2. Basic equations and assumptions

In this section, we propose the system of equations and the basic conditions under which we are going to work in this article. We consider a two-dimensional domain $B$ with a boundary smooth enough ( $C^1$ ) to apply our arguments (Divergence and Trace theorems).

We assume that our domain is occupied by a homogeneousFootnote ¹ strain gradient elastic material with two dissipative mechanisms that we could think as temperature ( $\theta _1$ ) and mass dissipation ( $\theta _2$ ).

Our system of equations can be written as

(2.1) \begin{align} \rho \ddot{u}_i&=\left (A_{ijrs}u_{r,s}+B_{ijpqr}u_{r,pq}+a_{ij}^l\theta _l\right )_{,j}\nonumber \\[5pt] & \quad -\left (B_{rsijk}u_{r,s}+C_{jkipqr}u_{r,pq}+C_{ijk}^l\theta _l\right )_{,jk}\!,\\[5pt] d_{lj}\dot{\theta }_j&=(k_{ij}^{lm}\theta _{m,j})_{,i}+a_{ij}^l \dot{u}_{i,j}+C_{ijk}^l\dot{u}_{i,jk}, \nonumber \end{align}

where $i,j,k,m,p,q,r,s,l=1,2$ .

We are going to assume the initial conditions, for a.e. ${\boldsymbol{x}}\in B$ ,

(2.2) \begin{equation} u_i({\boldsymbol{x}},0)=u_i^0({\boldsymbol{x}}),\quad \dot{u}_i({\boldsymbol{x}},0)=v_i^0({\boldsymbol{x}}),\quad \theta _j({\boldsymbol{x}},0)=\theta _j^0({\boldsymbol{x}}), \end{equation}

and the boundary conditions:

(2.3) \begin{equation} u_i({\boldsymbol{x}},t)=u_{i,j}({\boldsymbol{x}},t)=\theta _i({\boldsymbol{x}},t)=0\quad{\boldsymbol{x}}\in \partial B, \,\, t\gt 0. \end{equation}

In system (2.1), $u_i$ is the two-dimensional displacement, $\theta _1$ is the temperature, $\theta _2$ is the mass dissipation variable, $A_{ijrs}$ , $B_{ijpqr}$ and $C_{ijkpqr}$ define the strain gradient elasticity, $\rho$ is the mass density, $d_{ij}$ plays a role similar to the thermal capacity in the classical heat equation, and $k_{ij}^{lm}$ has a role similar to the thermal conductivity, and $a_{ij}^l$ and $C_{ijk}^l$ are the coupling terms.

From now on, we are going to assume that

1. (i) $\rho \gt 0$ , $d_{ij}$ is a positive definite matrix.

2. (ii) There exists a positive constant $C$ such that

   (2.4) \begin{equation} \begin{array}{l} \displaystyle \int _B\Big ( A_{ijrs}u_{i,j}u_{r,s}+2B_{ijpqr}u_{i,j}u_{r,pq}+C_{ijlpqr}u_{l,ij}u_{r,pq}\Big )\, dA\\[5pt] \qquad \displaystyle \qquad \geqslant C\int _B \Big (u_{i,j}u_{i,j}+u_{r,pq}u_{r,pq}\Big )\, dA \end{array} \end{equation}
   for every vector function $u_i$ vanishing on the boundary.

3. (iii) There exists a positive constant $D$ such that

   (2.5) \begin{equation} k_{ij}^{lm}\theta _{l,i}\theta _{m,j}\geqslant D \theta _{i,j}\theta _{i,j}. \end{equation}

4. (iv) The terms $A_{ijrs}$ , $B_{ijpqr}$ , $C_{ijkpqr}$ , $C_{ijk}^l$ , $d_{ij}$ and $k_{ij}^{lm}$ are symmetric in the sense that

   (2.6) \begin{equation} \begin{array}{l} A_{ijrs}=A_{rsij},\quad B_{ijpqr}=B_{ijqpr},\quad C_{ijkpqr}=C_{jikpqr},\quad C_{ijk}^l=C_{ikj}^l,\\[5pt] d_{ij}=d_{ji},\quad k_{ij}^{lm}=k_{ji}^{ml}. \end{array} \end{equation}

We note that assumptions (i)–(iv) are natural in the study of strain gradient thermoelasticity. The meaning of (i) is obvious, the condition (ii) can be interpreted in the context of the elastic stability, condition (iii) guarantees that the dissipation is positive and conditions (iv) are also natural in the context of two dissipation mechanisms.

We are going to transform our problem as a Cauchy problem defined on a suitable Hilbert space. From now on, we denote by $\mathcal{H}$ the space:

\begin{equation*}\mathcal {H}={\boldsymbol{W}}_0^{2,2}(B)\times {\boldsymbol{L}}^2(B)\times {\boldsymbol{L}}^2(B),\end{equation*}

where ${\boldsymbol{W}}_0^{2,2}(B)=[W_0^{2,2}(B)]^2$ and ${\boldsymbol{L}}^2(B)=[L^2(B)]^2$ , with $W_0^{2,2}(B)$ and $L^2(B)$ being the usual Sobolev spaces [Reference Adams and Fournier1] (i.e. the boldface is used to define vectors). An element in our space can be determined by $(u_1,u_2,v_1,v_2,\theta _1,\theta _2)$ . Moreover, the vector ${\boldsymbol{v}}=(v_1,v_2)$ is considered for the time derivative of the displacement vector.

If we consider the vector $({\boldsymbol{u}},{\boldsymbol{v}},{\boldsymbol{\theta }})$ , we can define the square of the norm:

\begin{align*} \|({\boldsymbol{u}},{\boldsymbol{v}},{\boldsymbol{\theta }})\|^2&=\frac{1}{2}\int _B \Big (\rho v_i\overline{v_i}+A_{ijrs}u_{i,j}\overline{u_{r,s}}+ B_{ijpqr}(u_{i,j}\overline{u_{r,pq}}+\overline{u_{i,j}}u_{r,pq})\\[5pt] &\quad +C_{ijlpqr}u_{l,ij}\overline{u_{r,pq}}+d_{ij}\theta _i\overline{\theta _j}\Big )\,dA, \end{align*}

where the bar means the conjugated complex. We consider the inner product associated to this norm. In view of the previous assumptions, our inner product defines a norm which is equivalent to the usual product in the Hilbert space $\mathcal{H}$ defined previously.

We can define the following operator:

\begin{align*} \mathcal{A}\left (\begin{array}{c}{\boldsymbol{u}}\\[5pt] {\boldsymbol{v}}\\[5pt] {\boldsymbol{\theta }} \end{array}\right )= \left (\begin{array}{c} v_i\\[5pt] (A_{ijrs}u_{r,s}+B_{ijpqr}u_{r,pq})_{,j}-(B_{rsijk}u_{r,s}+C_{kjipqr}u_{r,pq})_{,jk} +(a_{ij}^l\theta _l)_{,j}-(C_{ijk}^l\theta _l)_{,jk}\\[5pt] e_{kl}[(k_{ij}^{lm}\theta _{m,j})_{,i}+a_{ij}^lv_{i,j}+C_{ijk}^lv_{i,jk}] \end{array}\right ), \end{align*}

where $e_{kl}$ is the inverse of the matrix $d_{lj}$ .

We remark that the domain of the operator $\mathcal{A}$ is the subspace made by elements $({\boldsymbol{u}},{\boldsymbol{v}},{\boldsymbol{\theta }})$ such that ${\boldsymbol{v}}\in{\boldsymbol{W}}_0^{2,2}(B)$ , $(k_{ij}^{lm}\theta _{,j})_{,i}\in{\boldsymbol{L}}^2(B)$ and $(A_{ijrs}u_{r,s}+B_{ijpqr}u_{r,pq})_{,j}-(B_{rsijk}u_{r,s}+C_{kjipqr}u_{r,pq})_{,jk} +(a_{ij}^l\theta _l)_{,j}-(C_{ijk}^l\theta _l)_{,jk}\in{\boldsymbol{L}}^2(B)$ . Obviously, it is a dense subspace in $\mathcal{H}$ .

It is relevant noting that we can write our problem in the form:

(2.7) \begin{equation} \frac{dU}{dt}=\mathcal{A}U,\quad U(0)=({\boldsymbol{u}}^0,{\boldsymbol{v}}^0,{\boldsymbol{\theta }}^0). \end{equation}

This is the Cauchy problem which is defined in the space $\mathcal{H}$ that we will study in the next sections.

3. Well-posedness

In this section, we will prove that the operator $\mathcal{A}$ generates a contractive semigroup of linear operators. Therefore, the existence, uniqueness and the continuous dependence with respect to the initial conditions will be concluded.Footnote ²

Lemma 1. For every $U=({\boldsymbol{u}},{\boldsymbol{v}},{\boldsymbol{\theta }})\in \mathrm{Dom}(\mathcal{A})$ , the following inequality

\begin{equation*}Re\langle \mathcal {A}U,U\rangle \leqslant 0\end{equation*}

holds.

Proof. If we apply the definition of the operator and we use boundary conditions (2.3), after the use of the divergence theorem we obtain that

\begin{equation*}Re\langle \mathcal {A}U,U\rangle =-\frac {1}{2}\int _B k_{ij}^{lm}\theta _{m,j}\overline {\theta _{l,i}}\, dA.\end{equation*}

In view of the assumptions (iii)–(iv), we see that the lemma is proved.

Proof. Let us consider $({\boldsymbol{f}}_1,{\boldsymbol{f}}_2,{\boldsymbol{f}}_3)\in \mathcal{H}$ . We need to solve the system:

\begin{align*} \mathcal{A}\left (\begin{array}{c}{\boldsymbol{u}}\\[5pt] {\boldsymbol{v}}\\[5pt] {\boldsymbol{\theta }} \end{array}\right )= \left (\begin{array}{c}{\boldsymbol{f}}_1\\[5pt] {\boldsymbol{f}}_2\\[5pt] {\boldsymbol{f}}_3 \end{array}\right ). \end{align*}

That is, we find that

(3.1) \begin{equation} \begin{array}{l} v_i=f_{1i},\\[5pt] (A_{ijrs}u_{r,s}+B_{ijpqr}u_{r,pq})_{,j} -(B_{rsijk}u_{r,s}+C_{kjipqr}u_{r,pq})_{,jk}\\[5pt] \qquad +\;(a_{ij}^l\theta _{,l})_{,j}-(C_{ijk}^l\theta _l)_{,jk}=f_{2i},\\[5pt] (k_{ij}^{lm}\theta _{m,j})_{,i}+a_{ij}^lv_{i,j}+C_{ijk}^lv_{i,jk}=e_{kl}f_{3k}. \end{array} \end{equation}

We can obtain $\boldsymbol{v}$ from the first equation. Then, we find that

\begin{equation*}(k_{ij}^{lm}\theta _{m,j})_{,i}=e_{kl}f_{3k}-a_{ij}^l\,f_{1i,j}-C_{ijk}^l\,f_{1i,jk}.\end{equation*}

It is clear that we can solve this system assuming null Dirichlet boundary conditions. In fact, the solution satisfies the conditions for the third component of the domain of the operator. If we substitute $\boldsymbol{\theta }$ into the second component of the system (3.1), we can get the solution after the use of the Lax-Milgram lemma (see [Reference Ieşan13]). Indeed, we can prove that

\begin{equation*}\|({\boldsymbol{u}},{\boldsymbol{v}},{\boldsymbol {\theta }})\|\leqslant K \|({\boldsymbol{f}}_1,{\boldsymbol{f}}_2,{\boldsymbol{f}}_3)\|,\end{equation*}

where $K$ is independent of the solution.

Since the domain of the operator is dense, we can apply the Lumer-Phillips corollary to the Hille-Yosida theorem to prove the following.

In view of this theorem, we can conclude that our problem is well-posed in the sense of Hadamard.

4. Exponential stability

The aim of this section is to provide a sufficient condition on the coupling terms to guarantee the exponential stability of the solutions to our problem.

We start by considering the operator

\begin{equation*}P_l({\boldsymbol{u}})=a_{ij}^lu_{i,j}+C_{ijk}^lu_{i,jk}\quad \hbox {for}\quad l=1,2.\end{equation*}

From now on, we will assume that

1. (v) The operator ${\boldsymbol{P}}=(P_1,P_2)$ is elliptic in $B$ .

It is relevant to note that, in view of the boundary conditions (2.3), we can conclude that (see [Reference Liu and Zheng17, p. 14])

(4.1) \begin{equation} \|{\boldsymbol{u}}\|_{{\boldsymbol{W}}^{2,2}(B)}\leqslant C\|{\boldsymbol{P}}({\boldsymbol{u}})\|_{{\boldsymbol{L}}^2(B)}, \end{equation}

where $C$ is a positive constant which is independent of $\boldsymbol{u}$ .

With the proposed assumption, and thanks to estimate (4.1), we can obtain the following result.

Theorem 4.1. Let us assume that the operator $\boldsymbol{P}$ satisfies condition (v). Then, there exist two positive constants $M$ and $\omega$ such that

\begin{equation*}\|U(t)\|\leqslant M e^{-\omega t}\|U(0)\|\end{equation*}

for every $U(0)$ in the domain of the operator $\mathcal{A}$ .

The proof of this theorem is based in the following lemmas.

Proof. In order to prove this lemma, we start by following the arguments used in the book of Liu and Zheng [Reference Liu and Zheng17, p. 25]. There, the proof uses a contradiction procedure. It is assumed that the thesis of the lemma does not hold and then, in view of a series of comments, and that zero belongs to the resolvent of $\mathcal{A}$ , we can find (see, again, [Reference Liu and Zheng17, p. 25]) a sequence of real numbers $\beta _n\in \mathbb{R}$ , with $\beta _n\rightarrow \beta \neq 0$ , and a sequence of elements $U_n=({\boldsymbol{u}}_n,{\boldsymbol{v}}_n,{\boldsymbol{\theta }}_n)$ in the domain of the operator, with unit norm, such that

(4.2) \begin{equation} \|(i\beta _n I-\mathcal{A})U_n\|_{\mathcal{H}}\rightarrow 0\quad \hbox{as}\quad n\rightarrow \infty. \end{equation}

This condition is equivalent to assume that

(4.3) \begin{align} i\beta u_i-v_i\rightarrow 0 \quad \hbox{in}\quad W^{2,2}_0(B), \end{align}

(4.4) \begin{align} &i\beta v_i-\Big ((A_{ijrs}u_{r,s}+B_{ijpqr}u_{r,pq})_{,j}-(B_{rsijk}u_{r,s}+C_{kjipqr}u_{r,pq})_{,jk}\nonumber \\[5pt] &\quad +(a_{ij}^l\theta _{,l})_{,j}-(C_{ijk}^l\theta _l)_{,jk}\Big ) \rightarrow 0 \quad \hbox{in}\quad L^2(B), \end{align}

(4.5) \begin{align} i\beta d_{lk}\theta _k-\Big ((k_{ij}^{lm}\theta _{m,j})_{,i}+a_{ij}^lv_{i,j}+C_{ijk}^lv_{i,jk}\Big ) \rightarrow 0 \quad \hbox{in}\quad L^2(B). \end{align}

We note that we have omitted the sub-index $n$ in convergences (4.3)–(4.5) for the sake of simplicity in the notation.

Now, from the dissipation inequality and the assumption on the thermal variables, we have that

\begin{equation*}{\boldsymbol {\theta }}\rightarrow {\textbf{0}} \quad \hbox {in}\quad {\boldsymbol{W}}^{1,2}(B).\end{equation*}

If we multiply convergence (4.5) by $\displaystyle \frac{1}{\beta }{\boldsymbol{P}}({\boldsymbol{u}})$ , we find that

\begin{equation*}\frac {1}{\beta } \langle (k_{ij}^{lm}\theta _{m,j})_{,i}, P_l({\boldsymbol{u}})\rangle +i\|{\boldsymbol{P}}({\boldsymbol{u}})\|^2\rightarrow 0,\end{equation*}

and we can apply the divergence theorem to obtain that

(4.6) \begin{align} \frac{1}{\beta } \langle (k_{ij}^{lm}\theta _{m,j})_{,i}, P_l({\boldsymbol{u}})\rangle & = \frac{1}{\beta }\int _{\partial B} (k_{ij}^{lm}\theta _{m,j}) \overline{P_l({\boldsymbol{u}})}n_i\, dl \nonumber \\[5pt] & +\frac{1}{\beta }\int _B k_{ij}^{lm}\theta _{m,j} (P_l({\boldsymbol{u}}))_{,i}\,dA. \end{align}

where $n_i$ is the normal vector to $\partial B$ .

We note that $\frac{1}{\beta }({\boldsymbol{P}}({\boldsymbol{u}}))_{,i}$ is bounded in view of convergences (4.4) and (4.5) and the fact that the operator defining the elastic part is elliptic. Therefore, the last integral in (4.6) vanishes since ${\boldsymbol{\theta }}\rightarrow{\textbf{0}}$ in ${\boldsymbol{W}}^{1,2}(B)$ . To show that the first integral tends to zero, we note that we can apply the following estimates

\begin{equation*}\frac {1}{\beta }\int _{\partial B} \theta _{i,j}\overline {u_{k,lm}}\, dl\leqslant \frac {1}{\beta }\left (\int _{\partial B}\theta _{i,j}\overline {\theta _{i,j}}\,dl\right )^{1/2} \left (\int _{\partial B}u_{k,lm}\overline {u_{k,lm}}\,dl\right )^{1/2}.\end{equation*}

But we also have

\begin{equation*}\frac {1}{\beta ^{1/2}}\left (\int _{\partial B}\theta _{i,j}\overline {\theta _{i,j}}\,dl\right )^{1/2} \leqslant C_1 \left (\int _{B}\nabla \theta _{i}\overline {\nabla \theta _{i}}\,dA\right )^{1/4} \frac {1}{\beta ^{1/2}}\left (\int _{B}\Delta \theta _{i}\overline {\Delta \theta _{i}}\,dA\right )^{1/4}\end{equation*}

since $\theta _i$ vanishes at the boundary. At the same time, we find that

\begin{equation*}\frac {1}{\beta ^{1/2}} \left (\int _{\partial B}u_{k,lm}\overline {u_{k,lm}}\,dl\right )^{1/2}\leqslant C_2 \left (\int _B \Delta u_k \overline {\Delta u_k}\, dA\right )^{1/4} \frac {1}{\beta ^{1/2}} \left (\int _B \Delta ^2 u_k\overline {\Delta ^2 u_k}\, dA\right )^{1/4}\end{equation*}

whenever $u_i$ and $\nabla u_i$ vanish at the boundary. The parameters $C_1$ and $C_2$ are independent of the functions.

But we note that $\displaystyle \beta ^{-1}\Big (\int _B\Delta \theta _i\overline{\Delta \theta _i}\, dA\Big )^{1/2}$ and $\displaystyle \beta ^{-1}\Big (\int _B\Delta ^2 u_k \overline{\Delta ^2 u_k}\, dA\Big )^{1/2}$ are boundedFootnote ³ as well as $\displaystyle \int _B \Delta u_k\Delta u_k \, dA$ . Since

\begin{equation*}\int _B \nabla \theta _{i}\overline {\nabla \theta _{i}}\, dA\rightarrow 0,\end{equation*}

we can conclude that the first integral in (4.6) also tends to zero, and therefore, it follows that $\|{\boldsymbol{P}}({\boldsymbol{u}})\|\rightarrow 0$ . But, as we can see the estimate (4.1) we also conclude that ${\boldsymbol{u}}\rightarrow{\textbf{0}}$ in ${\boldsymbol{W}}^{2,2}(B)$ . Now, an easy argument also shows that ${\boldsymbol{v}}\rightarrow{\textbf{0}}$ in ${\boldsymbol{L}}^2(B)$ and we arrive to a contradiction. It comes from the fact that we assume that there would exist an element of the imaginary axis at the spectrum. Therefore, the lemma is proved.

Lemma 4. The asymptotic condition

\begin{equation*}\overline {\lim _{|\beta |\rightarrow \infty }} \|(i\beta I-\mathcal {A})^{-1}\| \lt \infty \end{equation*}

holds.

We have proved the exponential decay result for the solutions to our problem. It is worth saying that an easy example (for a given operator $\boldsymbol{P}$ to be elliptic) corresponds to assume

\begin{equation*}C_{1,ij}^1 u_{1,ij}=\Delta u_1,\quad C_{2,ij}^2u_{2,ij}=\Delta u_2,\end{equation*}

meanwhile the remaining coefficients $\beta _{ij}^l$ and $C_{ijk}^l$ vanish.

5. Analyticity of the semigroup

The aim of this section is to prove the analyticity of the solutions to problem (2.7) whenever the conditions (i)–(v) hold. As a consequence, we will obtain the uniqueness of the backward in time problem, and so, it will be proved that the unique solution vanishing for every time, after a certain value, is the null solution.

Proof. Since we have shown that the imaginary axis is contained at the resolvent of the operator, the theorem will be concluded if we see that

(5.1) \begin{equation} \overline{\lim _{|\beta |\rightarrow \infty }}\|\beta ^{-1}(i\beta I-\mathcal{A})^{-1}\|\lt \infty. \end{equation}

We will prove condition (5.1) by contradiction. Let us assume the existence of a sequence $\beta _n\rightarrow \infty$ and a sequence of unit norm vectors $U_n$ (at the domain of the operator) such that

(5.2) \begin{equation} i U_n-\beta _n^{-1}\mathcal{A}U_n\rightarrow 0\quad \hbox{in}\quad \mathcal{H}, \end{equation}

and we will conclude that $U_n\rightarrow 0$ , which is a contradiction. We note that condition (5.2) is equivalent to the convergences:

(5.3) \begin{align} i u_i-\beta ^{-1}v_i \rightarrow 0 \quad \hbox{in}\quad W^{2,2}(B), \end{align}

(5.4) \begin{align} & i v_i-\beta ^{-1}\Big ((A_{ijrs}u_{r,s}+B_{ijpqr}u_{r,pq})_{,j}-(B_{rsijk}u_{r,s}+C_{kjipqr}u_{r,pq})_{,jk}\nonumber \\[5pt] &\qquad +(a_{ij}^l\theta _{,l})_{,j}-(C_{ijk}^l\theta _l)_{,jk}\Big ) \rightarrow 0 \quad \hbox{in}\quad L^{2}(B), \end{align}

(5.5) \begin{align} i d_{lk}\theta _k-\beta ^{-1}\Big ((k_{ij}^{lm}\theta _{m,j})_{,i}+a_{ij}^lv_{i,j}+C_{ijk}^lv_{i,jk}\Big ) \rightarrow 0 \quad \hbox{in}\quad L^{2}(B), \end{align}

where we omit again the sub-index ‘ $n$ ’ to simplify the notation.

If we multiply convergence (5.2) by $U_n$ , we obtain that

(5.6) \begin{equation} \beta ^{-1/2}{\boldsymbol{\theta }} \rightarrow{\textbf{0}}\quad \hbox{in}\quad{\boldsymbol{W}}^{1,2}(B). \end{equation}

Gagliardo-Niremberg’s inequality implies that

\begin{equation*}\|\beta ^{-1/2}{\boldsymbol{v}}\|_{{\boldsymbol{W}}^{1,2}(B)}^2\leqslant C \beta ^{-1}\|{\boldsymbol{v}}\|_{{\boldsymbol{W}}^{2,2}(B)}\|{\boldsymbol{v}}\|_{{\boldsymbol{L}}^2(B)}.\end{equation*}

As $\beta ^{-1}\|{\boldsymbol{v}}\|_{{\boldsymbol{W}}^{2,2}(B)}$ and $\|{\boldsymbol{v}}\|_{{\boldsymbol{L}}^2(B)}$ are bounded, we see that $\|\beta ^{-1/2}{\boldsymbol{v}}\|_{{\boldsymbol{W}}^{1,2}(B)}$ is bounded.

If we multiply convergence (5.5) by $\boldsymbol{\theta }$ , we get

\begin{equation*}i\langle d_{kl}\theta _k,\theta _l\rangle +\langle \beta ^{-1/2} C_{ijk}^lv_{i,j},\beta ^{-1/2}\theta _{l,k}\rangle \rightarrow 0.\end{equation*}

In view of convergence (5.6) and that $\|\beta ^{-1/2}{\boldsymbol{v}}\|_{{\boldsymbol{W}}^{1/2}(B)}$ is bounded, we conclude that

\begin{equation*}{\boldsymbol {\theta }}\rightarrow {\textbf{0}} \quad \hbox {in}\quad {\boldsymbol{L}}^2(B).\end{equation*}

It follows that

\begin{equation*}i{\boldsymbol{P}}({\boldsymbol{u}})+\beta ^{-1}(k_{ij}^{lm}\theta _{m,j})_{,i}\rightarrow 0 \quad \hbox {in} \quad {\boldsymbol{L}}^2(B).\end{equation*}

Since $\boldsymbol{u}$ is bounded in ${\boldsymbol{W}}^{2,2}(B)$ and, in view of condition (iii), we see that

\begin{equation*}\|\beta ^{-1} {\boldsymbol {\theta }}\|_{{\boldsymbol{W}}^{2,2}(B)}\lt \infty .\end{equation*}

From convergence (5.4) and condition (iv) we have

\begin{equation*}\|\beta ^{-1} {\boldsymbol{u}}\|_{{\boldsymbol{W}}^{4,2}(B)}\lt \infty .\end{equation*}

We use again the Gagliardo-Niremberg’s inequality to see

\begin{equation*}\|\beta ^{-1/2}{\boldsymbol{u}}\|^2_{{\boldsymbol{W}}^{3,2}(B)}\leqslant C\beta ^{-1}\|{\boldsymbol{u}}\|_{{\boldsymbol{W}}^{4,2}(B)}\|{\boldsymbol{u}}\|_{{\boldsymbol{W}}^{2,2}(B)}.\end{equation*}

In the next step, we multiply convergence (5.5) by ${\boldsymbol{P}}({\boldsymbol{u}})$ to find that

(5.7) \begin{equation} i\|{\boldsymbol{P}}({\boldsymbol{u}})\|^2-\langle \beta ^{-1/2}k_{ij}^{lm}\theta _{m,j},\beta ^{-1/2}(P_l({\boldsymbol{u}}))_{,i}\rangle +\beta ^{-1} \int _{\partial B}k_{ij}^{lm}\theta _{m,j}\overline{P_l({\boldsymbol{u}})}n_i\, dl\rightarrow 0. \end{equation}

Now, we want to see that this last integral tends to zero. We note that

\begin{equation*}\beta ^{-1} \int _{\partial B}k_{ij}^{lm}\theta _{m,j}\overline {P_l({\boldsymbol{u}})}n_i\, dl \leqslant C\beta ^{-3/4}\|\nabla {\boldsymbol {\theta }}\|_{{\boldsymbol{L}}^2(\partial B)}\beta ^{-1/4}\|\Delta {\boldsymbol{u}}\|_{{\boldsymbol{L}}^2(\partial B)}.\end{equation*}

But we have

(5.8) \begin{align} \displaystyle \beta ^{-3/4}\|\nabla{\boldsymbol{\theta }}\|_{{\boldsymbol{L}}^2(\partial B)} \leqslant C\left ( \beta ^{-1}\|{\boldsymbol{\theta }}\|_{{\boldsymbol{W}}^{2,2}(B)}\right )^{1/2} \left ( \beta ^{-1/2}\|{\boldsymbol{\theta }}\|_{{\boldsymbol{W}}^{1,2}(B)}\right )^{1/2}, \end{align}

(5.9) \begin{align} \beta ^{-1/4}\|\Delta{\boldsymbol{u}}\|_{{\boldsymbol{L}}^2(\partial B)} \leqslant C\left ( \beta ^{-1/2}\|\Delta{\boldsymbol{u}}\|_{{\boldsymbol{W}}^{1,2}(B)}\right )^{1/2} \|\Delta{\boldsymbol{u}}\|_{{\boldsymbol{L}}^2(B)}^{1/2}. \end{align}

We note that the right hand-side of (5.9) is bounded and the right hand-side of (5.8) tends to zero. Then, we obtain that the last integral of convergence (5.7) converges to zero. We also note that the inner product in (5.8) tends to zero thanks to convergence (5.6) and the fact that $\beta ^{-1}\|{\boldsymbol{u}}\|_{{\boldsymbol{W}}^{4,2}(B)}$ is bounded. Thus, we see that

\begin{equation*}{\boldsymbol{u}}\rightarrow {\textbf{0}} \quad \hbox {in}\quad {\boldsymbol{W}}^{2,2}(B).\end{equation*}

If we multiply convergence (5.4) by $\boldsymbol{v}$ , we also obtain

\begin{equation*}{\boldsymbol{v}}\rightarrow {\textbf{0}}\quad \hbox {in}\quad {\boldsymbol{L}}^2(B).\end{equation*}

We have arrived to a contradiction because we assumed that $\|U_n\|=1$ , and so, the theorem is proved.

In view of the fact that the semigroup generated by operator $\mathcal{A}$ is analytic, we can conclude the following.

6. Hyperbolic case

In this section, we sketch how to extend the exponential stability of the solutions to the case of the Green-Lindsay dissipation mechanisms (see [Reference Green and Lindsay10] for further details regarding this theory).

In this case, the system of equations is (see [Reference Ieşan and Quintanilla11]):

(6.1) \begin{equation} \begin{array}{l} \rho \ddot{u}_i=\Big (A_{ijrs} u_{r,s}+B_{ijpqr}u_{r,pq}+a_{ij}^l(\theta _l+\alpha \dot{\theta }_l)\Big )_{,j}\\[5pt] \qquad \qquad -\Big ( B_{rsijk}u_{r,s}+C_{kjipqr}u_{r,pq}+C_{ijk}^l(\theta _l+\alpha \dot{\theta }_l)\Big )_{,jk},\\[5pt] m_{lp}\ddot{\theta }_p+d_{lp}\dot{\theta }_p=(k_{ij}^{lm}\theta _{m,j})_{,i}+a_{ij}^l\dot{{u}}_{i,j}+C_{ijk}^l\dot{{u}}_{i,jk} +b^{jl}_i\dot{\theta }_{j,i}+(b_{i}^{jl}\dot{\theta }_j)_{,i}. \end{array} \end{equation}

We note that, in this system, $\rho$ , $u_i$ , $A_{ijrs}$ , $B_{ijpqr}$ , $C_{ijkpqr}$ , $a_{ij}^l$ , $C_{ijk}^l$ , $d_{lp}$ , $k_{ij}^{lm}$ and $\theta _l$ were defined in the previous sections. However, we have now introduced several new elements. Here, $\alpha$ is a positive constant which is considered as a relaxation parameter. We also have a new symmetric matrix $m_{lp}$ which is assumed to satisfy that

\begin{equation*}\alpha d_{ij}-m_{ij}\quad \hbox {and}\quad m_{ij}\end{equation*}

are positive definite matrices. Another new component is the matrix $b_i^{jl}$ which is assumed to satisfy the following symmetry:

\begin{equation*}b_i^{jl}=b_i^{lj}.\end{equation*}

It is also worth to note that we need to modify assumption (iii), which is now replaced by the following one:

1. (iii^*) There exists a positive constant $D$ such that

   \begin{equation*}k_{ij}^{lm}\theta _{l,i}\theta _{m,j}+2 b_i^{jl}\dot {\theta }_j\theta _{l,i}+(d_{ij}\alpha -m_{ij})\dot {\theta }_i\dot {\theta }_j \geqslant D(\theta _{i,j}\theta _{i,j}+\dot {\theta }_i\dot {\theta }_i).\end{equation*}

In this case, we also need to impose the initial conditions (2.2) and

(6.2) \begin{equation} \dot{\theta }_j({\boldsymbol{x}},0)=\vartheta _j^0({\boldsymbol{x}})\quad \hbox{for a.e. }{\boldsymbol{x}}\in B, \end{equation}

and boundary conditions (2.3).

We can consider the problem defined by the system (6.1), the initial conditions (2.2) and (6.2), and the boundary conditions (2.3).

This problem can be written in the abstract form (2.7), where

\begin{equation*}U(0)=({\boldsymbol{u}}^0,{\boldsymbol{v}}^0,{\boldsymbol {\theta }}^0,{\boldsymbol {\vartheta }}^0),\end{equation*}

and

\begin{align*} \mathcal{A}\left (\begin{array}{c}{\boldsymbol{u}}\\[5pt] {\boldsymbol{v}}\\[5pt] {\boldsymbol{\theta }}\\[5pt] {\boldsymbol{\vartheta }} \end{array}\right )= \left (\begin{array}{c} v_i\\[5pt] \Big (A_{ijrs} u_{r,s}+B_{ijpqr}u_{r,pq}+a_{ij}^l(\theta _l+\alpha \dot{\theta }_l)\Big )_{,j}\\[5pt] \qquad \qquad -\Big ( B_{rsijk}u_{r,s}+C_{kjipqr}u_{r,pq}+C_{ijk}^l(\theta _l+\alpha \dot{\theta }_l)\Big )_{,jk}\\[5pt] \vartheta _i\\[5pt] n_{dl}[(k_{ij}^{lm}\theta _{m,j})_{,i}+a_{ij}^lv_{i,j}+C_{ijk}^lv_{i,jk}+b_i^{jl}\vartheta _{j,i}+(b_i^{jl}\vartheta _j)_{,i} -d_{lp}\vartheta _p] \end{array}\right ), \end{align*}

where $n_{dl}$ is the inverse of the matrix $m_{lp}$ .

This problem is defined in the modified Hilbert space:

\begin{equation*}\mathcal {H}={\boldsymbol{W}}^{2,2}_0(B)\times {\boldsymbol{L}}^2(B)\times {\boldsymbol{W}}_0^{1,2}(B)\times {\boldsymbol{L}}^2(B).\end{equation*}

We note that we can consider the inner product with the following square of the norm:

\begin{align*} \begin{array}{lll} \|({\boldsymbol{u}},{\boldsymbol{v}},{\boldsymbol{\theta }},{\boldsymbol{\vartheta }})\|^2&=&\dfrac{1}{2}\int _B \Big (\rho v_i\overline{v_i}+A_{ijrs}u_{i,j}\overline{u_{r,s}}+ B_{ijpqr}(u_{i,j}\overline{u_{r,pq}}+\overline{u_{i,j}}u_{r,pq})\\[9pt] && +C_{ijkpqr}u_{k,ij}\overline{u_{r,pq}}+\alpha ^{-1}m_{lp}(\theta _l+\alpha \vartheta _l)\overline{(\theta _p +\alpha \vartheta _p)}\\[5pt] && +(d_{lp}-\alpha ^{-1}m_{lp})\theta _l\overline{\theta _p}+ \alpha k_{ij}^{lm}\theta _{l,i}\overline{\theta _{m,j}}\Big )\,dA. \end{array} \end{align*}

This norm is equivalent to the usual one in the Hilbert space $\mathcal{H}$ .

We also note that the domain of the operator $\mathcal{A}$ is dense and it can be obtained as the elements of the Hilbert space $\mathcal{H}$ such that

\begin{align*} \begin{array}{l}{\boldsymbol{v}}\in{\boldsymbol{W}}_0^{2,2}(B),\quad{\boldsymbol{\vartheta }}\in{\boldsymbol{W}}_0^{1,2}(B),\\[5pt] (B_{ijpqr}u_{r,pq})_{,j}+(B_{rsijk}u_{r,s}+C_{ijkpqr}u_{r,pq}+C_{ijk}^l(\theta _l+\alpha \vartheta _l))_{,jk}\in{\boldsymbol{L}}^2(B),\\[5pt] (k_{ij}^{lm}\theta _{m,j})_{,i}\in{\boldsymbol{L}}^2(B). \end{array} \end{align*}

We note that, if $U=({\boldsymbol{u}},{\boldsymbol{v}},{\boldsymbol{\theta }},{\boldsymbol{\vartheta }})$ belongs to the domain of the operator $\mathcal{A}$ , we have

\begin{equation*}Re\langle \mathcal {A}U,U\rangle =-\frac {1}{2}\int _B \Big ( k_{ij}^{lm}\theta _{l,i}\overline {\theta _{m,j}} +b_i^{jl}(\vartheta _j\overline {\theta _{l,i}}+\overline {\vartheta _j}\theta _{l,i}) +(d_{ij}\alpha -m_{ij})\vartheta _i\overline {\vartheta _j}\Big )\, dA,\end{equation*}

which is less or equal than zero thanks to condition (iii $^*$ ).

We can also follow a similar argument to the one used previously to show that zero belongs to the resolvent of the operator. Therefore, we can obtain the same result provided in Theorem 3.1 in our case.

We could also prove that Theorem 4.1 also holds in this hyperbolic case. Following the same strategy used in the proof of Lemma 3, we arrive to a sequence $U_n=({\boldsymbol{u}}_n,{\boldsymbol{v}}_n,{\boldsymbol{\theta }}_n,{\boldsymbol{\vartheta }}_n)$ of unit norm vectors in the domain of the operator and a sequence $\beta _n\rightarrow \beta$ (finite or infinite but different from zero) such that condition (4.2) holds.

This is equivalent to write

(6.3) \begin{align} i\beta u_i-v_i\rightarrow 0 \quad \hbox{in}\quad W^{2,2}_0(B), \end{align}

(6.4) \begin{align} & i\beta v_i-\Big (A_{ijrs} u_{r,s}+B_{ijpqr}u_{r,pq}+a_{ij}^l(\theta _l+\alpha \dot{\theta }_l)\Big )_{,j}\nonumber \\[5pt] &\qquad -\Big ( B_{rsijk}u_{r,s}+C_{kjipqr}u_{r,pq}+C_{ijk}^l(\theta _l+\alpha \dot{\theta }_l)\Big )_{,jk} \rightarrow 0 \quad \hbox{in}\quad L^2(B), \end{align}

(6.5) \begin{align} i\beta \theta _i-\vartheta _i\rightarrow 0 \quad \hbox{in}\quad W^{1,2}_0(B), \end{align}

(6.6) \begin{align} & i\beta d_{lk}\theta _k-\Big ((k_{ij}^{lm}\theta _{m,j})_{,i}+a_{ij}^lv_{i,j}+C_{ijk}^lv_{i,jk} +b^{jl}_i\vartheta _{j,i}+(b_{i}^{jl}\vartheta _j)_{,i}\nonumber \\[5pt] &\qquad -m_{lk}\vartheta _k\Big ) \rightarrow 0 \quad \hbox{in}\quad L^2(B). \end{align}

Again, the dissipation inequality implies that

\begin{equation*}{\boldsymbol {\theta }}\rightarrow {\textbf{0}} \quad \hbox {in} \quad {\boldsymbol{W}}_0^{1,2}(B),\quad {\boldsymbol {\vartheta }}\rightarrow {\textbf{0}} \quad \hbox {in} \quad {\boldsymbol{L}}^{2}(B).\end{equation*}

The remaining part of the proof follows the same points as in the proof of Lemma 3. Therefore, we have obtained the following.

Theorem 6.1. The problem proposed by system (6.1) with the initial conditions (2.2) and (6.2), and the boundary conditions (2.3) admits a unique solution. Furthermore, if $U(0)=({\boldsymbol{u}}^0,{\boldsymbol{v}}^0,{\boldsymbol{\theta }}^0,{\boldsymbol{\vartheta }}^0)$ belongs to the domain of the operator $\mathcal{A}$ , then there exist two positive constants $M$ and $\omega$ such that

\begin{equation*}\|U(t)\|\leqslant Me^{-\omega t}\|U(0)\|.\end{equation*}

7. Further comments

In this paper, we have proved the analyticity and the exponential decay of the solutions to the two-dimensional strain gradient thermoelasticity, with two dissipative mechanisms, in the case that we assume that the coupling terms define an elliptic operator and the dissipation is of the type of Fourier. Exponential decay has been also obtained when the dissipative structure is defined by the Green and Lindsay theory. However, it is suitable to say that the extension of these arguments to the three-dimensional case is obvious and we have restricted to the two-dimensional one for the sake of simplicity in the notation. It is important to remark that, in order to guarantee that the coupling terms define an elliptic operator, it is needed that we consider chiral materials since the tensor $C_{ijk}^l$ cannot vanish. Therefore, in the case of isotropic materials our analysis cannot be applied. It is also remarkable that, in the case of the strain gradient theory, we need a number of dissipative mechanisms smaller than in the case of the usual elasticity. This is because the coupling in this case is stronger.

One could ask if the arguments and results presented in the previous sections can be applied to other thermoelastic theories. We want to express that we have tried to extend them to the Lord-Shulmann theory but we have not been able to obtain the exponential decay in this case. In view of the mathematical similarity of our problem with the plate problem, and taking into account that we cannot prove the exponential decay for the Lord-Shulmann plate [Reference Quintanilla and Racke22] but it is possible to obtain the exponential decay for the Green-Lindsay plate [Reference Quintanilla, Racke and Ueda23], one suspects that, in the general case, we cannot expect the exponential decay for the Lord-Shulmann theory of strain gradient elasticity. However, this is an open question that we hope to address in the near future.