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

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.

and the constitutive equations 1 In the above system of equations ρ is the mass density, u is the displacement, t is the stress, h is the equilibrated stress, g is the equilibrated body force, η 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), is the first moment of the energy, Q is the microheat flux average, P is the first heat flux moment, φ is the volume fraction, θ is the temperature, T is the microtemperature, and τ 1 and τ 2 are the relaxation parameters. The constitutive parameters, μ, μ 0 , β 0 , β 1 , a 0 , μ 2 , ξ, k and k i define the characteristics of the material, and in particular, they define the couplings 2 . We will assume that 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: 1 A general formulation of the constitutive equations would allow that the relaxation parameters for the macroscopic structure could be different from the ones corresponding to the microstructure. However, as our contribution try to be a pioneering work in the study concerning dual-phase-lag at the microstructure, we consider the easier case from the mathematical aspect and it corresponds to the case when they agree. The general case is much more difficult from a mathematical point of view and the analysis of the general case is not clear even for the rigid solid. 2 We have assumed that the Onsager postulate is satisfied (see Eringen 1999, p.55).
If we denote byf = f + τ 1ḟ + τ 2 1 2f , we can write our system as From now on, we will omit the hats on the mechanical variables to simplify the notation.
Since we assume homogeneous Dirichlet boundary conditions, it follows that: In this situation, the energy of the system is given by The dissipation is given by As we said before, under the assumptions we imposed previously, the energy and the dissipation are positive definite.
We have Therefore, we can expect the stability of the solutions of the problem. In fact, we will show the exponential stability (or polynomial).

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 As usual, W 1,2 0 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.

Defining an operator
We then can write our problem as The main aim of this section is to prove that the operator A generates a C 0 semigroup of contractions on H. Given U = (u, v, φ, ϕ, θ, ϑ, ζ, T , S, R) and U * = (u * , v * , φ * , ϕ * , θ * , ϑ * , ζ * , T * , S * , R * ), we consider the inner product defined as 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 H (see Liu et al. 2020).

Theorem 1 Assume that the conditions (1)-(2) hold. Then, operator
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.
has a solution in the domain of operator A. The solution for v, ϕ, ϑ, ζ, S, and R is directly obtained. Therefore, we obtain the system of equations Here, F i , i = 1, · · · , 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 It is bounded and coercive in W 1,2 0 × W 1,2 0 . By the Lax-Milgram lemma, we see the existence of θ 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 φ 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 H.
Thus, we have proved the following result.

Theorem 2 Assume that conditions (1)-(2) hold, then, for every U 0 ∈ D(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.

Exponential decay: case of 2 2 > 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 β 0 = 0 and μ 2 = 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 A of the semigroup and the condition lim 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 2τ 2 > τ 1 , and that β 0 , μ 2 = 0, then the semigroup generated by the operator A is exponentially stable. That is, there exist two positive constants which are independent of the initial data N , η, such that 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 λ n with λ n → , |λ n | < | | and a sequence of vectors U n = (u n , v n , φ n , ϕ n , θ n , ϑ n , ζ n , T n , S n , R n ) in D(A), and with unit norm, such that lim n→∞ (iλ n I − A)U n = 0.
From (6) and the dissipation inequality and the assumptions on the coefficients, we see that θ n,x , ϑ n,x , T n,x and S n,x tend to zero in L 2 . Taking the L 2 inner product of (13) with ϑ n and (16) with S n , respectively, we conclude that ζ n and R n also tend to zero in L 2 .
A similar argument to the one used to prove that u n,x and v n tend to zero can be used to prove that φ n,x and ϕ n 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 λ −1 n φ n,x in place of λ −1 n u n,x . However, this contradicts the assumption that U n H = 1. We conclude that the intersection of the imaginary axis with the spectrum of the operator 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 λ n , such that |λ n | → ∞ 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 U n → 0 contradiction. We point out that the only difference is that now λ n → ∞.

Polynomial decay: case of 2 2 = 1
In the case that 2τ 2 = τ 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 β 0 = 0 and μ 2 = 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 A of the semigroup and the condition holds; the semigroup is polynomially stable. Furthermore, the estimate can be obtained for every U (0) ∈ 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 θ n,x and T n,x tend to zero in L 2 . In the case that λ n is bounded, we also see that ϑ 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 U n = (u n , v n , φ n , ϕ n , θ n , ϑ n , ζ n , T n , S n , R n ) in D(A), and with unit norm, such that that is λ 2 n (iρλ n v n − (μu n,x x + μ 0 φ n,x − β 0 (θ n,x + τ 1 ϑ n,x + τ 2 1 2 ζ n,x ))) → 0 in L 2 , (20) By (18) and dissipation inequality, we obtain that λ n θ n,x and λ n T n,x tend to zero (also in L 2 ). We can obtain from (23) and (26) that ϑ 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 2τ 2 = τ 1 , and that β 0 , μ 2 = 0, then the semigroup generated by the operator A is polynomially stable. That is, there exists a positive constant N , such that ||U (t)|| ≤ N t −1/2 ||U (0)|| D(A) for every U (0) ∈ D(A).

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 2τ 2 > τ 1 , the exponential decay of the solutions holds, and when 2τ 2 = τ 1 , the polynomial decay of order 2 of the solution holds. Both results have been obtained under the assumption that β 0 and μ 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-phaselag thermal effects proposed in the case that 2τ 2 > τ 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.