On the hyperbolic thermoelasticity with several dissipation mechanisms
- Open Access
- 26-04-2023
- Original
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Abstract
1 Introduction
The temporal decay of the solutions to thermomechanical problems has deserved much attention and it has led to many studies. If we restrict ourselves only the thermoelasticity, we can cite the work of Dafermos [4] as a pioneering research in the study of the decay of the solutions. In fact, he proved the asymptotic stability for the one-dimensional case as well as the existence of undamped isothermal solutions. Later, several authors shown [18, 26] that, in the one-dimensional case, we can obtain the exponential decay. However, for a dimension greater than one, other authors [12, 13] (see also [11]) proved that we cannot expect such behaviour. This is due to the fact that the coupling terms do not translate, in a relevant form, the damping produced by the heat equation to the mechanical part. Therefore, if we want to achieve the exponential decay in a dimension greater than one, we shall introduce more dissipation mechanisms and, at the same time, to propose coupling mechanisms which translate, in an effective way, the thermal dissipation to the mechanical part. Recently, we have performed this kind of studies in the case of the heat conduction of parabolic type for the two-dimensional setting (see [5, 6]). The extension to the three-dimensional case was straightforward.
It is convenient to recall that it is well-known the exponential decay of the solutions to different theories of the thermoelasticity in the one-dimensional case [20‐22, 25].
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By other hand, the parabolic theories of the heat conduction have received a lot of criticisms from the physical point of view because, indeed, they allow the instantaneous propagation of the thermal waves, which lead to the violability of the causality principle. For this reason, several authors have been interested to propose alternative theories for the heat conduction. The most well-known is the damping hyperbolic heat equation introduced by Cattaneo and Maxwell [3]. From this heat equation, two thermoelastic theories have been derived: the one proposed by Lord and Shulman [17] and the one proposed by Green and Lindsay [8]. We note that both theories have deserved much attention and they have been deeply studied. In particular, the exponential decay is known for the one-dimensional case [1, 25].
As we said before, first cases where time decay of thermoelasticity were considered correspond to the parabolic heat conduction. Several authors developed a similar study for hyperbolic thermoelasticity and similar conclusions were obtained and published. However, it is also known that, in the case of a thermoelastic plate, the behavior of the solutions is different depending on the type of heat conduction theory. In the case of the parabolic theories, we know that the solutions generate an analytic semigroup satisfying the exponential decay [14, 15], but the behavior in the case of hyperbolic heat conduction is very different. It is known that, for the case of the Green and Lindsay theory, we can obtain the exponential decay, but we lose the analyticity [24]. In the case of the Lord and Shulmann theory, even the exponential decay is not found [23]. Another interesting case can be seen in the study of the strain gradient thermoelasticity. It is known that we can obtain the exponential stability in the case of the isotropic one-dimensional parabolic heat conduction, but the decay is again slow for Green and Lindsay or Lord and Shulmann counterpart [24]. Moreover, in a recent article [7], the authors proved that, for the chiral strain gradient thermoelasticity, we could also find several differences. In this sense, for suitable coupling terms the solutions of the parabolic problem (for the heat conduction) generate an analytic semigroup and then, we find the exponential stability, but we could not expect the exponential decay for the Lord and Shulmann theory. Therefore, we believe that it is of interest, from a thermomechanical point of view, to clarify if the behavior of the solutions changes in a relevant way when hyperbolic heat conduction theories are proposed. In this work, our aim is to make a similar study to the one proposed in the parabolic case [5, 6] for a couple of hyperbolic heat conduction theories of thermoelasticity with several dissipative mechanisms. We will see that there are not relevant changes in the solutions when we change the kind of the heat conduction.
The paper is structured as follows. In Sect. 2, we describe the two-dimensional problem involving a thermoelastic material of Lord–Shulman type with four dissipation mechanisms. The basic assumptions are also recalled. Then, in Sect. 3 we write our problem as a Cauchy problem in an adequate Hilbert space and we prove the existence of solutions. The exponential decay of the solutions is shown in the next section whenever a certain assumption (which implies that the coupling terms are not isotropic) is hold. Section 5 is devoted to the Green and Lindsay theory. We make a short sketch of the results and the basic arguments used in the previous sections to obtain similar results as the ones of the Lord and Shulman theory. Finally, we conclude the work with some further comments regarding the difficulty to weaken the required conditions to obtain the exponential decay.
2 Basic equations of the thermoelastic model
In this work, we consider a two-dimensional domain B with a boundary \(\partial B\) assumed smooth enough to guarantee the use of the divergence theorem.
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We will study in B the problem determined by the following system of partial differential equations:where \(i,j=1,2\) and \(p,q,l=1,\ldots ,4\), and we have used the notation:It is worth noting that this system corresponds to the thermoelasticity of Lord–Shulman with four dissipative mechanisms.
$$\begin{aligned} \begin{array}{l} \rho \ddot{u}_i=\mu \Delta u_i+(\lambda +\mu )u_{j,ji}+A_{ik}^l\theta _{l,k},\\ m_{lp}\dot{\hat{\theta }}_p=K_{lp}\Delta \theta _p+A_{ik}^l\dot{{\hat{u}}}_{i,k}+\xi _{pq}^l(\theta _p-\theta _q), \end{array} \end{aligned}$$
(1)
$$\begin{aligned} {\hat{f}}=f+d{\dot{f}} \quad \hbox {for a given}\quad d>0. \end{aligned}$$
(2)
As usual, here we denote by \(u_i\) the displacement vector, \(\theta _1\) and \(\theta _2\) are two different temperatures, \(\theta _3\) and \(\theta _4\) determine two dissipative mechanisms (which can be associated to the mass diffusion), \(\rho \) is the mass density, \(\lambda \) and \(\mu \) are the Lamé coefficients, \(K_{lp}\) is the matrix of thermal conductivity (or the mass dissipation), \(m_{lp}\) is the matrix of thermal capacity (or the mass dissipation), \(\xi _{pq}^l\) are the coefficients related to the relative temperatures (concentrations) and \(A_{pq}^l\) are the coupling coefficients.
Due to the type of problem we propose, we assume that \(\xi _{12}^2=-\xi _{12}^1=l_1\), \(\xi _{34}^2=-\xi _{34}^1=\xi _{12}^4=-\xi _{12}^3=l_2\), \(\xi _{34}^4=-\xi _{34}^3=l_3\) and the remaining \(\xi _{pq}^l=0\) (see [10]).
It is usual to find processes where two temperatures \(\theta _1,\theta _2\) act at the same time as we can see in the classical works of Ieşan [10] or Gurtin and de La Penha [9]. They are usually understood in the context of mixtures. At the same time, the mass diffusion is a dissipative mechanism which is considered and studied for a long time ago [19]. Of course, we can also consider it in a similar way of the two temperatures in order to obtain two mechanisms of mass diffusion (for \(\theta _3,\theta _4\)).
We can note that, using notation (2), system (1) can be written as:From now on, in order to simplify the writing, we will omit the hat over all the variables of system (3).
$$\begin{aligned} \begin{array}{l} \rho \ddot{{\hat{u}}}_i=\mu \Delta {\hat{u}}_i+(\lambda +\mu ){\hat{u}}_{j,ji}+A_{ik}^l(\theta _{l,k}+d{\dot{\theta }}_{l,k}),\\ m_{lp}({\dot{\theta }}_p+d\ddot{\theta }_p)=K_{lp}\Delta \theta _p+A_{ik}^l\dot{{\hat{u}}}_{i,k}+\xi _{pq}^l(\theta _p-\theta _q). \end{array} \end{aligned}$$
(3)
We will also assume the boundary conditions:
where \(i,j=1,2\), \(l=1,\ldots ,4\) and
represents the outward unit normal vector to \(\partial B\) at point \(\varvec{x}\). Moreover, we also prescribe some initial conditions:
where \(i=1,2\) and \(j=1,\ldots ,4\).
(4)
represents the outward unit normal vector to \(\partial B\) at point \(\varvec{x}\). Moreover, we also prescribe some initial conditions:
(5)
We will study problem (3)-(5) under the following assumptions: The meaning of the condition on the mass density is clear, meanwhile the assumptions on the Lamé constants guarantee that the elastic part is positive definite (and then, the stability is deduced). Assumptions (ii) are natural to work with several dissipation mechanisms, although the symmetry of the term \(K_{ij}\) is not a consequence of the thermomechanical axioms. Condition (iii) is related with the property of the heat (mass diffusion) conductor and it leads to the dissipation of the variables \(\theta _i\).
(i)
\(\rho >0\), \(\mu >0\) and \(\lambda +\mu >0\).
(ii)
The matrices \(m_{ij}\) and \(K_{ij}\) are symmetric; that is, \(m_{ij}=m_{ji}\) and \(K_{ij}=K_{ji}\) for \(i,j=1,\ldots ,4\).
(iii)
The matrices \(m_{ij}\), \(K_{ij}\) and \(l_i\) are positive definite; that is, there exist three positive constants \(C_1\), \(C_2\) and \(C_3\) such that
$$\begin{aligned} \begin{array}{l} m_{ij}\xi _i\xi _j\ge C_1 \xi _i\xi _i,\\ K_{ij}\xi _i\xi _j\ge C_2 \xi _i\xi _i,\\ l_1\xi _1^2+2l_2\xi _1\xi _2+l_3\xi _2^2\ge C_3(\xi _1^2+\xi _2^2). \end{array} \end{aligned}$$
It is relevant to point out that the solutions to problem (3)–(5) satisfy the equality:whereand
$$\begin{aligned} E(t)+2\int _0^t D(s)\, ds=E(0), \end{aligned}$$
(6)
$$\begin{aligned} \begin{array}{l} \displaystyle E(t)=\int _B\Big (\rho {\dot{u}}_i{\dot{u}}_i+2\mu e_{ij}e_{ij}+\lambda e_{ii}e_{jj}+m_{lk}(\theta _l+d{\dot{\theta }}_l)(\theta _k+d{\dot{\theta }}_k)+dK_{lk}\nabla \theta _l\nabla \theta _k\\ \qquad \displaystyle \qquad +d \left[ l_1(\theta _1-\theta _2)^2+2l_2(\theta _1- \theta _2)(\theta _3-\theta _4)+l_3(\theta _3-\theta _4)^2 \right] \Big )\, da, \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{l} \displaystyle D(t)=\int _B\Big (K_{lp}\nabla \theta _l\nabla \theta _p+l_1(\theta _1-\theta _2)^2+2l_2(\theta _1-\theta _2)(\theta _3-\theta _4)\\ \qquad \displaystyle \qquad +l_3(\theta _3-\theta _4)^2\Big )\, da. \end{array} \end{aligned}$$
3 Existence of solutions
In this section, we will obtain an existence and uniqueness result to problem (3)–(5), under the assumptions (i)-(iii), in the following Hilbert space:where \(H^1_0(B)\) and \(L^2(B)\) are the classical Sobolev spaces, andIn this space, we will consider the inner product with the norm defined byAs usual, a bar over a variable denotes the conjugated complex. It is also important to note that this norm is equivalent to the usual one in the Hilbert space \(\mathcal {H}\) by using assumptions (i)–(iii).
$$\begin{aligned} \mathcal {H}=[H^1_0(B)]^2\times [L^2(B)]^2\times [H^1_*(B)]^4\times [L^2_*(B)]^4, \end{aligned}$$
$$\begin{aligned} L^2_*(B)=\left\{ f\in L^2(B); \, \int _B f\,da=0 \right\} ,\quad H_*^1(B)=H^1(B)\cap L^2_*(B). \end{aligned}$$
$$\begin{aligned} \begin{array}{l} \displaystyle \Vert (\varvec{u},\varvec{v},\varvec{\theta },\varvec{\vartheta })\Vert ^2=\int _B \Big (\rho v_i\overline{v_i}+2\mu e_{ij}\overline{e_{ij}} +\lambda e_{ii}\overline{e_{jj}}+m_{lk}(\theta _l+d\vartheta _l)\overline{(\theta _k+d\vartheta _k)}\\ \qquad \qquad \qquad \qquad \quad \; \displaystyle +dK_{lk}\nabla \theta _l\overline{\nabla {\theta _k}}+d\xi _{pq}^l(\theta _p-\theta _q)\overline{\theta _l}\Big )\, da. \end{array} \end{aligned}$$
We can write our problem in the following form:where the matrix differential operator \(\mathcal {A}\) is defined asHere, \((n_{dl})\) is the inverse matrix of \((m_{lp})\). It is convenient to remark that the domain of the operator \(\mathcal {A}\) are the elements of the Hilbert space \(\mathcal {H}\) such that \(v_i\in H^1_0(B),\) \(\vartheta _l\in H^1_*(B)\), \(\mu \Delta u_i+(\lambda +\mu )u_{j,ji}\in L^2(B)\) and \(\theta _l\in H^2(B)\cap L^2_*(B)\).
$$\begin{aligned} \frac{d}{dt} U(t)=\mathcal {A} U(t), \quad U(0)=(\varvec{u}^0,\varvec{v}^0,\varvec{\theta }^0,\varvec{\vartheta }^0), \end{aligned}$$
(7)
$$\begin{aligned} \mathcal {A}\left( \begin{array}{c} u_i\\ v_i\\ \theta _d\\ \vartheta _d \end{array}\right) =\left( \begin{array}{c} v_i\\ \rho ^{-1}(\mu \Delta u_i+(\lambda +\mu )u_{j,ji}+A_{ik}^l(\theta _{l,k}+d\vartheta _{l,k}))\\ \vartheta _d\\ d^{-1}n_{dl}(K_{lp}\Delta \theta _p+A_{ik}^lv_{i,k}+\xi _{pq}^l(\theta _p-\theta _q))-d^{-1}\vartheta _d \end{array}\right) . \end{aligned}$$
Remark 1
Although, as usual we assume the second condition of (4) at the domain of the operator, we actually state the suitable ones. That should be more given in the form:For the same reason, it is implicitly assumed that \(\theta _{p,j}n_j=0\) on the boundary.
$$\begin{aligned} K_{lp}\theta _{p,j}n_j=0\quad \hbox {on}\quad \partial B. \end{aligned}$$
Lemma 1
For each element U of the domain of the operator \(\mathcal {A}\) we find that
$$\begin{aligned} Re<\mathcal {A}U,U>\le 0. \end{aligned}$$
(8)
Proof
If we apply the divergence theorem and we use the boundary conditions (4) we obtainFrom the assumptions (iii) we can see that condition (8) is satisfied. \(\square \)
$$\begin{aligned}\begin{array}{l} \displaystyle Re <\mathcal {A}U,U>=-\int _B \Big ( K_{lm} \nabla \theta _l \overline{\nabla \theta _m}+l_1(\theta _1-\theta _2)\overline{(\theta _1-\theta _2)}\\ \qquad \qquad \qquad \qquad \qquad \;\; + l_3(\theta _3-\theta _4)\overline{(\theta _3-\theta _4)}+l_2[(\theta _1-\theta _2)\overline{(\theta _3-\theta _4)} + (\theta _3-\theta _4)\overline{(\theta _1-\theta _2)}]\Big )\, da. \end{array} \end{aligned}$$
Lemma 2
Zero belongs to the resolvent of the operator \(\mathcal {A}\).
Proof
Let us consider \(F=(f_1,f_2,f_3,f_4,g_1,g_2,g_3,g_4,h_1,h_2,h_3,h_4)\) an element of the Hilbert space. We must prove the existence of an element U in the domain of \(\mathcal {A}\) such thatWe can observe that it implies thatHere, \(i,j,k=1,2\) and \(p,q,l=1,\ldots , 4\).
$$\begin{aligned} \mathcal {A}U=F. \end{aligned}$$
$$\begin{aligned} \begin{array}{l} v_i=f_i,\\ \mu \Delta u_i+(\lambda +\mu )u_{j,ji}+A_{ik}^l(\theta _{l,k}+d\vartheta _{l,k})=\rho f_{2+i},\\ \vartheta _p=g_p,\\ K_{lp}\Delta \theta _p+A_{ik}^lv_{i,k}+\xi _{pq}^l(\theta _p-\theta _q)-m_{lp}\vartheta _p=d m_{lp}h_p. \end{array} \end{aligned}$$
Clearly, we can obtain \(v_i\) and \(\vartheta _p\) and therefore, it leads to the following linear system:Keeping in mind that the right-hand side of the second system of equations of the previous system is in \(L^2(B)\) and the required assumptions on \(K_{lp}\) and \(\xi _{pq}^l\), we can conclude the existence of a solution \(\theta _p\in H^1(B)\). Thus, introducing this solution in the first equation of system (9) we can find the solution for \(u_i\). Moreover, we can obtain the existence of a positive constant C, independent of F, such that\(\square \)
$$\begin{aligned} \begin{array}{l} \mu \Delta u_i+(\lambda +\mu )u_{j,ji}+A_{ik}^l\theta _{l,k}=\rho f_{2+i}-d A_{ik}^lg_{l,k},\\ K_{lp}\Delta \theta _p+\xi _{pq}^l(\theta _p-\theta _q)=-A_{ik}^lf_{i,k}+m_{lp}g_p+d m_{lp}h_p. \end{array} \end{aligned}$$
(9)
$$\begin{aligned} \Vert U\Vert \le C \Vert F\Vert . \end{aligned}$$
If we apply the corollary of Lumer–Phillips to the Hille–Yosida theorem, we have shown the following.
Theorem 1
The operator \(\mathcal {A}\) generates a \(C^0\)-semigroup of contractions in the space \(\mathcal {H}\).
Remark 2
Since the operator \(\mathcal {A}\) generates a \(C^0\)-semigroup of contractions, we see that, if \((\hat{\varvec{u}}(0),\hat{\varvec{v}}(0),\varvec{\theta }(0),\varvec{\vartheta }(0))\) belongs to the domain of the operator, we find thatTherefore, we obtain the regularity for the variables \(\varvec{\theta }\) and \(\varvec{\vartheta }\). With respect to the displacement field \(\varvec{u}(t)\), we note thatand so,Thus, we haveIn a similar way, we can see that \(\varvec{v}(t)\in C^1([0,t];[H^1_0(B)]^2)\).
$$\begin{aligned} (\hat{\varvec{u}}(t),\hat{\varvec{v}}(t),\varvec{\theta }(t),\varvec{\vartheta }(t))\in C^0([0,t]; \mathcal {D(A)})\cap C^1([0,t];\mathcal {H}). \end{aligned}$$
$$\begin{aligned} \varvec{u}(t)+d{\dot{\varvec{u}}}(t)\in C^1([0,t];[H^1_0(B)]^2), \end{aligned}$$
$$\begin{aligned} \frac{d}{dt}\left( e^{d^{-1}t}\varvec{u}(t)\right) \in C^1([0,t];[H^1_0(B)]^2). \end{aligned}$$
$$\begin{aligned} \varvec{u}(t)\in C^2([0,t];[H^1_0(B)]^2). \end{aligned}$$
However, we point out that we must also impose strict conditions on the initial data.
4 Exponential decay for a particular choice of the coupling coefficients
In this section, we aim to prove the exponential decay of the solutions obtained in the previous section whenever the coupling matrix \(\left( A_{ij}^l \right) \) given byhas rank 4. It is stated in the following.
$$\begin{aligned} \left( \begin{array}{c c c c} A_{11}^1&{}A_{12}^1&{}A_{21}^1&{}A_{22}^1 \\ A_{11}^2&{}A_{12}^2&{}A_{21}^2&{}A_{22}^2 \\ A_{11}^3&{}A_{12}^3&{}A_{21}^3&{}A_{22}^3 \\ A_{11}^4&{}A_{12}^4&{}A_{21}^4&{}A_{22}^4 \end{array}\right) \end{aligned}$$
Theorem 2
If the coupling matrix \(\left( A_{ij}^l \right) \) has rank 4, then the solutions to problem (7) decay in an exponential form.
Proof
In order to prove this result, we will use the characterization of the semigroup of contractions exponentially stable. From [16] we know that this is equivalent to show that the imaginary axis is within its resolvent and that the conditionholds.
$$\begin{aligned} \overline{\lim _{\mid \beta \mid \rightarrow \infty }}\Vert (i\beta I-\mathcal {A})^{-1}\Vert <\infty \end{aligned}$$
(10)
First, we will prove the condition about the resolvent. It will be done by contradiction. If such condition is not satisfied, then there exist a sequence of real numbers \(\beta _n\rightarrow \beta \ne 0\) and a sequence of vectors in the domain of the operator, with unit norm, such thatThis convergence is equivalent to the following convergences:From the dissipation inequality we find thatand so, it follows that \(\beta _n^{-1}\nabla \vartheta _{ln}\rightarrow 0\). If we multiply convergence (15) by \(\beta _n^{-1}\vartheta _{qn}\) we find that \(\vartheta _{pn}\rightarrow 0\). If we divide now convergence (15) by \(\beta _n^{-1}\) we havefor \(q=1,\ldots ,4\). If we multiply now q-th convergence by \(A_{ik}^qu_{in,k}\) and we take into account thatbecause \(\beta _n^{-1} u_{in}\) is bounded in \(H^2(B)\) since the elastic part is positive definite, we conclude thatSince we have assumed that the matrix of coefficients has rank four, we can conclude that \(u_{in}\rightarrow 0\) in \(H^1(B)\). If we multiply now convergence (13) by \(u_{in}\) we also conclude that \(v_{in}\rightarrow 0\), and we arrive to a contradiction because we assumed that \(i\beta \ne 0\) did not belong to the resolvent.
$$\begin{aligned} \Vert (i\beta _n I-\mathcal {A})U_n\Vert \rightarrow 0. \end{aligned}$$
(11)
$$\begin{aligned}{} & {} i\beta _n u_{jn}-v_{jn}\rightarrow 0 \quad \hbox {in}\quad H^1(B), \end{aligned}$$
(12)
$$\begin{aligned}{} & {} i\beta _n\rho v_{jn}-(\mu \Delta u_{jn}+(\lambda +\mu )u_{kn,kj}+A_{jk}^l\theta _{ln,k} +dA_{jk}^l\vartheta _{ln,k}) \rightarrow 0\quad \hbox {in}\quad L^2(B), \end{aligned}$$
(13)
$$\begin{aligned}{} & {} i\beta _n\theta _{qn}-\vartheta _{qn}\rightarrow 0 \quad \hbox {in}\quad H^1(B), \end{aligned}$$
(14)
$$\begin{aligned}{} & {} i\beta _n d m_{qp}\vartheta _{pn}-(K_{qp}\Delta \theta _{pn}+A_{ik}^qv_{in,k} +\xi _{rs}^q(\theta _{rn}-\theta _{sn}) -m_{qp}\vartheta _{pn})\rightarrow 0 \quad \hbox {in}\quad L^2(B). \end{aligned}$$
(15)
$$\begin{aligned} \nabla \theta _{ln}\rightarrow 0, \end{aligned}$$
$$\begin{aligned} \beta _n^{-1}K_{qp}\Delta \theta _{pn}+iA_{ik}^q u_{in,k}\rightarrow 0 \quad \hbox {in}\quad L^2(B) \end{aligned}$$
$$\begin{aligned}<\beta _n^{-1}\Delta \theta _{pn},A_{ik}^q u_{in,k}>=-<\nabla \theta _{pn},\beta _n^{-1}A_{ik}^q \nabla u_{in,k}>\rightarrow 0, \end{aligned}$$
$$\begin{aligned} A_{ik}^q u_{in,k}\rightarrow 0 \quad \hbox {in}\quad L^2(B) \hbox { for }q=1,\ldots ,4. \end{aligned}$$
In order to show the asymptotic condition (10), we can follow a similar argument. Hence, we assume that this condition does not hold. Therefore, there will exist a sequence of real numbers \(\beta _n\rightarrow \infty \) and a sequence of unit norm vectors, belonging to the domain of the operator \(\mathcal {A}\), such that convergence (11) is fulfilled. Now, we can repeat the same argument used previously, since the key point was to assume that \(\beta _n\) did not tend to zero, and we arrive again to a contradiction. Therefore, the asymptotic condition is satisfied and the theorem is proved. \(\square \)
5 An application to the Green-Lindsay theory
We can also consider the case when the dissipative mechanisms are given by the Green and Linday theory. The system of equations is now the following:For this system, we impose again the boundary and initial conditions given in (4) and (5), respectively. We also assume conditions (i)–(iii) but we will also need that the matrixis positive definite; that is,
$$\begin{aligned} \begin{array}{l} \rho \ddot{u}_i=\mu \Delta u_i+(\lambda +\mu )u_{j,ji}+A_{ik}^l(\theta _{l,k}+\alpha {\dot{\theta }}_{l,k}),\\ m_{lp}\ddot{\theta }_p+d_{lp}{\dot{\theta }}_p=K_{lp}\Delta \theta _p+A_{ik}^l{\dot{u}}_{i,k}+\xi _{pq}^l(\theta _p-\theta _q). \end{array} \end{aligned}$$
(16)
$$\begin{aligned} (d_{lp}\alpha -m_{lp}) \end{aligned}$$
(iv) there exists a positive constant \(C_4\) such that \((d_{lp}\alpha -m_{lp})\xi _l\xi _p\ge C_4 \xi _i\xi _i.\)
We note that condition (iv) is compatible with the usual assumptions within the Green-Lindsay theory.
We can observe that the relation (6) is satisfied whenandWe can consider this problem in the Hilbert space given in Sect. 3 and to define an inner product with the norm:We can write the problem in the form (7), whereIt is clear that the domain of the operator \(\mathcal {A}\) is made of the elements of the Hilbert space \(\mathcal {H}\) such that \(v_i\in H^1_0(B)\), \(\vartheta _l\in H^1_*(B)\), \(\mu \Delta u_i+(\lambda +\mu )u_{j,ji}\in L^2(B)\) and \(\theta _l\in H^2(B)\cap L^2_*(B)\). Therefore, it is dense in \(\mathcal {H}\).
$$\begin{aligned} \begin{array}{l} \displaystyle E(t)=\int _B \Big (\rho {\dot{u}}_i{\dot{u}}_i+2\mu e_{ij}e_{ij}+\lambda e_{ii}e_{jj}+\alpha ^{-1}m_{lp}(\theta _l+\alpha {\dot{\theta }}_l)(\theta _p+\alpha {\dot{\theta }}_p)\\ \qquad \qquad \displaystyle +\alpha \left[ l_1(\theta _1-\theta _2)^2 +2l_2(\theta _1-\theta _2)(\theta _3-\theta _4)+l_3(\theta _3-\theta _4)^2 \right] \\ \qquad \qquad \displaystyle +(d_{lp}-\alpha ^{-1}m_{lp})\theta _l\theta _p+\alpha K_{lp}\nabla \theta _l\nabla \theta _p\Big )\, da, \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{l} \displaystyle D(t)=\int _B\Big (l_1(\theta _1-\theta _2)^2 +2l_2(\theta _1-\theta _2)(\theta _3-\theta _4)+l_3(\theta _3-\theta _4)^2\\ \qquad \qquad \displaystyle +K_{lp}\nabla \theta _l\nabla \theta _p+ (d_{lp}\alpha -m_{lp}){\dot{\theta }}_l{\dot{\theta }}_p\Big )\, da. \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{l} \displaystyle \Vert (\varvec{u},\varvec{v},\varvec{\theta },\varvec{\vartheta })\Vert ^2=\int _B\Big ( \rho v_i\overline{v_i}+2\mu e_{ij}\overline{e_{ij}}+\lambda e_{ii}\overline{e_{jj}}\\ \qquad \qquad \qquad \qquad \;\;\; +\alpha \xi _{pq}^l(\theta _p-\theta _q)\overline{\theta _l}+\alpha ^{-1}m_{lp}(\theta _l+\alpha \vartheta _l)\overline{(\theta _p+\alpha \vartheta _p)}\\ \qquad \qquad \qquad \qquad \;\;\; +( d_{lp}-\alpha ^{-1}m_{lp})\theta _l\overline{\theta _p}+\alpha K_{lp}\nabla \theta _l\overline{\nabla \theta _p}\Big )\,da. \end{array} \end{aligned}$$
$$\begin{aligned} \mathcal {A}\left( \begin{array}{c} u_i\\ v_i\\ \theta _d\\ \vartheta _d \end{array}\right) =\left( \begin{array}{c} v_i\\ \rho ^{-1}(\mu \Delta u_i+(\lambda +\mu )u_{j,ji}+A_{ik}^l(\theta _{l,k}+\alpha \vartheta _{l,k}))\\ \vartheta _d\\ n_{dl}(K_{lp}\Delta \theta _p+A_{ik}^lv_{i,k}+\xi _{pq}^l(\theta _p-\theta _q)-d_{lp}\vartheta _p) \end{array}\right) . \end{aligned}$$
We can see thatfor every \(U\in \mathcal {D(A)}\). Therefore, \(\mathcal {A}\) is a dissipative operator.
$$\begin{aligned} \begin{array}{l} \displaystyle Re <\mathcal {A}U,U>=-\int _B\Big (K_{lp}\nabla \theta _l\overline{\nabla \theta _p}+ (d_{lp}\alpha -m_{lp})\vartheta _l\overline{\vartheta _p}+l_1(\theta _1-\theta _2)\overline{(\theta _1-\theta _2)}\\ \qquad \qquad \qquad \qquad \qquad \displaystyle +l_3(\theta _3-\theta _4)\overline{(\theta _3-\theta _4)} +l_2 \left[ (\theta _1-\theta _2)\overline{(\theta _3-\theta _4)}+(\theta _3-\theta _4)\overline{(\theta _1-\theta _2)}\right] \Big )\, da \end{array} \end{aligned}$$
By other hand, if \(F=(f_1,f_2,f_3,f_4,g_1,g_2,g_3,g_4,h_1,h_2,h_3,h_4)\) and we try to solve the system:This system leads to the following coupled equations:which can be solved similarly as in the proof of Lemma 2. Again, we can conclude an existence and uniqueness result for the solutions as in Theorem 1.
$$\begin{aligned} \begin{array}{l} v_i=f_i,\\ \mu \Delta u_i+(\lambda +\mu )u_{j,ji}+A_{ik}^l(\theta _{l,k}+\alpha \vartheta _{l,k})=\rho f_{2+i},\\ \vartheta _p=g_p,\\ K_{lp}\Delta \theta _p+A_{ik}^lv_{i,k}+\xi _{pq}^l(\theta _p-\theta _q)-d_{lp}\vartheta _p= m_{lp}h_p. \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{l} \mu \Delta u_i+(\lambda +\mu )u_{j,ji}+A_{ik}^l\theta _{l,k}=\rho f_{2+i}-\alpha A_{ik}^lg_{l,k},\\ K_{lp}\Delta \theta _p+\xi _{pq}^l(\theta _p-\theta _q)=m_{lp}h_p-A_{ik}^lf_{i,k}+d_{lp}g_p, \end{array} \end{aligned}$$
Finally, we can show the exponential decay of the solutions. Following the same arguments used in the proof of Theorem 2 we find thatIn this case, the energy inequality leads toFrom now on, we can follow the same procedure as in the proof of Theorem 2 and therefore, we can also obtain the exponential decay of the solutions whenever the coupling matrix has rank 4, that we state in the following.
$$\begin{aligned}{} & {} i\beta _n u_{jn}-\vartheta _{jn}\rightarrow 0 \quad \hbox {in}\quad H^1(B),\\{} & {} i\beta _n\rho v_{jn}-(\mu \Delta u_{jn}+(\lambda +\mu )u_{kn,kj}+A_{jk}^l (\theta _{ln,k}+\alpha \vartheta _{ln,k}) \rightarrow 0\quad \hbox {in}\quad L^2(B),\\{} & {} i\beta _n\theta _{qn}-\vartheta _{qn}\rightarrow 0 \quad \hbox {in}\quad H^1(B),\\{} & {} \nonumber i\beta _n m_{lp}\vartheta _{pn}-(K_{qp}\Delta \theta _{pn}+A_{ik}^qv_{in,k} +\xi _{pq}^l(\theta _{p}-\theta _{q}) -d_{lp}\vartheta _{p})\rightarrow 0 \quad \hbox {in}\quad L^2(B). \end{aligned}$$
$$\begin{aligned} \nabla \theta _{pn}\rightarrow 0,\quad \vartheta _{pn}\rightarrow 0. \end{aligned}$$
6 Further comments
It is relevant to note that Theorems 2 and 3 have been shown assuming that the coupling matrix had maximum rank. Clearly, this requirement was a key point in the proof. However, in order to satisfy this condition it is needed that its terms are not isotropic. In fact, in the isotropic case we can follow the example provided by Dafermos [4] and we can obtain undamped isothermal solutions. This example could be generalized to domains satisfying the following condition (see [2]):
CONDITION D. There exists a nonzero vector field \(\varvec{\phi }\in [H^1(B)]^2\) such thatIn the case \(\lambda +\mu =0\) we can consider the system:If we impose the initial conditions:we also obtain undamped isothermal solutions. In this case, the matrix of coupling coefficients has rank 2.
$$\begin{aligned} \phi _{i,jj}+\gamma ^2 \phi _i=0 \hbox { for a given }\gamma \ne 0,\quad \phi _{i,i}=0. \end{aligned}$$
$$\begin{aligned} \begin{array}{l} \rho \ddot{u}_1=\mu \Delta u_1+A_{11}\theta _{1,1}+A_{22}\theta _{2,2},\\ \rho \ddot{u}_2=\mu \Delta u_2-A_{11}\theta _{1,1}-A_{22}\theta _{2,2},\\ m_1({\dot{\theta }}_1+d\ddot{\theta }_1)=K_1\Delta \theta _1-l_1(\theta _1-\theta _2)-l_2(\theta _3-\theta _4) +A_{11}{\dot{u}}_{1,1}-A_{11}{\dot{u}}_{2,1},\\ m_2({\dot{\theta }}_2+d\ddot{\theta }_2)=K_2\Delta \theta _2+l_1(\theta _1-\theta _2)+l_2(\theta _3-\theta _4) +A_{22}{\dot{u}}_{1,2}-A_{22}{\dot{u}}_{2,2},\\ m_3({\dot{\theta }}_3+d\ddot{\theta }_3)=K_3\Delta \theta _3-l_2(\theta _1-\theta _2)-l_3(\theta _3-\theta _4),\\ m_4({\dot{\theta }}_4+d\ddot{\theta }_4)=K_4\Delta \theta _4+l_2(\theta _1-\theta _2)+l_3(\theta _3-\theta _4). \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{l} u_1(\varvec{x},0)=u_2(\varvec{x},0)=u^0(\varvec{x}),\quad {\dot{u}}_1(\varvec{x},0)={\dot{u}}_2(\varvec{x},0)=v^0(\varvec{x}),\\ \theta _i(\varvec{x},0)=\vartheta _i(\varvec{x},0)=0 \quad i=1,\ldots ,4, \end{array} \end{aligned}$$
Finally, we point out that we can extend the results of the previous sections to the case where the elasticity tensor is positive definite, or to the three-dimensional case but, in this case, we shall consider nine dissipative mechanisms and the coupling matrix must have maximum rank. Moreover, it is worth remarking that similar examples can be also considered for the Green-Lindsay theories.
Acknowledgements
This paper is part of the project PID2019-105118GB-I00, funded by the Spanish Ministry of Science, Innovation and Universities and FEDER “A way to make Europe”. The authors would like to express their gratitude to the anonymous referees, whose comments improved the quality of the paper.
Declarations
Conflict of interest
The authors declare there is no conflict of interest.
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