Dual-phase-lag one-dimensional thermo-porous-elasticity with microtemperatures

Abstract

This paper is devoted to studying the linear system of partial differential equations modelling a one-dimensional thermo-porous-elastic problem with microtemperatures in the context of the dual-phase-lag heat conduction. Existence, uniqueness, and exponential decay of solutions are proved. Polynomial stability is also obtained in the case that the relaxation parameters satisfy a certain equality. Our arguments are based on the theory of semigroups of linear operators.

1 Introduction

At the beginning of the last century, Cosserat brothers (Cosserat and Cosserat 1909) proposed the study of the micropolar elastic solids; that is a type of material such that the material points can rotate and, therefore, microstructure is taken into account. In the second part of the last century, there were extensive study about elastic materials with microstructure based on the axioms of thermomechanics. In the work of Goodman and Cowin (1972), the authors proposed the foundations of a continuum theory for granular materials with interstitial voids. Their basic idea consists in writing the bulk density as the product of the density matrix by the volume fraction. Based on this point of view, (Cowin 1985; Cowin and Nunziato 1983; Nunziato and Cowin 1979) set down the theory of elastic solids with voids. The intention was to model the deformations of solids with pores or small voids distributed within them. Thermal effects were also included (Ieşan 1986, 2004; Ieşan and Nappa 2004). It is worth recalling that this theory has received a lots of attention in the past years (Feng and Apalara 2019; Feng and Yin 2019; Leseduarte et al. 2010; Magaña and Quintanilla 2006, 2007; Miranville and Quintanilla 2019, 2020; Pamplona et al. 2011; Santos et al. 2017) to understand the relevance of the microstructural component in the whole material. In fact, the microstructure has deserved much attention and one of its possible components is the microtemperature (Bazarra et al. 2019; Casas and Quintanilla 2005; Feng et al. 2020; Grot 1969; Ieşan 2007; Ieşan and Quintanilla 2000; Magaña and Quintanilla 2018; Passarella et al. 2017; Riha 1975, 1976).

Heat conduction is usually based on the Fourier law. In this case, the heat flux vector is expressed as a linear form of the gradient of temperature. However, this assumption brings us to a paradox, because the thermal waves propagates instantaneously and, therefore, the causality principle is violated. It has been natural to look for an alternative law for the heat flux vector. Cattaneo and Maxwell proposed the introduction of a relaxation parameter which brings to a hyperbolic damped equation. Other authors, such as Green and Naghdi, have proposed alternative laws. In this paper, we recover the one proposed by Tzou (1995). The introduction of two delay parameters is considered, and the Cattaneo and Maxwell law can be seen as a particular case. It is worth recalling that the first contribution concerning the decay of solutions for the dual-phase-lag thermoelasticity was presented by Quintanilla and Racke (2006). Many results dealing this thermoelatic theory have been obtained later. Recently, Liu et al. (2020) proposed the study of the dual-phase-lag heat conduction with microtemperatures and gave sufficient conditions to guarantee the stability of the problem (Borgmeyer et al. 2014; Liu et al. 2017; Liu and Quintanilla 2018; Magaña and Quintanilla 2018).

In this short note, we want to focus our attention to porous-thermo-elastic materials with microtemperatures in the context of the dual-phase-lag theory. That is to consider the elastic materials with voids where the heat conduction and the microheat conduction are determined by the dual-phase-lag theory. In this sense, the present paper develops three main objectives: the first one is to propose the one-dimensional thermo-porous-elasticity with microtemperatures in the context of the dual-phase-lag heat conduction. The second one is to provide of a family of conditions on the parameters of the system guaranteeing the well-posedness of the problem in a suitable Hilbert space. The third one is to prove an exponential stability result for the solutions. We also obtain the polynomial stability of the solutions when the relaxation parameters satisfy a certain relation (limit case).

2 Basic equations

This section is devoted to set down the basic equations for the one-dimensional problem of the thermo-porous-elasticity with microtemperatures for isotropic and homogeneous materials in the context of the dual-phase-lag theory. As we consider the case of a rod, we could think that it occupies the interval of length $\pi$. In this case, the evolution equations are

$$\begin{aligned}&\rho \ddot{u}=t_x, \quad J \ddot{\phi }=h_x+ g ,\\&\rho T_0 {\dot{\eta }}=q_x, \quad \rho {\dot{\epsilon }}=P_x+q-Q, \end{aligned}$$

and the constitutive equations¹

$$\begin{aligned}&t=\mu u_x+\mu _0 \phi -\beta _0 \theta ,\quad h=a_0\phi _x-\mu _2 T,\\&g=-\mu _0u_x-\xi \phi +\beta _1 \theta ,\quad \rho \eta =\beta _0 u_x+\beta _1 \phi +a \theta ,\\&\rho \epsilon =-\mu _2 \phi _x-b T\\&q+\tau _1 {\dot{q}} +\frac{\tau _1^2}{2}\ddot{q} =(k\theta _x+k_1T)+\tau _2 (k{\dot{\theta }}_x+k_1{\dot{T}}),\\&P+\tau _1 {\dot{P}} +\frac{\tau _1^2}{2}{\ddot{P}} =-k_4(S_x+\tau _2 {\dot{S}}_x),\\&Q+\tau _1 {\dot{Q}} +\frac{\tau _1^2}{2}{\ddot{Q}} =(k-k_1)\theta _x+(k_1-k_2)T+\tau _2 ((k-k_1)\dot{\theta }_x+(k_1-k_2){\dot{T}}). \end{aligned}$$

In the above system of equations $\rho$ is the mass density, u is the displacement, t is the stress, h is the equilibrated stress, g is the equilibrated body force, $\eta$ is the entropy, q is the heat flux, J is the equilibrated inertia, $T_0$ is the reference temperature at the equilibrium state (that we will assume equal to one), $\epsilon$ is the first moment of the energy, Q is the microheat flux average, P is the first heat flux moment, $\phi$ is the volume fraction, $\theta$ is the temperature, T is the microtemperature, and $\tau _1$ and $\tau _2$ are the relaxation parameters. The constitutive parameters, $\mu , \mu _0, \beta _0,\beta _1,a_0, \mu _2,\xi ,k$ and $k_i$ define the characteristics of the material, and in particular, they define the couplings².

We will assume that

$$\begin{aligned}&\mu>0,\quad \mu \xi>\mu _0^2,\quad a_0>0, \quad k>0, \quad k_4>0, \quad \rho >0, \end{aligned}$$

(1)

$$\begin{aligned}&J>0, \quad a>0, \quad b>0,\quad k k_2>k_1^2,\quad 2\tau _2\ge \tau _1>0. \end{aligned}$$

(2)

These assumptions are natural in the context of the theory. In particular, they imply that the internal energy and the dissipation are positive definite bilinear forms which are related to the elastic stability. We also mention that the last condition on the relaxation parameters implies that the heat conduction is stable and dissipative (Liu et al. 2020) (see also the numerical treatment (Bazarra et al. 2021)). When the relaxation parameters do not satisfy this condition, the instability of solutions holds (Quintanilla 2003).

If we substitute the constitutive equations into the evolution equations, we obtain the following linear system:

$$\begin{aligned}&\rho \ddot{u}=\mu u_{xx}+\mu _0 \phi _x-\beta _0\theta _x,\\&J \ddot{\phi }=a_0 \phi _{xx}-\mu _0 u_x-\mu _2 T_x +\beta _1\theta -\xi \phi ,\\&a({\dot{\theta }}+\tau _1 \ddot{\theta }+ \frac{\tau _1^2}{2} \dddot{\theta })=-\beta _0 ({\dot{u}}_x+\tau _1 \ddot{u}_{x}+ \frac{\tau _1^2}{2} \dddot{u}_x)\\&\qquad - \beta _1 ({\dot{\phi }}+\tau _1 \ddot{\phi }+ \frac{\tau _1^2}{2} \dddot{\phi })+k(\theta _{xx}+\tau _2 {\dot{\theta }}_{xx})+k_1 (T_x+\tau _2 {\dot{T}}_x),\\&b ({\dot{T}}+\tau _1 {\ddot{T}}+ \frac{\tau _1^2}{2} \dddot{T})=-\mu _2 ({\dot{\phi }}_x+\tau _1 \ddot{\phi }_x+ \frac{\tau _1^2}{2} \dddot{\phi }_x)+k_4(T_{xx}+\tau _2 {\dot{T}}_{xx}) \\&\qquad -k_2(T+\tau _2 {\dot{T}})-k_1(\theta _{x}+\tau _2 {\dot{\theta }}_{x} ). \end{aligned}$$

If we denote by ${\hat{f}}=f+\tau _1 {\dot{f}}+\frac{\tau _1^2}{2} \ddot{f}$, we can write our system as

$$\begin{aligned}&\rho \ddot{{\hat{u}}}=\mu {\hat{u}}_{xx}+\mu _0 \hat{\phi }_x-\beta _0(\theta _x+\tau _1 {\dot{\theta }}_x+\frac{\tau _1^2}{2} \ddot{\theta }_x),\\&J \ddot{{\hat{\phi }}}=a_0 {\hat{\phi }}_{xx}-\mu _0 {\hat{u}}_x-\mu _2 (T_x+\tau _1 {\dot{T}}_x+\frac{\tau _1^2}{2} {\ddot{T}}_x) +\beta _1(\theta +\tau _1 {\dot{\theta }} +\frac{\tau _1^2}{2} \ddot{\theta }) -\xi {\hat{\phi }},\\&a({\dot{\theta }}+\tau _1 \ddot{\theta }+ \frac{\tau _1^2}{2} \dddot{\theta })=-\beta _0 \dot{{\hat{u}}}_x- \beta _1 \dot{\hat{\phi }}+k(\theta _{xx}+\tau _2 {\dot{\theta }}_{xx})+k_1 (T_x+\tau _2 {\dot{T}}_x),\\&b ({\dot{T}}+\tau _1 {\ddot{T}}+ \frac{\tau _1^2}{2} \dddot{T})=-\mu _2 \dot{{\hat{\phi }}}_x+k_4(T_{xx}+\tau _2 {\dot{T}}_{xx}) -k_2(T+\tau _2 {\dot{T}})-k_1(\theta _{x}+\tau _2 {\dot{\theta }}_{x} ). \end{aligned}$$

From now on, we will omit the hats on the mechanical variables to simplify the notation.

To propose the well-posed problem, we will need to impose the initial conditions

$$\begin{aligned} \begin{array}{l} u(x,0)=u^0(x),\quad {\dot{u}}(x,0)=v^0(x),\quad \phi (x,0)=\phi ^0(x),\quad {\dot{\phi }}(x,0)=\varphi ^0(x),\\ \theta (x,0)=\theta ^0(x),\quad T(x,0)=T^0(x),\quad {\dot{\theta }}(x,0)=\vartheta (x),\quad {\dot{T}}(x,0)=S^0(x)\\ \ddot{\theta }(x,0)=\zeta ^0(x),\quad {\ddot{T}}(x,0)=R^0(x), \end{array} \end{aligned}$$

where $u^0$, $v^0$, $\phi ^0$, $\varphi ^0$, $\theta ^0$, $\vartheta ^0$, $\zeta ^0$, $T^0$, $S^0$, and $R^0$ are given functions.

Since we assume homogeneous Dirichlet boundary conditions, it follows that:

$$\begin{aligned} u(x,t)=\phi (x,t)= \theta (x,t)=T(x,t)=0, \quad t\in [0,\infty ), \quad x=0,\pi . \end{aligned}$$

In this situation, the energy of the system is given by

$$\begin{aligned} E(t)= & {} \frac{1}{2} \int _0^{\pi } (\rho \vert {\dot{u}}\vert ^2+J \vert \dot{\phi }\vert ^2 +\mu \vert u_x \vert ^2+2 \mu _0u_x \phi +\xi \vert \phi \vert ^2+ a \vert \phi _x\vert ^2) dx \\&+\frac{1}{2} \int _0^{\pi } (a \vert {\hat{\theta }}\vert ^2+ b \vert {\hat{T}}\vert ^2+k(\tau _1+\tau _2)\vert \theta _x \vert ^2+ \frac{k\tau _1^2 \tau _2}{2}\vert {\dot{\theta }}_x\vert ^2+k_2 (\tau _1+\tau _2)\vert T \vert ^2 \\&+\frac{k_2\tau _1^2 \tau _2}{2}\vert {\dot{T}}\vert ^2+ k \tau _1^2 \theta _x {\dot{\theta }}_x) dx \\&+\frac{1}{2} \int _0^{\pi } (k_2 T{\dot{T}} +2 (\tau _1+\tau _2)k_1 \theta _x T +k_1 \tau _1^2(\theta _x {\dot{T}}+{\dot{\theta }}_xT)\\&+k_1 \tau _1^2\tau _2{\dot{\theta }}_x {\dot{T}}+k_4(\tau _1+\tau _2) \vert T_x \vert ^2 \\&+\frac{\tau _1^2}{2} k_4\vert {\dot{T}}_x \vert ^2+\tau _1^2 k_4 T_x {\dot{T}}_x) dx. \end{aligned}$$

The dissipation is given by

$$\begin{aligned} D(t)= & {} \int _0^{\pi } \left[ k\vert \theta _{x}\vert ^2+ k(\tau _1 \tau _2 -\frac{\tau _1^2}{2})\vert {\dot{\theta }}_{x}\vert ^2 +k_2 \vert T\vert ^2+k_2(\tau _1 \tau _2 -\frac{\tau _1^2}{2}) \vert \dot{T}\vert ^2\right. \\&\left. +2k_1 \theta _x T+2k_1 (\tau _1 \tau _2 -\frac{\tau _1^2}{2}){\dot{\theta }}_x{\dot{T}}\right] dx \\&+\int _0^{\pi } \left[ k_4\vert T_x \vert ^2+ (\tau _1 \tau _2 -\frac{\tau _1^2}{2})\vert {\dot{T}}_x \vert ^2\right] dx. \end{aligned}$$

As we said before, under the assumptions we imposed previously, the energy and the dissipation are positive definite.

We have

$$\begin{aligned} E(t)+\int _0^t D(\xi ) d\xi =E(0). \end{aligned}$$

Therefore, we can expect the stability of the solutions of the problem. In fact, we will show the exponential stability (or polynomial).

3 Existence and uniqueness

This section is devoted to show the well-posedness of the problem proposed previously. Therefore, our intention is to transform our problem into a Cauchy problem in a suitable Hilbert space.

We will propose the problem in the Hilbert space

$$\begin{aligned} {{\mathcal {H}}}=W_0^{1,2} \times L^2 \times W_0^{1,2} \times L^2 \times W_0^{1,2} \times W_0^{1,2} \times L^2 \times W_0^{1,2} \times W_0^{1,2} \times L^2. \end{aligned}$$

As usual, $W_0^{1,2}$ and $L^2$ are the usual Hilbert spaces. It is worth noting that now we consider that the elements take values in the complex field.

An element in this space will be denoted by $U=(u, v, \phi , \varphi , \theta , \vartheta , \zeta , T, S, R)$.

Defining an operator ${\mathcal {A}}: {\mathcal {H}} \rightarrow {\mathcal {H}}$ by

$$\begin{aligned} {\mathcal {A}}U = \left\{ \begin{array}{l} v \\ \frac{1}{\rho }[\mu u_x + \mu _0 \phi - \beta _0( \theta + \tau _1 \vartheta + \frac{\tau _1^2}{2}\zeta )]_x \\ \varphi \\ \frac{1}{J} [(a_0\phi _x - \mu _2 (T_x +\tau _1 S_x +\frac{\tau _1^2}{2}R_x))_x - \mu _0 u_x - \xi \phi + \beta _1(\theta +\tau _1 \vartheta +\frac{\tau _1^2}{2}\zeta ) ] \\ \vartheta \\ \zeta \\ \frac{2}{a\tau _1^2} [-\beta _0 v_x- \beta _1 {\varphi }+k(\theta _{x}+\tau _2\vartheta _{x})_x +k_1 (T_x+ \tau _2 S_x)-a\vartheta -a\tau _1 \zeta ]\\ S \\ R \\ \frac{2}{b\tau _1^2}[-\mu _2 \varphi _x+k_4(T_{x}+\tau _2S_{x})_x-k_2(T+\tau _2S)-k_1 (\theta +\tau _2 \vartheta _x)-aS-a\tau _1 R] \end{array} \right. \end{aligned}$$

(3)

with the domain

$$\begin{aligned} {\mathcal {D}}({\mathcal {A}})=\{ U \in {\mathcal {H}} \;|\; v,\varphi , \zeta ,R\in W_0^{1,2},\quad u_{xx},\;\phi _{xx},\; \theta _{xx}+\tau _2 \vartheta _{xx},\; T_{xx}+\tau _2S_{xx}\in L^2 \}. \end{aligned}$$

We then can write our problem as

$$\begin{aligned} \frac{dU}{dt}={{\mathcal {A}}}U,\, \, \, \, U(0)=U^0, \end{aligned}$$

(4)

where $U^0=(u^0, v^0, \phi ^0, \varphi ^0, \theta ^0, \vartheta ^0,\zeta ^0,T^0,S^0,R^0)$.

The main aim of this section is to prove that the operator ${{\mathcal {A}}}$ generates a $C_0$ semigroup of contractions on $\mathcal {H}$.

Given $U=(u,v,\phi ,\varphi ,\theta ,\vartheta ,\zeta , T,S,R)$ and $U^*=(u^*,v^*,\phi ^*,\varphi ^*,\theta ^*,\vartheta ^*,\zeta ^*, T^*,S^*, R^*)$, we consider the inner product defined as

$$\begin{aligned}&\langle U,U^*\rangle _{\mathcal {H}}=\frac{1}{2} \int _0^{\pi }(\rho v \bar{v}^*+J\varphi {\bar{\varphi }}^*+\mu u_x {\bar{u}}_x^*+\mu _0(u_x \bar{\phi }^*+{\bar{u}}_x^* \phi )+\xi \phi {\bar{\phi }}^*\\&\quad +a_0\phi _x \bar{\phi }_x^*+c\theta \bar{{\hat{\theta }}}^*+bT\bar{{\hat{T}}}^*) dx \\&\quad +\frac{1}{2} \int _0^{\pi }(k(\tau _1+\tau _2)\theta _x\bar{\theta }_x^*+\frac{k\tau _1^2 \tau _2}{2}\vartheta _x\bar{\vartheta }_x^*+k_2(\tau _1+\tau _2) T{\bar{T}}^*+\frac{k_2\tau _1^2 \tau _2}{2}T{\bar{T}}^*\\&\quad +\frac{k\tau _1^2}{2}(\theta _x{\bar{\vartheta }}_x^*+ {\bar{\theta }}_x^* \vartheta _x ) )dx \\&\quad +\frac{1}{2} \int _0^{\pi }(\frac{k_2}{2}(T{\bar{S}}^*+{{\bar{T}}}^*S) +k_1(\tau _1+\tau _2) (\theta _x {\bar{T}}^*+{\bar{\theta }}_x^* T)+\frac{k_1 \tau _1^2\tau _2}{2} (\vartheta _x {\bar{S}}^*+\bar{\vartheta }_x^* S) ) dx \\&\quad +\frac{1}{2} \int _0^{\pi }( \frac{k_1 \tau _1^2}{2}(\theta _x \bar{S}^*+{\bar{\theta }}_x^* S) +k_4(\tau _1+\tau _2) T_x \bar{T}^*_x+\frac{k_4 \tau _1^2}{2} S_x {\bar{S}}^*_x \\&\quad +\frac{k_4 \tau _1^2}{2} (T_x {\bar{S}}^*_x+{\bar{T}}^*_x S_x) )dx \\&\quad +\frac{1}{2} \int _0^{\pi } \frac{k_1 \tau _1^2}{2}(\vartheta _x \bar{T}^*+{\bar{\vartheta }}_x^* T) dx. \end{aligned}$$

Here, and from now on, the bar denotes the conjugated complex. It is clear that this inner product is equivalent to the usual one in the Hilbert space ${{\mathcal {H}}}$ (see Liu et al. 2020).

Theorem 1

Assume that the conditions (1)–(2) hold. Then, operator ${{\mathcal {A}}}$ generates a $C_0$ semigroup of contractions.

Proof

It is clear that ${\mathcal {D}}({\mathcal {A}})$ is dense in ${\mathcal {H}}$. We can easily show that

$$\begin{aligned}&{\hbox {Re}}\langle {{\mathcal {A}}}U,U\rangle _{{\mathcal {H}}}=-\frac{1}{2} \int _0^{\pi } [k\vert \theta _{x}\vert ^2+ k(\tau _1 \tau _2 -\frac{\tau _1^2}{2})\vert {\vartheta }_{x}\vert ^2 +k_2 \vert T\vert ^2\\&\quad +k_2(\tau _1 \tau _2 -\frac{\tau _1^2}{2}) \vert S\vert ^2+2k_1{\hbox {Re}}\theta _x {\bar{T}}] dx \\&\quad -\frac{1}{2} \int _0^{\pi } [2k_1 (\tau _1 \tau _2 -\frac{\tau _1^2}{2}){\hbox {Re}} {\vartheta }_x{\bar{S}}+k_4 \vert T_x \vert ^2+ k_4 (\tau _1 \tau _2 -\frac{\tau _1^2}{2})\vert {S}_x \vert ^2]dx. \end{aligned}$$

In view of the assumptions we have imposed on the constitutive coefficients, we see that this is less or equal to zero (see Liu et al. 2020).

Now, we prove that zero belongs to the resolvent of the operator. Given $(f_1,f_2,f_3,f_4,f_5, f_6,f_7,f_8,f_9,f_{10})\in {\mathcal {H}}$, we must show that the system

$$\begin{aligned}&v=f_1,\quad \varphi =f_3,\quad \vartheta =f_5,\quad \zeta =f_6, \quad S=f_8,\quad R=f_9 \\&\mu u_{xx}+\mu _0 \phi _x-\beta _0(\theta _x +\tau _1 \vartheta _x+\frac{\tau _1^2}{2}\zeta _x)=\rho f_2,\\&a_0 \phi _{xx}-\mu _0 u_x- \mu _2 (T_x +\tau _1 S_x +\frac{\tau _1^2}{2}R_x)+\beta _1 (\theta +\tau _1 \vartheta +\frac{\tau _1^2}{2}\zeta )-\xi \phi =Jf_4,\\&\quad -\beta _0 v_x- \beta _1 {\varphi }+k(\theta _{xx}+\tau _2\vartheta _{xx}) +k_1 (T_x+ \tau _2 S_x)-a\vartheta -a\tau _1 \zeta =\frac{a \tau _1^2}{2} f_5,\\&\quad -\mu _2 \varphi _x+k_4(T_{xx}+\tau _2S_{xx})-k_2(T+\tau _2S)-k_1 (\theta +\tau _2 \vartheta _x)-aS-a\tau _1 R=\frac{b \tau _1^2}{2}f_6, \end{aligned}$$

has a solution in the domain of operator ${\mathcal {A}}$. The solution for v, $\varphi$, $\vartheta , \zeta , S$, and R is directly obtained. Therefore, we obtain the system of equations

$$\begin{aligned}&\mu u_{xx}+\mu _0 \phi _x-\beta _0\theta _x =F_1,\\&a_0 \phi _{xx}-\mu _0 u_x- \mu _2 T_x +\tau _1 S_x +\beta _1 \theta -\xi \phi =F_2,\\&k\theta _{xx} +k_1 T_x =F_3\\&k_4T_{xx} -k_2T -k_1 \theta _x =F_4. \end{aligned}$$

Here, $F_i$, $i=1,\cdots ,4$ can be obtained in terms of the $f_i,\, i=1...10$. What it is relevant is that $F_i$ belongs to $W^{-1,2}$ for every $i=1...4$. If we look to the last two equations, we can define the bilinear form

$$\begin{aligned} {{\mathcal {B}}} [(\theta _1,T_1), (\theta _2,T_2)]=\int _0^{\pi }\left( (k\theta _{1,xx}+k_1 T_{1,x})\bar{\theta }_2+ (k_4T_{1,xx} -k_2T_1 -k_1 \theta _{1,x} )\bar{T}_2\right) dx. \end{aligned}$$

It is bounded and coercive in $W^{1,2}_0\times W^{1,2}_0$. By the Lax–Milgram lemma, we see the existence of $\theta$ and T satisfying the last two equations. Moreover, they belong to $W^{1,2}_0$. Thus, we can substitute them into the first two equations and use again the Lax–Milgram lemma to prove the existence of u and $\phi$ also in $W^{1,2}_0$.

In view of the Lumer–Phillips corollary to the Hille–Yosida theorem, we find that our operator generates a $C_0$ semigroup of contractions on $\mathcal {H}$. $\square$

Thus, we have proved the following result.

Theorem 2

Assume that conditions (1)–(2) hold, then, for every $U^0 \in {{\mathcal {D}}({\mathcal {A}})}$, there exists a unique solution to problem (4).

We note that, since the operator generates a contractive semigroup, the problem is stable and well-posed in the sense of Hadamard. Furthermore, in case that we impose supply terms with suitable regularity conditions, the solutions will depend continuously on the supply terms.

4 Exponential decay: case of $2\tau _2>\tau _1$

The aim of this section is to prove that the solutions of our problem decay in an exponential way to the equilibrium solution whenever we assume that $\beta _0\ne 0$ and $\mu _2\ne 0$. To prove this result, we will use the characterization of the exponentially stable semigroups that we can find for instance in the book of Liu and Zheng (1999). In this sense, we recall that whenever the imaginary axis is contained in the resolvent of the generator ${{\mathcal {A}}}$ of the semigroup and the condition

$$\begin{aligned} {\overline{\lim }}_{\!\!\!\!\!\!\!\!\!\!\!\!\!{}_{{}_{{\lambda \in R}, |\lambda |\rightarrow \infty }}} \Vert (i\lambda {{\mathcal {I}}}-{\mathcal A})^{-1}\Vert _{{\mathcal {L}}({\mathcal {H}})} < \infty , \end{aligned}$$

(5)

holds, the semigroup is exponentially stable.

Then, we prove the following result which states the exponential decay of the energy system.

Theorem 3

Assume that (1)–(2) hold when $2\tau _2>\tau _1$, and that $\beta _0, \mu _2\ne 0$, then the semigroup generated by the operator ${\mathcal {A}}$ is exponentially stable. That is, there exist two positive constants which are independent of the initial data $N,\eta$, such that

$$\begin{aligned} \vert \vert U(t) \vert \vert _{{\mathcal {H}}} \le N \exp (-\eta t) \vert \vert U(0) \vert \vert _{{\mathcal {H}}} \end{aligned}$$

for every $U(0)\in {{\mathcal {H}}}$.

Proof

We shall employ the arguments used in the book of (Liu and Zheng (1999), page 25). In the case that the intersection of the imaginary axis and the spectrum is non-empty, then there will exist of real numbers $\lambda _n$ with $\lambda _n \rightarrow \varpi , \, \, \vert \lambda _n \vert < \vert \varpi \vert$ and a sequence of vectors ${U}_n=(u_n, v_n,\phi _n,\varphi _n,\theta _n,\vartheta _n,\zeta _n,T_n,S_n,R_n)$ in ${\mathcal {D(A)}}$, and with unit norm, such that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert ( i \lambda _n {{\mathcal {I}}}- {{\mathcal {A}}} ) {U}_n\Vert =0 . \end{aligned}$$

(6)

This can be written as

$$\begin{aligned}&i\lambda _n u_{n}-v_{ n}\rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(7)

$$\begin{aligned}&i\rho \lambda _n v_{n}-(\mu u_{n,xx}+\mu _0 \phi _{n,x}-\beta _0 (\theta _{n,x}+\tau _1\vartheta _{n,x}+\frac{\tau _1^2}{2}\zeta _{n,x})) \rightarrow 0 \hbox { in } L^2, \end{aligned}$$

(8)

$$\begin{aligned}&i\lambda _n \phi _{n}-\varphi _{n}\rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(9)

$$\begin{aligned}&i\rho J \lambda _n \varphi _{n}-(a_0 \phi _{n,xx}-\mu _0 u_{n,x}- \mu _2 (T_{n,x}+\tau _1 S_{n,x}+\frac{\tau _1^2}{2} R_{n,x} )\nonumber \\&\quad +\beta _1(\theta _n+\tau _1\vartheta _{n}+\frac{\tau _1^2}{2}\zeta _n) -\xi \phi _n) \rightarrow 0 \hbox { in }L^2, \end{aligned}$$

(10)

$$\begin{aligned}&i\lambda _n\theta _n-\vartheta _n \rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(11)

$$\begin{aligned}&i\lambda _n\vartheta _n-\zeta _n \rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(12)

$$\begin{aligned}&\frac{ia\tau _1^2\lambda _n}{2}\zeta _n+\beta _0 v_{n,x}+\beta _1\varphi _n -k(\theta _{n,xx}+\tau _2\vartheta _{n,xx}) \nonumber \\&\quad -k_1 (T_{n,x}+ \tau _2 S_{n,x})+a\vartheta _n +a\tau _1 \zeta _n\rightarrow 0 \hbox { in }L^2, \end{aligned}$$

(13)

$$\begin{aligned}&i\lambda _nT_n-S_n \rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(14)

$$\begin{aligned}&i\lambda _nS_n-R_n \rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(15)

$$\begin{aligned}&\frac{ib\tau _1^2\lambda _n}{2} R_n+ \mu _2\varphi _{n,x} - k_4(T_{n,xx}+\tau _2S_{n,xx})+k_2(T_n+\tau _2S_n) \nonumber \\&\quad +k_1 (\theta _{n,x}+\tau _2 \vartheta _{n,x})+aS_n+a\tau _1 R_n \rightarrow 0 \hbox { in } L^2. \end{aligned}$$

(16)

From (6) and the dissipation inequality and the assumptions on the coefficients, we see that $\theta _{n,x},\vartheta _{n,x}, T_{n,x}$ and $S_{n,x}$ tend to zero in $L^2$. Taking the $L^2$ inner product of (13) with $\vartheta _n$ and (16) with $S_n$, respectively, we conclude that $\zeta _n$ and $R_n$ also tend to zero in $L^2$.

Next, in view of (7), we take the $L^2$ inner product of (13) with $\lambda _n^{-1}u_{n,x}$ to obtain that

$$\begin{aligned} i\beta _0 \Vert u_{n,x}\Vert ^2+k \lambda _n^{-1}\langle \theta _{n,x}+\tau _2\vartheta _{n,x},u_{n,xx}\rangle + \lambda _n^{-1} k\left( \theta _{n,x}+\tau _2\vartheta _{n, x})\bar{u}_{n,x}\vert _0^{\pi } \right) \rightarrow 0.\end{aligned}$$

Hereafter, we use the notation $\Vert \cdot \Vert$ and $\langle \cdot ,\;\cdot \rangle$ to denote the $L^2$ norm and inner product, respectively. Since $\Vert \lambda _n^{-1} u_{n,xx}\Vert$ is bounded (because (13)), we see that $k \lambda _n^{-1} \langle \theta _{n,x}+\tau _2\vartheta _{n,x},u_{n,xx}\rangle$ tends to zero. On the other hand, in view of the Gagliardo–Nirenberg inequality, we know that

$$\begin{aligned}&\vert \lambda _n\vert ^{-1/2}(\theta _{n,x}+\tau _2\vartheta _{n,x} )\Vert _{L^{\infty }} \le K_1 \vert \lambda _n\vert ^{-1/2}\Vert \theta _{n,x}+\tau _2\vartheta _{n,x} \Vert ^{1/2} \Vert \theta _{n,xx}+\tau _2\vartheta _{n,xx} \Vert ^{1/2} \\&\quad +K_2 \vert \lambda _n\vert ^{-1/2}\Vert \theta _{n,x}+\tau _2\vartheta _{n,x} \Vert \rightarrow 0. \end{aligned}$$

In a similar way

$$\begin{aligned} \vert \lambda _n\vert ^{-1/2}u_{n,x}\Vert _{L^{\infty }} \le K_1 \vert \lambda _n\vert ^{-1/2}\Vert u_{n,x} \Vert ^{1/2} \Vert u_{n,xx} \Vert ^{1/2}+K_2 \vert \lambda _n\vert ^{-1/2}\Vert u_{n,x} \Vert \rightarrow 0. \end{aligned}$$

We then conclude that

$$\begin{aligned} \lambda _n^{-1} k(\theta _{n,x}+\tau _2\vartheta _{n, x})\bar{u}_{n,x}\vert _0^{\pi } \rightarrow 0. \end{aligned}$$

Therefore, we obtain that $\Vert u_{n,x}\Vert \rightarrow 0$, and therefore, from (8), we also conclude that $\Vert v_n\Vert$ tends to zero.

A similar argument to the one used to prove that $\Vert u_{n,x}\Vert$ and $\Vert v_n\Vert$ tend to zero can be used to prove that $\Vert \phi _{n,x}\Vert$ and $\Vert \varphi _n\Vert$ also tend to zero. The only key point is that we now work with (16) in place of (13) and to take $L^2$ inner product with $\lambda _n^{-1}\phi _{n,x}$ in place of $\lambda _n^{-1}u_{n,x}$. However, this contradicts the assumption that $\Vert U_n\Vert _{\mathcal {H}}=1$. We conclude that the intersection of the imaginary axis with the spectrum of the operator ${\mathcal {A}}$ is void.

To conclude the proof of the theorem, we only need to show that condition (5) also holds. Again, we shall use the contradiction argument. It follows that there exist a sequence of real numbers $\lambda _n$, such that $\vert \lambda _n \vert \rightarrow \infty$ and a sequence of unitary vectors ${U}_n$ in the domain of the operator satisfying (7)– (16). We repeat the arguments used to show that the imaginary axis is contained in the resolvent of the operator to arrive the $\Vert U_n\Vert \rightarrow 0$ contradiction. We point out that the only difference is that now $\lambda _n \rightarrow \infty$. $\square$

5 Polynomial decay: case of $2\tau _2=\tau _1$

In the case that $2\tau _2=\tau _1$, we cannot expect that the decay of the solutions is controlled by a exponential function. In fact, we could adapt to this situation the result obtained in this sense in the case that we do not consider the microtemperatures (see Liu and Quintanilla 2018) . In this section, we prove that the solutions of our problem decay in a polynomial way to the equilibrium solution whenever we assume that $\beta _0\ne 0$ and $\mu _2\ne 0$. Our result will be a consequence of the characterization proposed by Borichev and Tomilov (2010). In this sense, we recall that whenever the imaginary axis is contained in the resolvent of the generator ${{\mathcal {A}}}$ of the semigroup and the condition

$$\begin{aligned} {\overline{\lim }}_{\!\!\!\!\!\!\!\!\!\!\!\!\!{}_{{}_{{\lambda \in R, |\lambda |\rightarrow \infty }}}} \lambda ^{-2}\Vert (i\lambda {\mathcal I}-{{\mathcal {A}}})^{-1}\Vert _{{\mathcal {L}}({\mathcal {H}})} < \infty , \end{aligned}$$

(17)

holds; the semigroup is polynomially stable. Furthermore, the estimate

$$\begin{aligned} \vert \vert U(t) \vert \vert _{{\mathcal {H}}} \le N t^{-1/2}\vert \vert U(0) \vert \vert _{ {\mathcal {D(A)}}} \end{aligned}$$

can be obtained for every $U(0)\in {\mathcal {D(A)}}$.

To prove that the imaginary axis is contained in the resolvent, we can follow the same way of the previous section. The dissipation inequality implies that $\theta _{n,x}$ and $T_{n,x}$ tend to zero in $L^2$. In the case that $\lambda _n$ is bounded, we also see that $\vartheta _{n,x}$ and $S_{n,x}$ also tend to zero in $L^2$. At this point, we can follow point-by-point the arguments used in the previous section to conclude that the imaginary axis is contained in the resolvent.

Now, we want to prove that the asymptotic condition (17) also holds. Assuming that this is not true, we can find a sequence of vectors

$$\begin{aligned} {U}_n=(u_n, v_n,\phi _n,\varphi _n,\theta _n,\vartheta _n,\zeta _n,T_n,S_n,R_n) \end{aligned}$$

in ${\mathcal {D(A)}}$, and with unit norm, such that

$$\begin{aligned} {\overline{\lim }}_{\!\!\!\!\!\!\!\!\!\!\!\!{}_{{}_{{\lambda _n\in R, |\lambda _n|\rightarrow \infty }}}} \lambda _n^2\Vert (i\lambda _n{\mathcal I}-{{\mathcal {A}}})U_n\Vert _{{\mathcal {H}}} = 0, \end{aligned}$$

(18)

that is

$$\begin{aligned}&\lambda _n^2(i\lambda _n u_{n}-v_{ n})\rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(19)

$$\begin{aligned}&\lambda _n^2(i\rho \lambda _n v_{n}-(\mu u_{n,xx}+\mu _0 \phi _{n,x}-\beta _0 (\theta _{n,x}+\tau _1\vartheta _{n,x}+\frac{\tau _1^2}{2}\zeta _{n,x}) ))\rightarrow 0 \hbox { in } L^2, \end{aligned}$$

(20)

$$\begin{aligned}&\lambda _n^2(i\lambda _n \phi _{n}-\varphi _{n})\rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(21)

$$\begin{aligned}&\lambda _n^2(i\rho J \lambda _n \varphi _{n}-(a_0 \phi _{n,xx}-\mu _0 u_{n,x}- \mu _2 (T_{n,x}+\tau _1 S_{n,x}+\frac{\tau _1^2}{2} R_{n,x} )\nonumber \\&\quad +\beta _1(\theta _n+\tau _1\vartheta _{n}+\frac{\tau _1^2}{2}\zeta _n) -\xi \phi _n)) \rightarrow 0 \hbox { in }L^2, \end{aligned}$$

(22)

$$\begin{aligned}&\lambda _n^2(i\lambda _n\theta _n-\vartheta _n )\rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(23)

$$\begin{aligned}&\lambda _n^2(i\lambda _n\vartheta _n-\zeta _n) \rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(24)

$$\begin{aligned}&\lambda _n^2(\frac{ia\tau _1^2\lambda _n}{2}\zeta _n+\beta _0 v_{n,x}+\beta _1\varphi _n -k(\theta _{n,xx}+\tau _2\vartheta _{n,xx}) \nonumber \\&\quad -k_1 (T_{n,x}+ \tau _2 S_{n,x})+a\vartheta _n +a\tau _1 \zeta _n)\rightarrow 0 \hbox { in }L^2, \end{aligned}$$

(25)

$$\begin{aligned}&\lambda _n^2(i\lambda _nT_n-S_n) \rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(26)

$$\begin{aligned}&\lambda _n^2(i\lambda _nS_n-R_n )\rightarrow 0 \hbox { in } W^{1,2}, \end{aligned}$$

(27)

$$\begin{aligned}&\lambda _n^2(\frac{ib\tau _1^2\lambda _n}{2} R_n+ \mu _2\varphi _{n,x} - k_4(T_{n,xx}+\tau _2S_{n,xx})+k_2(T_n+\tau _2S_n) \nonumber \\&\quad +k_1 (\theta _{n,x}+\tau _2 \vartheta _{n,x})+aS_n+a\tau _1 R_n) \rightarrow 0 \hbox { in } L^2. \end{aligned}$$

(28)

By (18) and dissipation inequality, we obtain that $\lambda _n\theta _{n,x}$ and $\lambda _nT_{n,x}$ tend to zero (also in $L^2$). We can obtain from (23) and (26) that $\vartheta _{n,x},S_{n,x}$ also tend to zero in $L^2$. From this point, we can follow the same argument to the one used in the previous section.

We have proved

Theorem 4

Assume that (1)–(2) hold when $2\tau _2=\tau _1$, and that $\beta _0, \mu _2\ne 0$, then the semigroup generated by the operator ${\mathcal {A}}$ is polynomially stable. That is, there exists a positive constant N,  such that

$$\begin{aligned} \vert \vert U(t) \vert \vert \le N t^{-1/2}\vert \vert U(0) \vert \vert _{ {\mathcal {D(A)}}} \end{aligned}$$

for every $U(0)\in {\mathcal {D(A)}}$.

6 Some comments

In this paper, we have proved several results concerning the decay of the solutions for the porous elasticity when we add dual-phase-lag temperature and microtemperature. To be precise, we have proved that whenever $2\tau _2>\tau _1$, the exponential decay of the solutions holds, and when $2\tau _2=\tau _1$, the polynomial decay of order 2 of the solution holds. Both results have been obtained under the assumption that $\beta _0$ and $\mu _2$ are different from zero. In the case when one of this parameters vanishes, we cannot expect the exponential decay and a further study would be needed to clarify the behaviour.

From the previous results, one suspects that the phenomena of the second spectrum (see Ramos et al. 2020) in the case of the porous elasticity can be eliminated when the dual-phase-lag thermal effects proposed in the case that $2\tau _2>\tau _1$. However, this topic could be the aim of another paper.

In a recent paper, Ramos et al. showed that the solutions of a truncated version of the system of porous elasticity (see Ramos et al. 2020) decay in an exponential way when the porous dissipation is considered. We believe that the inclusion of the thermal effects (also dual-phase-lag) as the only dissipation mechanism brings the system to a similar behaviour.