Stability of non-classical thermoelasticity mixture problems

We discuss the stability problem for binary mixtures systems coupled with heat equations. The present manuscript covers the non-classical thermoelastic theories of Coleman-Gurtin and Gurtin-Pipkin - both theories overcome the property of infinite propagation speed (Fourier's law property). We first state the well-posedness and our main result is related to long-time behavior. More precisely, we show, under suitable hypotheses on the physical parameters, that the corresponding solution is stabilized to zero with exponential or rational rates.


1.
Introduction. Mathematical models for mixtures of elastic solids/fluids have attracted considerable attention in the past literature. We begin by mentioning the work of Truesdell in [25] that provides the balance/conservation equations for a continuum theory of mixtures -the mixture is considered as a distribution of different continuous media in the same physical space. Adkins in [1] proposed purely mechanical theory for mixtures of fluids and mixture of a fluid and an elastic solid. For an inviscid miscible mixture of fluids without chemical reactions or diffusion, we mention reference [14]. Concerning with immiscible mixtures, we cite the work of Bedford and Drumheller [7]. In connection with the present manuscript, we recall [17] where the model for mixtures of thermoviscoelastic bodies -Kelvin-Voigt material -explicitly appears. For the meaningfulness and applicability of the mixtures problems, Adkins and Craine [2] presented the constitutive theories and some applications for binary mixtures of fluids and a binary mixture of elastic solids. We also recall references [12,18] for applications to engineering and biological problems.
1.1. Problem formulation. In this manuscript, following the theories of Bedford and Stern [8] -also Ieşan and Quintanilla [17] -we consider a portion (0, ) ⊂ R occupied by a mixture of two interacting continua with particles occupying the same position at the time t. Let us consider u = u(x, t) and w = w(y, t) being the In what follows, by ρ 1 and ρ 2 we denote the mass densities of the two constituents (u, w) at time t. M and S are the partial stresses associated with the constituent u and w, respectively, and P is the internal diffusive force. Concerning the dissipative effect, we use the symbols (θ, ϑ, q, T 0 ) to denote the temperature deviation (difference to a fixed constant reference temperature), the entropy density, the heat flux vector and the absolute temperature in the reference configuration, respectively. Under this notation, the general form for the constitutive equations are as follows -motion equations: with a ij , α, β i , ρ 3 (i,j = 1,2) denoting the constitutive coefficients that define the mixture, the parameter σ ∈ [0,1] is responsible for characterization of the heat flux law and the function k is the memory kernel, whose properties are given in Section 2. Under these notations, the constitutive equations (1.3) combined with (1.1) and (1.2), yield the following system of differential partial equations defined in (0, )× (0, ∞) ρ 1 u tt − (a 11 u + a 12 w) xx + α(u − w) − β 1 θ x = 0, (1.4) ρ 2 w tt − (a 12 u + a 22 w) xx − α(u − w) − β 2 θ x = 0, (1.5) (1.6) We also complete the PDE system (1.4)-(1.6) with boundary conditions and initial data (u, u t , w, w t , θ) t=0 = (u 0 , u 1 , w 0 , w 1 , θ 0 ) and θ(−s) s>0 =θ 0 (s) on (0, ). (1.8)

1.2.
Previous results and present contribution. The mathematical studies of mixtures problems, from the point of view of the well-posedness and stability, have received considerable attention. Indeed, starting with [20], the asymptotic behavior of solutions to mixtures of thermoelastic solids was established. The energy decay rate for solutions to viscoelastic mixtures was studied in [24]. More recently, in [6], Alves et al. study the exponential decay rates for solution to thermoelastic mixture with second sound. Concerning existence of analytic solutions to viscoelastic mixtures of solids, we mention [13]. Next, for the sake of brevity, we shall concentrate only on references for thermoelastic mixtures problems in a one-dimensional configuration. As already mentioned, the role of the parameter σ ∈ [0, 1] is to characterize the heat flux law. The case when σ assumes values in (0, 1), the heat propagation law is corresponding with Coleman-Gurtin. Limit cases σ = 0 and σ = 1, the heat flux law coincides with the fully parabolic Fourier law and the fully hyperbolic Gurtin-Pipkin law, respectively. In this direction, we quote the works of Alves et al. [3,4,5] and Muñoz Rivera et al. [21]. Reference [3] studies the problem (1.4)-(1.6) with classical theory of thermoelasticity (σ = 1) and under suitable assumptions on the constitutive coefficients (α, a ij , β i , ρ i ), the authors establish both exponential and polynomial decay rates for the corresponding solution. In [4,5], the authors analyzed the problem (1.4)-(1.6) for Kelvin-Voigt materials with σ = 1. The first establishes decay rates for the solution and the second establishes the analyticity property of the corresponding semigroup. Both results also depend on the constitutive coefficients. In [21], the authors complete the analysis in [3] showing optimal rational decay rates for the problem with a particular boundary condition.
In this article, we complete the analysis concerning to the parameter σ. Here, we assume σ ∈ [0, 1). Following the same direction as the quoted results, we study the effects of (α, a ij , β i , ρ i ) on the decay rates. As an starting point, we begin introducing the stability numbers associated with (1.4)-(1.6). Let κ i (i = 1, 2) be as follows The critical case σ = 0 needs particular conditions. In this case, the stability numbers are defined by with γ 1 = (β 2 a 11 − β 1 a 12 )(ρ 1 β 2 ) −1 .
2. Assumptions and Well-posedness. In this section, we present the basic assumptions and we also post the well-posedness result.
Memory kernel assumptions. We first set the following notation for the memory kernel µ(s) = −(1 − σ)k (s).
Constitutive coefficients assumptions. We assume that α, ρ i (i = 1, 2, 3) are positive constants, β i (i = 1, 2) are real numbers which do not vanish simultaneously and A = [a ij ] 2×2 is a positive definite symmetric matrix. The last, in particular, implies that a 11 a 22 − a 2 12 > 0. Semigroup Formulation. In order to transform (1.4)-(1.6) into an equivalent autonomous problem, we introduce a new variable η ≡ η t (x, s) [11] defined by Taking these into account and assuming (without loss of generality) T 0 = 1, we find the following representation for (1.4)-(1.8) with boundary conditions x ( , s) = 0, (2.7) and initial data (u, u t , w, w t , θ, η t ) t=0 = (u 0 , u 1 , w 0 , w 1 , θ 0 , η 0 ) on (0, ). (2.8) The next step is to state the Cauchy problem associated with (2.3)-(2.8). To this end, we introduce the space of the well-posedness H as follows and M is the µ-weighted Lebesgue space L 2 (0, ) of all functions on (0, ∞) with values in H 1 The space M carries the inner product and corresponding norm given by We also introduce the operator B : where η s stands for the distributional derivative of η with respect to the internal variable s. Following reference [15], one can show that operator B is the infinitesimal generator of a C 0 -semigroup and For Z(t) = (u, v, w, ω, θ, η), we consider H endowed with energy norm where · L 2 and ., . L 2 denote the usual norm and inner product L 2 (0, ) space, respectively. Now, let us consider the variable Z(t) = (u, v, w, ω, θ, η) with v = u t and ω = w t . Then, problem (2.3)-(2.8) can be expressed as a first-order equation We end this section with the well-posedness result.

Remark 2.
In the present manuscript, we are facing a combination of mixtures problems with classical thermoelastic theory and memory in past history. As already mentioned the reference [3] studies the well-posedness of solutions when σ = 1. The well-posedness for problems with memory in the past history framework has been treated in [10]. Therefore, Theorem 2.1 can be proved by adapting and applying essentially the same technique as the cited works. Here, we limit to give a brief sketch.
3. Statement of main results. The first main result is the exponential stability. To state the corresponding result, we recall the stability characterization due to Pruss [23] (See Theorem 1.3.2 in [19]). The last establishes the exponential stability for a C 0 -semigroup of contractions {S(t)} t≥0 , with infinitesimal generator A acting on a Hilbert space H , if and only if where · L (H ) denotes the norm in the space of continuous linear functions in H .
Our main result reads as follows: Theorem 3.1. Let the memory kernel and coefficients assumptions having validity.
In case of lack of exponential stability, we obtain rational decay rates for the corresponding semigroup. Here, we make use of recent results obtained by Borichev and Tomilov (See Theorem 2.4 in [9]). In a short form, the result establishes the following: Let {S(t)} t≥0 a bounded semigroup on a Hilbert space H , with generator A , such that iR ⊂ (A ). Then, for a fixed parameter r > 0 the following inequalities are equivalent and for some positive constant C.

3.1.
Proof of the Main Results. In the following, to further simplify notations, we will omit dependence on "σ" if no confusion arises. In this case, the symbol A will play the role of A σ . We first show the condition (3.1).

MARGARETH S. ALVES AND RODRIGO N. MONTEIRO
In this case the complex numbers iR λ n = i a 11 + a 12 ρ 1 are eigenvalues of operator A with eigenvector Z n = sin nπx , iλ n sin nπx , sin nπx , iλ n sin nπx , 0, 0 .
In case of ρ 2 β 1 = ρ 1 β 2 , we can find, for n ∈ N sufficiently large, which are eigenvalues of operator A with corresponding eigenvector is an eigenvalue of A with eigenvector Z n0 given by where n 0 is a fixed natural number.
As a consequence, we have the following result.
To conclude, let us assume that assumption (ii) is enforced. To this end, we back to (3.22), (3.23) and (3.24) to conclude This implies that u = 0. As before, we arrive at a contradiction.

Preparation for the proof of (3.2) and (3.3):
Our present goal is to establish resolvent operator estimate with r ≥ 0 depending on (1.9) and (1.10).
In order to show condition (3.25) several preliminary estimates are needed. We start with the resolvent equation: let iλ ∈ (A ), Z ∈ D(A ) and F ∈ H , such that, (iλI − A )Z = F. The last and the definition (2.10) imply The first estimate is obtained combining the assumptions on the function µ(·) with (2.9) and (2.11) Throughout this manuscript, we routinely use Young, Hölder, Poincaré inequalities and dissipative estimate (3.32), often without explicit mention. By the symbol C, we denote a generic positive constant that may differ from line to line and independent on σ ∈ [0, 1). We also set, for the convenience of notation, F as a complex number that satisfies the estimate for some positive constant C also independent on σ ∈ [0, 1). We observe that, when σ = 0, estimate (3.32) for θ is non longer valid. Estimates for component θ are given by the following Lemmas.
Proof. Taking the M inner product of θ with equation (3.31), we obtain

MARGARETH S. ALVES AND RODRIGO N. MONTEIRO
The next step is to estimate the product defined by I. Via integration by parts, we find Therefore, the above identity leads to Combining the fact that Now, the conclusion follows from the above estimate and from (3.32).
Proof. Taking the L 2 (0, ) inner product of θ with equation (3.30), we arrive at The last combined with the definition of norm in H implies the conclusion.

STABILITY OF MIXTURE PROBLEMS 4891
Proof. Using equations (3.26) and (3.28) in (3.30), we find the following identity Taking the L 2 (0, ) inner product of the above identity with (β 1 u x + β 2 w x ) and using resolvent equations (3.26), (3.28), we obtain The next step is to estimate the right-hand side of (3.33). Note that Now, setting ∆ = a 11 a 22 − a 2 12 , we can write Using equations (3.27) and (3.29), we find The above implies the following estimate Inserting (3.34) and (3.35) into (3.33), we find the desired conclusion.
Via above identities and resolvent equations (3.26) and (3.28), the real part of the first product on the right-hand side of (3.36) satisfies Here, we have used ψ x L 2 ≤ C θ x L 2 . Next, using the definition of H norm, we find Combination of previous inequalities leads the desired estimate.
Proof. We recall (3.37) to find Invoking now Young inequality, we conclude that for any > 0 there exists a positive constant C such that The last, when combined with inequality K 1 v + K 2 ω L 2 ≤ C Z H yields the conclusion.

Proof of the
We shall estimate J. From assumption χ 0 = 0, we find and These imply that Combination of (3.39) with (3.40) implies Recalling Lemmas 3.7 and 3.9, for > 0 small and |λ| large, we obtain Finally, we invoke estimate (3.32) if σ ∈ (0, 1) or Lemmas 3.5-3.6 for the limit case σ = 0, to conclude This concludes the proof.
Proof of the Lack of Exponential Decay: The proof of the present statement makes use of Lemma 5.1 in [16]. Notice that the memory kernel assumptions, in particular, imply lim s→0 √ sµ(s) = 0. Then, under this condition the result in [16] states that √ λ (3.42) Let us consider F n = (0, aρ −1 1 sin nπx , 0, bρ −1 2 sin nπx , 0, 0) ∈ H , n ∈ N and a, b ∈ R. With reference to the boundary conditions (2.7), one can assume solutions of the resolvent equation as following u n = A n sin nπx , w n = B n sin nπx , θ n = C n cos nπx , η n = D n (s) cos nπx .
The last combined with assumption (1.11) also implies that This concludes the proof.
Proof of the Polynomial Decay Rate-Theorem 3.2. In line with Borichev and Tomilov result, we only need to verify (3.25) with r = 4.
Collecting estimates (3.53), (3.56) and (3.59) for a positive constant C and |λ| large enough. Now, we recall estimate (3.32) (or Lemmas 3.5 and 3.6 for σ = 0) to conclude Here, we have used the inequality This concludes the proof.