs}{\alpha }_l{\mathrm{e}}_{rs}{\alpha }_l\right)}^{1/2}\\ & +{\left(b_{mnk}{a}_{mnk}\right)}^{1/2}{\left({\epsilon }_{rs}{\alpha }_l{\epsilon }_{rs}{\alpha }_l\right)}^{1/2}\le \\ & \le \frac{A_1{p}^2}{2c_1}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+\frac{A_1}{2c_3{p}^2}{\mu }_{mn}{\alpha }_m{\alpha }_n+\\ & +\frac{A_2{p}^2}{2c_2}{C}_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+\frac{A_2}{2c_3{p}^2}{\mu }_{mn}{\alpha }_m{\alpha }_n,\mathrm{\forall }p,\end{array}$(37) where we used the notations: (38) $\begin{array}{r}{A}_1=\underset{x\in \overline{D}}{max}{\left(a_{mnk}{a}_{mnk}\right)}^{1/2},A_2=\underset{x\in \overline{D}}{max}{\left(b_{mnk}{b}_{mnk}\right)}^{1/2}.\end{array}$(38) On the other hand, we consider the notation: (39) $\begin{array}{r}K=\underset{x\in \overline{D}}{max}{\left(k_{ij}{k}_{ij}\right)}^{1/2},\end{array}$(39) and use the Cauchy–Schwarz inequality and the arithmetic-geometric mean inequality, we obtain the estimate: (40) $\begin{array}{r}\left|k_{ij}{\alpha }_i{\alpha }_j\right|\le {\left(k_{ij}{k}_{ij}\right)}^{1/2}{\left({\alpha }_i{\alpha }_j{\alpha }_i{\alpha }_j\right)}^{1/2}\le \frac{K}{c_3}{\mu }_{ij}{\alpha }_i{\alpha }_j.\end{array}$(40) Now, we consider the estimates (37(37) $\begin{array}{rl}& |a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+b_{mnk}{\epsilon }_{mn}{\alpha }_k|\le \\ & \le {\left(a_{mnk}{a}_{mnk}\right)}^{1/2}{\left({\mathrm{e}}_{rs}{\alpha }_l{\mathrm{e}}_{rs}{\alpha }_l\right)}^{1/2}\\ & +{\left(b_{mnk}{a}_{mnk}\right)}^{1/2}{\left({\epsilon }_{rs}{\alpha }_l{\epsilon }_{rs}{\alpha }_l\right)}^{1/2}\le \\ & \le \frac{A_1{p}^2}{2c_1}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+\frac{A_1}{2c_3{p}^2}{\mu }_{mn}{\alpha }_m{\alpha }_n+\\ & +\frac{A_2{p}^2}{2c_2}{C}_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+\frac{A_2}{2c_3{p}^2}{\mu }_{mn}{\alpha }_m{\alpha }_n,\mathrm{\forall }p,\end{array}$(37) ) and (40(40) $\begin{array}{r}\left|k_{ij}{\alpha }_i{\alpha }_j\right|\le {\left(k_{ij}{k}_{ij}\right)}^{1/2}{\left({\alpha }_i{\alpha }_j{\alpha }_i{\alpha }_j\right)}^{1/2}\le \frac{K}{c_3}{\mu }_{ij}{\alpha }_i{\alpha }_j.\end{array}$(40) ) so that from (35(35) $\begin{array}{rl}\mathcal{F}(t)& ={\int }_{D(t)}(A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\\ & +2B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}+C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+\\ & +2a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+2b_{mnk}{\epsilon }_{mn}{\alpha }_k\\ & +k_{mn}{\alpha }_m{\alpha }_n)\mathrm{d}V+\\ & +2{\int }_0^t{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,t\in [0,\mathrm{\infty }),\end{array}$(35) ) we get: (41) $\begin{array}{rl}\mathcal{F}(t)& \ge {\int }_{D(t)}[(1-\frac{A_1{p}^2}{c_1})A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\\ & -\frac{1}{c_3}(K+\frac{A_1}{p^2}){\mu }_{mn}{\alpha }_m{\alpha }_n+\\ & +(1-\frac{A_2{p}^2}{c_2})C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}\\ & -\frac{1}{c_3}(K+\frac{A_2}{p^2}){\mu }_{mn}{\alpha }_m{\alpha }_n]\mathrm{d}V\\ & +2{\int }_0^t{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }p.\end{array}$(41) To simplify the writing in estimate (41(41) $\begin{array}{rl}\mathcal{F}(t)& \ge {\int }_{D(t)}[(1-\frac{A_1{p}^2}{c_1})A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\\ & -\frac{1}{c_3}(K+\frac{A_1}{p^2}){\mu }_{mn}{\alpha }_m{\alpha }_n+\\ & +(1-\frac{A_2{p}^2}{c_2})C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}\\ & -\frac{1}{c_3}(K+\frac{A_2}{p^2}){\mu }_{mn}{\alpha }_m{\alpha }_n]\mathrm{d}V\\ & +2{\int }_0^t{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }p.\end{array}$(41) ), we introduce the notations: (42) $\begin{array}{rl}{C}_1& =\frac{1}{2c_3}(K+\frac{A_1}{p^2}),D_1=1-\frac{A_1{p}^2}{c3},\\ C_2& =\frac{1}{2c_3}(K+\frac{A_2}{p^2}),D_2=1-\frac{A_2{p}^2}{c3},\end{array}$(42) and we take $p^2$ so that $\frac{A_1{p}^2}{c_1}\le 1,\frac{A_2{p}^2}{c_1}\le 1.$So, we can write (41(41) $\begin{array}{rl}\mathcal{F}(t)& \ge {\int }_{D(t)}[(1-\frac{A_1{p}^2}{c_1})A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\\ & -\frac{1}{c_3}(K+\frac{A_1}{p^2}){\mu }_{mn}{\alpha }_m{\alpha }_n+\\ & +(1-\frac{A_2{p}^2}{c_2})C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}\\ & -\frac{1}{c_3}(K+\frac{A_2}{p^2}){\mu }_{mn}{\alpha }_m{\alpha }_n]\mathrm{d}V\\ & +2{\int }_0^t{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }p.\end{array}$(41) ) in the following form: (43) $\begin{array}{rl}\mathcal{F}(t)& \ge D_1{\int }_{D(t)}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V\\ & +D_2{\int }_{D(t)}{C}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V+\\ & +2{\int }_0^t{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\\ & -2(C_1+C_2){\int }_{D(t)}{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V.\end{array}$(43) Considering the geometric relations (1(1) $\begin{array}{rl}{\mathrm{e}}_{mn}& =v_{n,m}+{ϵ}_{mnk}{\varphi }_k,\\ {\epsilon }_{mn}& ={\varphi }_{n,m},\\ {\alpha }_k& ={\vartheta }_{,}k.\end{array}$(1) ) and taking into account that $\dot{\vartheta }=T$, we obtain that the condition (23(23) $\begin{array}{r}{\alpha }_k(0,x)=0,\mathrm{\forall }x\in \overline{D}.\end{array}$(23) ) from Proposition 3.2 is satisfied. As a consequence, we deduce that inequality (24(24) $\begin{array}{rl}& {\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\\ & \ge c_4{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,\mathrm{\forall }t\in [0,\mathrm{\infty }),\end{array}$(24) ) is satisfied and from (43(43) $\begin{array}{rl}\mathcal{F}(t)& \ge D_1{\int }_{D(t)}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V\\ & +D_2{\int }_{D(t)}{C}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V+\\ & +2{\int }_0^t{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\\ & -2(C_1+C_2){\int }_{D(t)}{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V.\end{array}$(43) ) it is possible to draw the conclusion that $\mathcal{F}(t)\ge 0$, for any $t\in [0,\mathrm{\infty })$. The proof of Proposition 3.3 is thus concluded.

Remark

Using calculations similar to the above, we can determine two positive constants $K_1$, $K_2$ and $K_3$ so that we have: (44) $\begin{array}{rl}& {\int }_{D(t)}(A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+2B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}\\ & +C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+2a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+\\ & +2b_{mnk}{\epsilon }_{mn}{\alpha }_k+k_{mn}{\alpha }_m{\alpha }_n)\mathrm{d}V\\ & +{\int }_0^t{\int }_{D(s)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\ge \\ & \ge K_1{\int }_{D(t)}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V\\ & +K_2{\int }_{D(t)}{C}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V+\\ & +K_3{\int }_{D(t)}{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,t\in [0,\mathrm{\infty }).\end{array}$(44) Based on the results from Propositions 3.1–3.3, we can approach the first result regarding the conditions in which the solution of the problem $\mathcal{P}$ is unique.

Theorem 3.1

Assume that a>0 and the conditions from hypothesis (iii) are satisfied. Then the mixed initial-boundary value problem $\mathcal{P}$ could not have more than one solution.

Proof.

We suppose that problem $\mathcal{P}$ admits two possible solutions and the difference between these two solutions is denoted by $(v_m,{\varphi }_m,\vartheta )$. Because problem $\mathcal{P}$ is linear, this difference satisfies the problem $\mathcal{P}$ and it is proper to null initial data and zero boundary values. Then, from (20(20) $\begin{array}{rl}E(t)& ={\int }_D[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]\\ & +{\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(20) ) we deduce (45) $\begin{array}{r}E(t)=0,\mathrm{\forall }t\ge 0,\end{array}$(45) so that, considering the expression of $E(t)$ we find that: (46) $\begin{array}{r}{\dot{v}}_m(t,x)=0,{\dot{\varphi }}_m(x,t)=0,\text{for any}(x,t)\in \overline{D}\times [0,\mathrm{\infty }),\end{array}$(46) and considering that the initial data are zero, we immediately find that (47) $\begin{array}{r}{v}_m(t,x)=0,{\varphi }_m(x,t)=0,\text{for any}(x,t)\in \overline{D}\times [0,\mathrm{\infty }).\end{array}$(47) On the other hand, if we take into account the relations (20(20) $\begin{array}{rl}E(t)& ={\int }_D[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]\\ & +{\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(20) ), (45(45) $\begin{array}{r}E(t)=0,\mathrm{\forall }t\ge 0,\end{array}$(45) ) and (46(46) $\begin{array}{r}{\dot{v}}_m(t,x)=0,{\dot{\varphi }}_m(x,t)=0,\text{for any}(x,t)\in \overline{D}\times [0,\mathrm{\infty }),\end{array}$(46) ), we are led to (48) $\begin{array}{rl}& \frac{1}{2}{\int }_{D(t)}(a{\dot{\vartheta }}^2+k_{mn}{\alpha }_m{\alpha }_n)\\ & +{\int }_0^t{\int }_{D(s)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s=0,t\ge 0.\end{array}$(48) But $k_{mn}{\alpha }_m{\alpha }_n\ge 0$ and a>0, from the statement of the theorem, so that from (7(7) $\begin{array}{r}\varrho (\dot{\eta }-r)-q_{m,m}=0;\end{array}$(7) ) it results: (49) $\begin{array}{r}\dot{\vartheta }(x,t)=0,\Rightarrow \vartheta (x,t)=0,\text{for any}(x,t)\in \overline{D}\times [0,\mathrm{\infty }),\end{array}$(49) obviously, the last implication is due to the null initial value of ϑ.

Clearly, from (47(47) $\begin{array}{r}{v}_m(t,x)=0,{\varphi }_m(x,t)=0,\text{for any}(x,t)\in \overline{D}\times [0,\mathrm{\infty }).\end{array}$(47) ) and (49(49) $\begin{array}{r}\dot{\vartheta }(x,t)=0,\Rightarrow \vartheta (x,t)=0,\text{for any}(x,t)\in \overline{D}\times [0,\mathrm{\infty }),\end{array}$(49) ) it is deduced that the above defined difference between the two supposed solutions to the problem $\mathcal{P}$ is zero, and thus the demonstration of Theorem 3.1 is finished.

In the theorem that follows we replace the basic hypothesis from Theorem 3.1 and obtain another uniqueness result.

Theorem 3.2

If the basic hypothesis (i) is satisfied, then our problem P can have only a single solution.

Proof.

We suppose that problem $\mathcal{P}$ admits two possible solutions and the difference between these two solutions is denoted by $(v_m,{\varphi }_m,\vartheta )$. Because the problem $\mathcal{P}$ is linear, it follows that this difference satisfies the problem $\mathcal{P}$ and it is proper to null initial data and zero boundary values. If we analyse the expression of $E(t,x)$ from (20(20) $\begin{array}{rl}E(t)& ={\int }_D[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]\\ & +{\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(20) ) and take into account these null data, then we get: $E(t,x)=0,\mathrm{\forall }t\ge 0,$so that if we consider the relations (20(20) $\begin{array}{rl}E(t)& ={\int }_D[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]\\ & +{\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(20) ), (17(17) $\begin{array}{rl}a& >0,\varrho >0,\\ A_{mnkl}{x}_{mn}{x}_{kl}& \ge c_1{x}_{mn}{x}_{mn},C_{mnkl}{x}_{mn}{x}_{kl}\\ & \ge c_2{x}_{mn}{x}_{mn},\mathrm{\forall }{x}_{mn}=x_{nm},\\ {\mu }_{mn}{y}_m{y}_n& \ge c_3{y}_m,\mathrm{\forall }{y}_m.\end{array}$(17) ), (23(23) $\begin{array}{r}{\alpha }_k(0,x)=0,\mathrm{\forall }x\in \overline{D}.\end{array}$(23) ), (24(24) $\begin{array}{rl}& {\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\\ & \ge c_4{\int }_D{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,\mathrm{\forall }t\in [0,\mathrm{\infty }),\end{array}$(24) ), (35(35) $\begin{array}{rl}\mathcal{F}(t)& ={\int }_{D(t)}(A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\\ & +2B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}+C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+\\ & +2a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+2b_{mnk}{\epsilon }_{mn}{\alpha }_k\\ & +k_{mn}{\alpha }_m{\alpha }_n)\mathrm{d}V+\\ & +2{\int }_0^t{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,t\in [0,\mathrm{\infty }),\end{array}$(35) ) and (36(36) $\begin{array}{r}\mathcal{F}(t)\ge 0.\end{array}$(36) ) we find: $\begin{array}{rl}& {\dot{v}}_m(x,t)=0,\\ & {\dot{\varphi }}_m(x,t)=0,\dot{\vartheta }(x,t)=0,\text{for any}(x,t)\in \overline{D}\times [0,\mathrm{\infty }),\end{array}$and based on the null initial data we deduce that: $\begin{array}{rl}& v_m(x,t)=0,\\ & {\varphi }_m(x,t)=0,\vartheta (x,t)=0,\text{for any}(x,t)\in \overline{D}\times [0,\mathrm{\infty }).\end{array}$In this way, the proof of Theorem 3.2 is completed.

The last two results of our study refer to the continuous data dependence of solutions of the mixed problem $\mathcal{P}$. Based on the results of Theorems 3.1 and 3.2, we deduce that we can use the function $E(t)$, defined in (20(20) $\begin{array}{rl}E(t)& ={\int }_D[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]\\ & +{\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(20) ), as a measure for a solution $(v_m,{\varphi }_m,\vartheta )$ of the problem $\mathcal{P}$. In the following theorem, we address the continuous dependence in relation to the initial data. First, we introduce the notation: (50) $\begin{array}{r}G(t)={\left[{\int }_{D(t)}\varrho (f_m{f}_m+g_m{g}_m+\frac{1}{a}{r}^2)\mathrm{d}V\right]}^{1/2}.\end{array}$(50)

Theorem 3.3

If one of the hypotheses (i) and (iii) with a>0 takes place, then for all solution $(v_m,{\varphi }_m,\vartheta )$ of our problem $\mathcal{P}$ the following estimate is satisfied: (51) $\begin{array}{r}\sqrt{E(t)}\le \sqrt{E(0)}+\frac{1}{\sqrt{2}}{\int }_0^{t}G(s)\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(51)

Proof.

We will use the Cauchy–Schwarz inequality in relation (22(22) $\begin{array}{rl}E(t)& =E(0)+{\int }_0^t{\int }_D\varrho (f_m{\dot{v}}_m+g_m{\dot{\varphi }}_m+r\dot{\vartheta })\mathrm{d}V\mathrm{d}s+\\ & +{\int }_0^t{\int }_{\mathrm{\partial }D}({\tau }_{nk}{\dot{v}}_n+{\sigma }_{nk}{\dot{\varphi }}_n+q_k\dot{\vartheta })n_k\mathrm{d}A\mathrm{d}s.\end{array}$(22) ), considering that the boundary data are vanishing, so that we obtain: (52) $\begin{array}{rl}E(t)& \le E(0)+\\ & +{\int }_0^t\sqrt{{\int }_{D(s)}\varrho (f_m{f}_m+g_m{g}_m+\frac{1}{a}{r}^2)\mathrm{d}V}\\ & \times \sqrt{{\int }_{D(s)}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n+a{\dot{\vartheta }}^2)\mathrm{d}V}\mathrm{d}s.\end{array}$(52) Now we consider again the definition (20(20) $\begin{array}{rl}E(t)& ={\int }_D[\frac{1}{2}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_m)+\varrho e]\\ & +{\int }_0^t{\int }_D{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(20) ) and take into account the relation (23(23) $\begin{array}{r}{\alpha }_k(0,x)=0,\mathrm{\forall }x\in \overline{D}.\end{array}$(23) ) so that we obtain the following inequality, of Gronwall type: (53) $\begin{array}{r}E(t)\le E(0)+{\int }_0^{t}G(s)\sqrt{2E(s)}\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(53) Now, we can apply the lemma of Gronwall, which will result in the estimate (51(51) $\begin{array}{r}\sqrt{E(t)}\le \sqrt{E(0)}+\frac{1}{\sqrt{2}}{\int }_0^{t}G(s)\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(51) ).

In our last theorem, we address the problem of the continuous dependence of a solution of the problem $\mathcal{P}$ with respect to the supply terms.

First, we introduce some notations to simplify the writing: (54) $\begin{array}{rl}f(t)& =\sqrt{{\int }_0^t{\int }_{D(s)}(\varrho f_m{f}_m+I_{mn}{g}_m{g}_n)\mathrm{d}V\mathrm{d}s},\\ R(t)& =\frac{1}{c_3}{\int }_0^t{\int }_0^s{\int }_{D(z)}{\varrho }^2{r}^2\mathrm{d}V\mathrm{d}z\mathrm{d}s,\\ g(t)& =2R(t)+2{({\int }_0^{t}f(s)\mathrm{d}s)}^2,\\ \mathcal{G}(t)& =\frac{1}{2}{\int }_0^t{\int }_{D(s)}(A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+2B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}+\\ & +C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+2a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k++2b_{mnk}{\epsilon }_{mn}{\alpha }_k+\\ & +k_{mn}{\alpha }_m{\alpha }_n+\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\varphi }_n)\mathrm{d}V\mathrm{d}s+\\ & +{\int }_0^t{\int }_0^s{\int }_{D(z)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}z\mathrm{d}s,t\ge 0.\end{array}$(54) Also, we denote by ${\mathcal{P}}_0$ the problem which corresponds to the problem $\mathcal{P}$, in the case of vanishing initial and boundary data.

Theorem 3.4

Assume hypothesis (ii) takes place and we choose a solution $(v_m,{\varphi }_m,\vartheta )$ of the problem ${\mathcal{P}}_0$ corresponding to the supply terms $(f_m,g_m,r)$ which satisfies the condition: ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-s/{\omega }^2}h(s)\mathrm{d}s<\mathrm{\infty },$where ω is defined by: ${\omega }^2=\frac{1}{c3}\underset{\overline{D}}{max}|a|.$Consider $(v_m,{\varphi }_m,\vartheta )$ that solution of the problem ${\mathcal{P}}_0$ which satisfies the condition: (55) $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim}{\mathrm{e}}^{-t/2{\omega }^2}\mathcal{G}(t)=0.\end{array}$(55) Then, it is satisfied the following estimation: (56) $\begin{array}{r}0\le \mathcal{G}(t)\le \frac{1}{2{\omega }^2}{\int }_t^{\mathrm{\infty }}{\mathrm{e}}^{(t-s)/2{\omega }^2}h(s)\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(56)

Proof.

If we take into account hypothesis (ii), then by using relations (54(54) $\begin{array}{rl}f(t)& =\sqrt{{\int }_0^t{\int }_{D(s)}(\varrho f_m{f}_m+I_{mn}{g}_m{g}_n)\mathrm{d}V\mathrm{d}s},\\ R(t)& =\frac{1}{c_3}{\int }_0^t{\int }_0^s{\int }_{D(z)}{\varrho }^2{r}^2\mathrm{d}V\mathrm{d}z\mathrm{d}s,\\ g(t)& =2R(t)+2{({\int }_0^{t}f(s)\mathrm{d}s)}^2,\\ \mathcal{G}(t)& =\frac{1}{2}{\int }_0^t{\int }_{D(s)}(A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+2B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}+\\ & +C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+2a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k++2b_{mnk}{\epsilon }_{mn}{\alpha }_k+\\ & +k_{mn}{\alpha }_m{\alpha }_n+\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\varphi }_n)\mathrm{d}V\mathrm{d}s+\\ & +{\int }_0^t{\int }_0^s{\int }_{D(z)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}z\mathrm{d}s,t\ge 0.\end{array}$(54) )${}_4$ and (44(44) $\begin{array}{rl}& {\int }_{D(t)}(A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+2B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}\\ & +C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+2a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+\\ & +2b_{mnk}{\epsilon }_{mn}{\alpha }_k+k_{mn}{\alpha }_m{\alpha }_n)\mathrm{d}V\\ & +{\int }_0^t{\int }_{D(s)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\ge \\ & \ge K_1{\int }_{D(t)}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V\\ & +K_2{\int }_{D(t)}{C}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V+\\ & +K_3{\int }_{D(t)}{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,t\in [0,\mathrm{\infty }).\end{array}$(44) ), we obtain the estimate: (57) $\begin{array}{rl}\mathcal{G}(t)& \ge \frac{1}{2}{\int }_0^t{\int }_{D(s)}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\varphi }_n)\mathrm{d}V\mathrm{d}s++\frac{1}{2}{\int }_0^t\\ & \times {\int }_0^s{\int }_{D(z)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}z\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(57) Clearly, if $\mathcal{G}(t)=0$ then we immediate obtain: $\begin{array}{rl}& v_m(t,x)=0,\\ & {\varphi }_m(t,x)=0,\vartheta (t,x)=0,\text{for any}(t,x)\in [0,\mathrm{\infty })\times \overline{D}.\end{array}$On the other hand, based on hypothesis (ii), we can obtain: (58) $\begin{array}{r}{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\ge {\lambda }_0{c}_3{\int }_{D(t)}{\vartheta }^2\mathrm{d}V,\mathrm{\forall }t\in [0,\mathrm{\infty }),\end{array}$(58) where ${\lambda }_0>0$ is the smallest autovalue in the problem of a membrane in the region D.

Now we calculate the derivative of $\mathcal{G}(t)$ from (54(54) $\begin{array}{rl}f(t)& =\sqrt{{\int }_0^t{\int }_{D(s)}(\varrho f_m{f}_m+I_{mn}{g}_m{g}_n)\mathrm{d}V\mathrm{d}s},\\ R(t)& =\frac{1}{c_3}{\int }_0^t{\int }_0^s{\int }_{D(z)}{\varrho }^2{r}^2\mathrm{d}V\mathrm{d}z\mathrm{d}s,\\ g(t)& =2R(t)+2{({\int }_0^{t}f(s)\mathrm{d}s)}^2,\\ \mathcal{G}(t)& =\frac{1}{2}{\int }_0^t{\int }_{D(s)}(A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+2B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}+\\ & +C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+2a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k++2b_{mnk}{\epsilon }_{mn}{\alpha }_k+\\ & +k_{mn}{\alpha }_m{\alpha }_n+\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\varphi }_n)\mathrm{d}V\mathrm{d}s+\\ & +{\int }_0^t{\int }_0^s{\int }_{D(z)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}z\mathrm{d}s,t\ge 0.\end{array}$(54) )${}_4$ and with the help of relations (44(44) $\begin{array}{rl}& {\int }_{D(t)}(A_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}+2B_{mnkl}{\mathrm{e}}_{mn}{\epsilon }_{kl}\\ & +C_{mnkl}{\epsilon }_{mn}{\epsilon }_{kl}+2a_{mnk}{\mathrm{e}}_{mn}{\alpha }_k+\\ & +2b_{mnk}{\epsilon }_{mn}{\alpha }_k+k_{mn}{\alpha }_m{\alpha }_n)\mathrm{d}V\\ & +{\int }_0^t{\int }_{D(s)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}s\ge \\ & \ge K_1{\int }_{D(t)}{A}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V\\ & +K_2{\int }_{D(t)}{C}_{mnkl}{\mathrm{e}}_{mn}{\mathrm{e}}_{kl}\mathrm{d}V+\\ & +K_3{\int }_{D(t)}{\mu }_{mn}{\alpha }_m{\alpha }_n\mathrm{d}V,t\in [0,\mathrm{\infty }).\end{array}$(44) ) and (58(58) $\begin{array}{r}{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\ge {\lambda }_0{c}_3{\int }_{D(t)}{\vartheta }^2\mathrm{d}V,\mathrm{\forall }t\in [0,\mathrm{\infty }),\end{array}$(58) ) we obtain: (59) $\begin{array}{r}\dot{\mathcal{G}}(t)\ge \frac{{\lambda }_0{c}_3}{2}{\int }_0^t{\int }_{D(s)}{\vartheta }^2\mathrm{d}V\mathrm{d}s.\end{array}$(59) Now we remember that we considered that solution to the above problem $\mathcal{P}$ which corresponds to null values, both boundary and initial data, so that law (22(22) $\begin{array}{rl}E(t)& =E(0)+{\int }_0^t{\int }_D\varrho (f_m{\dot{v}}_m+g_m{\dot{\varphi }}_m+r\dot{\vartheta })\mathrm{d}V\mathrm{d}s+\\ & +{\int }_0^t{\int }_{\mathrm{\partial }D}({\tau }_{nk}{\dot{v}}_n+{\sigma }_{nk}{\dot{\varphi }}_n+q_k\dot{\vartheta })n_k\mathrm{d}A\mathrm{d}s.\end{array}$(22) ) takes the following form: (60) $\begin{array}{rl}\dot{\mathcal{G}}(t)& =\frac{1}{2}{\int }_0^t{\int }_{D(s)}-a{\vartheta }^2\mathrm{d}V\mathrm{d}s+\\ & +{\int }_0^t{\int }_0^s{\int }_{D(z)}(\varrho f_m{\dot{v}}_m\\ & +I_{mn}{\dot{\varphi }}_m+r\vartheta )\mathrm{d}V\mathrm{d}z\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(60) If we will use the Cauchy–Schwarz inequality, the arithmetic-geometric mean inequality and relations (58(58) $\begin{array}{r}{\int }_{D(t)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\ge {\lambda }_0{c}_3{\int }_{D(t)}{\vartheta }^2\mathrm{d}V,\mathrm{\forall }t\in [0,\mathrm{\infty }),\end{array}$(58) ) and (59(59) $\begin{array}{r}\dot{\mathcal{G}}(t)\ge \frac{{\lambda }_0{c}_3}{2}{\int }_0^t{\int }_{D(s)}{\vartheta }^2\mathrm{d}V\mathrm{d}s.\end{array}$(59) ), from (60(60) $\begin{array}{rl}\dot{\mathcal{G}}(t)& =\frac{1}{2}{\int }_0^t{\int }_{D(s)}-a{\vartheta }^2\mathrm{d}V\mathrm{d}s+\\ & +{\int }_0^t{\int }_0^s{\int }_{D(z)}(\varrho f_m{\dot{v}}_m\\ & +I_{mn}{\dot{\varphi }}_m+r\vartheta )\mathrm{d}V\mathrm{d}z\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(60) ) we deduce: (61) $\begin{array}{rl}\mathcal{G}(t)& \le {\omega }^2\dot{\mathcal{G}}(t)\\ & +\frac{1}{4}{\int }_0^t{\int }_0^s{\int }_{D(z)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}z\mathrm{d}s+R(t)+\\ & +[{\int }_0^t{\int }_{D(s)}(\varrho {\dot{v}}_m{\dot{v}}_m\\ & +{I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n)\mathrm{d}V\mathrm{d}s]}^{1/2}{\int }_0^{t}f(s)\mathrm{d}s.\end{array}$(61) In the last inequality, we apply again the arithmetic-geometric mean inequality, so we get the estimate: (62) $\begin{array}{rl}\mathcal{G}(t)& \le {\omega }^2\dot{\mathcal{G}}(t)+R(t)+{({\int }_0^{t}f(s)\mathrm{d}s)}^2+\\ & +\frac{1}{4}{\int }_0^t{\int }_0^s{\int }_{D(z)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}z\mathrm{d}s\\ & +\frac{1}{4}{\int }_0^t{\int }_{D(s)}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n)\mathrm{d}V\mathrm{d}s,\end{array}$(62) from where, with the help of estimate (57(57) $\begin{array}{rl}\mathcal{G}(t)& \ge \frac{1}{2}{\int }_0^t{\int }_{D(s)}(\varrho {\dot{v}}_m{\dot{v}}_m+I_{mn}{\dot{\varphi }}_m{\varphi }_n)\mathrm{d}V\mathrm{d}s++\frac{1}{2}{\int }_0^t\\ & \times {\int }_0^s{\int }_{D(z)}{\mu }_{mn}{\dot{\alpha }}_m{\dot{\alpha }}_n\mathrm{d}V\mathrm{d}z\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(57) ), it results: $\mathcal{G}(t)\le 2{\omega }^2\dot{\mathcal{G}}(t)+h(t),\mathrm{\forall }t\in [0,\mathrm{\infty }),$which can be rewritten in the form: (63) $\begin{array}{rl}& \frac{\mathrm{d}}{\mathrm{d}t}({\mathrm{e}}^{-t/(2{\omega }^2)}\mathcal{G}(t)\\ & +\frac{1}{2{\omega }^2}{\int }_0^t{\mathrm{e}}^{-s/(2{\omega }^2)}h(s)\mathrm{d}s)\ge 0,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(63) After a simple integration in (63(63) $\begin{array}{rl}& \frac{\mathrm{d}}{\mathrm{d}t}({\mathrm{e}}^{-t/(2{\omega }^2)}\mathcal{G}(t)\\ & +\frac{1}{2{\omega }^2}{\int }_0^t{\mathrm{e}}^{-s/(2{\omega }^2)}h(s)\mathrm{d}s)\ge 0,\mathrm{\forall }t\in [0,\mathrm{\infty }).\end{array}$(63) ) and taking into account hypothesis (55(55) $\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim}{\mathrm{e}}^{-t/2{\omega }^2}\mathcal{G}(t)=0.\end{array}$(55) ), we are led to the following estimate: $\begin{array}{rl}0& \le {\mathrm{e}}^{-t/(2{\omega }^2)}\mathcal{G}(t)\\ & +\frac{1}{2{\omega }^2}{\int }_0^t{\mathrm{e}}^{-s/(2{\omega }^2)}h(s)\mathrm{d}s\le \\ & \le \frac{1}{2{\omega }^2}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-s/(2{\omega }^2)}h(s)\mathrm{d}s,\mathrm{\forall }t\in [0,\mathrm{\infty }),\end{array}$that is, precisely the conclusion of the theorem, which concludes the proof.

4. An application

In this section, we will give a Kolosov–Muskhelishvili type solution for the mixed problem in thermoelasticity of type III for Cosserat media in the case of a plane deformation for an isotropic body.

As such, we have $v_3=0,{\varphi }_1=0,{\varphi }_2=0$, and the functions $v_1,v_2$ and ${\varphi }_3$ are independent of the $x_3$ coordinate.

The calculations are facilitated if we use the complex plane of the variables $z=x_1+i{x}_2$ and $\overline{z}=x_1-i{x}_2$.

We will use the following derivation operators: (64) $\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }z}& =\frac{1}{2}(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}-i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}),\\ \frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}& =\frac{1}{2}(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}+i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}),\\ {\mathrm{\nabla }}^2& =4\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\right).\end{array}$(64) Also, to simplify the writing, we will consider the following functions: (65) $\begin{array}{r}v=v_1+i{v}_2,\overline{v}=v_1-i{v}_2,w=\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{z}}.\end{array}$(65) Taking into account the above notations, according to Iesan and Nappa [41], the complex form of basic system of equations becomes: (66) $\begin{array}{rl}& 2(\mu +\alpha ){\mathrm{\nabla }}^{2}v+(\lambda +\mu -\alpha )\frac{\mathrm{\partial }w}{\mathrm{\partial }\overline{z}}-2\alpha i\frac{\mathrm{\partial }{\varphi }_3}{\mathrm{\partial }\overline{z}}=0,\\ & 2(\nu +\beta ){\mathrm{\nabla }}^2{\varphi }_3+\alpha i(w-2\frac{\mathrm{\partial }v}{\mathrm{\partial }z})-2\alpha {\varphi }_3=0,\end{array}$(66) in which $\alpha ,\beta ,\lambda ,\mu$ and ν are characteristic constants of the material.

Obviously, Equation (66(66) $\begin{array}{rl}& 2(\mu +\alpha ){\mathrm{\nabla }}^{2}v+(\lambda +\mu -\alpha )\frac{\mathrm{\partial }w}{\mathrm{\partial }\overline{z}}-2\alpha i\frac{\mathrm{\partial }{\varphi }_3}{\mathrm{\partial }\overline{z}}=0,\\ & 2(\nu +\beta ){\mathrm{\nabla }}^2{\varphi }_3+\alpha i(w-2\frac{\mathrm{\partial }v}{\mathrm{\partial }z})-2\alpha {\varphi }_3=0,\end{array}$(66) )${}_1$ can be rewritten in the form: (67) $\begin{array}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}[2(\mu +\alpha )\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+(\lambda +\mu -\alpha )w-2\alpha i{\varphi }_3]=0,\end{array}$(67) from where by integration and using the notation $k=(\lambda +3\mu )/(\lambda +\mu )$, we deduce (68) $\begin{array}{r}2(\mu +\alpha )\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+(\lambda +\mu -\alpha )w-2\alpha i{\varphi }_3=(k+1){\gamma }^{\mathrm{\prime }}(z),\end{array}$(68) where the arbitrary function $\gamma (z)$ is analytic for any z in the complex plane.

After applying the conjugate in (68(68) $\begin{array}{r}2(\mu +\alpha )\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+(\lambda +\mu -\alpha )w-2\alpha i{\varphi }_3=(k+1){\gamma }^{\mathrm{\prime }}(z),\end{array}$(68) ), the following equation is obtained: (69) $\begin{array}{r}2(\mu +\alpha )\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{z}}+(\lambda +\mu -\alpha )w+2\alpha i{\varphi }_3=(k+1)\overline{{\gamma }^{\mathrm{\prime }}(z)}.\end{array}$(69) If we add Equations (68(68) $\begin{array}{r}2(\mu +\alpha )\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+(\lambda +\mu -\alpha )w-2\alpha i{\varphi }_3=(k+1){\gamma }^{\mathrm{\prime }}(z),\end{array}$(68) ) and (69(69) $\begin{array}{r}2(\mu +\alpha )\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{z}}+(\lambda +\mu -\alpha )w+2\alpha i{\varphi }_3=(k+1)\overline{{\gamma }^{\mathrm{\prime }}(z)}.\end{array}$(69) ), member by member, we get: (70) $\begin{array}{r}w=\frac{1}{\lambda +\mu }({\gamma }^{\mathrm{\prime }}(z)+\overline{{\gamma }^{\mathrm{\prime }}(z)}),\end{array}$(70) in which w is defined in (65(65) $\begin{array}{r}v=v_1+i{v}_2,\overline{v}=v_1-i{v}_2,w=\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{z}}.\end{array}$(65) ).

Similarly, now we subtract member by member relations (68(68) $\begin{array}{r}2(\mu +\alpha )\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+(\lambda +\mu -\alpha )w-2\alpha i{\varphi }_3=(k+1){\gamma }^{\mathrm{\prime }}(z),\end{array}$(68) ) and (69(69) $\begin{array}{r}2(\mu +\alpha )\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{z}}+(\lambda +\mu -\alpha )w+2\alpha i{\varphi }_3=(k+1)\overline{{\gamma }^{\mathrm{\prime }}(z)}.\end{array}$(69) ) and get: (71) $\begin{array}{rl}i(\frac{\mathrm{\partial }v}{\mathrm{\partial }z}-\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{z}})& =\frac{k+1}{2(\mu +\alpha )}i({\gamma }^{\mathrm{\prime }}(z)-\overline{{\gamma }^{\mathrm{\prime }}(z)})\\ & -\frac{2\alpha }{\mu +\alpha }{\varphi }_3.\end{array}$(71) On the other hand, we can write Equation (66(66) $\begin{array}{rl}& 2(\mu +\alpha ){\mathrm{\nabla }}^{2}v+(\lambda +\mu -\alpha )\frac{\mathrm{\partial }w}{\mathrm{\partial }\overline{z}}-2\alpha i\frac{\mathrm{\partial }{\varphi }_3}{\mathrm{\partial }\overline{z}}=0,\\ & 2(\nu +\beta ){\mathrm{\nabla }}^2{\varphi }_3+\alpha i(w-2\frac{\mathrm{\partial }v}{\mathrm{\partial }z})-2\alpha {\varphi }_3=0,\end{array}$(66) )${}_2$ in the following form: (72) $\begin{array}{r}4{\mathrm{\nabla }}^2{\varphi }_3-\frac{2\alpha }{\nu +\beta }i(\frac{\mathrm{\partial }v}{\mathrm{\partial }z}-\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{z}})-\frac{4\alpha }{\mu +\beta }{\varphi }_3=0,\end{array}$(72) from which, by considering (71(71) $\begin{array}{rl}i(\frac{\mathrm{\partial }v}{\mathrm{\partial }z}-\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{z}})& =\frac{k+1}{2(\mu +\alpha )}i({\gamma }^{\mathrm{\prime }}(z)-\overline{{\gamma }^{\mathrm{\prime }}(z)})\\ & -\frac{2\alpha }{\mu +\alpha }{\varphi }_3.\end{array}$(71) ), can be deduced the equation: (73) $\begin{array}{r}{\mathrm{\nabla }}^2{\varphi }_3-C_1{\varphi }_3=\frac{\alpha (k+1)}{(\nu +\beta )(\mu +\alpha )}i({\gamma }^{\mathrm{\prime }}(z)-\overline{{\gamma }^{\mathrm{\prime }}(z)}),\end{array}$(73) in which we used the notation: $C_1=\frac{4\mu \alpha }{(\nu +\beta )(\mu +\alpha )}>0.$Let us denote by $\xi (z,\overline{z})$ a solution, in its general form, of the equation of Helmholtz's type: ${\mathrm{\nabla }}^2\xi -C_1\xi =0.$Then, the solution of Equation (73(73) $\begin{array}{r}{\mathrm{\nabla }}^2{\varphi }_3-C_1{\varphi }_3=\frac{\alpha (k+1)}{(\nu +\beta )(\mu +\alpha )}i({\gamma }^{\mathrm{\prime }}(z)-\overline{{\gamma }^{\mathrm{\prime }}(z)}),\end{array}$(73) ) can be written in the following form: (74) $\begin{array}{r}2\mu {\varphi }_3=\frac{4\mu }{\nu +\beta }\xi (z,\overline{z})-\frac{k+1}{2}i({\gamma }^{\mathrm{\prime }}(z)-\overline{{\gamma }^{\mathrm{\prime }}(z)}).\end{array}$(74) Now, by substituting the formula (74(74) $\begin{array}{r}2\mu {\varphi }_3=\frac{4\mu }{\nu +\beta }\xi (z,\overline{z})-\frac{k+1}{2}i({\gamma }^{\mathrm{\prime }}(z)-\overline{{\gamma }^{\mathrm{\prime }}(z)}).\end{array}$(74) ) into Equation (68(68) $\begin{array}{r}2(\mu +\alpha )\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+(\lambda +\mu -\alpha )w-2\alpha i{\varphi }_3=(k+1){\gamma }^{\mathrm{\prime }}(z),\end{array}$(68) ), the following relation is reached: (75) $\begin{array}{rl}2\mu \frac{\mathrm{\partial }v}{\mathrm{\partial }z}& =(k-\frac{\mu }{\lambda +\mu }){\gamma }^{\mathrm{\prime }}(z)\\ & -(1+\frac{\mu }{\lambda +\mu })\overline{{\gamma }^{\mathrm{\prime }}(z)}+4i\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(\frac{\mathrm{\partial }\xi (z,\overline{z})}{\mathrm{\partial }\overline{z}}\right).\end{array}$(75) Finally, Equation (75(75) $\begin{array}{rl}2\mu \frac{\mathrm{\partial }v}{\mathrm{\partial }z}& =(k-\frac{\mu }{\lambda +\mu }){\gamma }^{\mathrm{\prime }}(z)\\ & -(1+\frac{\mu }{\lambda +\mu })\overline{{\gamma }^{\mathrm{\prime }}(z)}+4i\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(\frac{\mathrm{\partial }\xi (z,\overline{z})}{\mathrm{\partial }\overline{z}}\right).\end{array}$(75) ) is integrated in relation to the variable z so that the following solution is obtained: (76) $\begin{array}{rl}2\mu v& =(k-\frac{\mu }{\lambda +\mu })\gamma (z)\\ & -(1+\frac{\mu }{\lambda +\mu })z\overline{{\gamma }^{\mathrm{\prime }}(z)}-\overline{\phi (z)}+4i\frac{\mathrm{\partial }\xi (z,\overline{z})}{\mathrm{\partial }\overline{z}},\end{array}$(76) in which the function $\phi (z)$ depends only on z and is analytic with respect to this variable.

In conclusion, the general solution of the system of Equation (66(66) $\begin{array}{rl}& 2(\mu +\alpha ){\mathrm{\nabla }}^{2}v+(\lambda +\mu -\alpha )\frac{\mathrm{\partial }w}{\mathrm{\partial }\overline{z}}-2\alpha i\frac{\mathrm{\partial }{\varphi }_3}{\mathrm{\partial }\overline{z}}=0,\\ & 2(\nu +\beta ){\mathrm{\nabla }}^2{\varphi }_3+\alpha i(w-2\frac{\mathrm{\partial }v}{\mathrm{\partial }z})-2\alpha {\varphi }_3=0,\end{array}$(66) ) is presented in (68(68) $\begin{array}{r}2(\mu +\alpha )\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+(\lambda +\mu -\alpha )w-2\alpha i{\varphi }_3=(k+1){\gamma }^{\mathrm{\prime }}(z),\end{array}$(68) ) and (76(76) $\begin{array}{rl}2\mu v& =(k-\frac{\mu }{\lambda +\mu })\gamma (z)\\ & -(1+\frac{\mu }{\lambda +\mu })z\overline{{\gamma }^{\mathrm{\prime }}(z)}-\overline{\phi (z)}+4i\frac{\mathrm{\partial }\xi (z,\overline{z})}{\mathrm{\partial }\overline{z}},\end{array}$(76) ).

5. Conclusions

Our study is dedicated to a linear theory for the thermoelasticity of type III in the particular case of Cosserat media. At the beginning we introduce the equations and conditions, specific for a mixed problem in this context, namely the motion and energy equations, the initial condition and boundary relations. After that, we establish two results regarding the solution unique to the formulated problem and two results on the continuous dependence of solutions, for the same mixed problem. In specialized literature, usually, the two types of results (of continuous dependence and uniqueness) are established by the logarithmic convexity method or by using Lagrange's identity. Both of these methods are based on the material symmetries of the medium, which does not happen in the case of our study. Moreover, our results concern the theory of type III thermoelasticity for Cosserat media in their most general form, namely anisotropic.

Acknowledgments

The authors are really grateful to the reviewers for their very useful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).