Exponential Stability in Hyperbolic Thermoelastic Diffusion Problem with Second Sound

Abstract

We consider a thermoelastic diffusion problem in one space dimension with second sound. The thermal and diffusion disturbances are modeled by Cattaneo's law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier's law. The system of equations in this case is a coupling of three hyperbolic equations. It poses some new analytical and mathematical difficulties. The exponential stability of the slightly damped and totally hyperbolic system is proved. Comparison with classical theory is given.

1. Introduction

The classical model for the propagation of heat turns into the well-known Fourier’s law 𝑞+𝑘∇𝜃=0, (1.1) where 𝜃 is temperature (difference to a fixed constant reference temperature), 𝑞 is the heat conduction vector, and 𝑘 is the coefficient of thermal conductivity. The model using classic Fourier's law inhibits the physical paradox of infinite propagation speed of signals. To eliminate this paradox, a generalized thermoelasticity theory has been developed subsequently. The development of this theory was accelerated by the advent of the second sound effects observed experimentally in materials at a very low temperature. In heat transfer problems involving very short time intervals and/or very high heat fluxes, it has been revealed that the inclusion of the second sound effects to the original theory yields results which are realistic and very much different from those obtained with classic Fourier’s law. The first theory was developed by Lord and Shulman [1]. In this theory, a modified law of heat conduction, the Cattaneo's law, 𝜏₀𝑞_𝑡+𝑞+𝑘∇𝜃=0 (1.2) replaces the classic Fourier's law. The heat equation associated with this a hyperbolic one and, hence, automatically eliminates the paradox of infinite speeds. The positive parameter 𝜏₀ is the relaxation time describing the time lag in the response of the heat flux to a gradient in the temperature.

The development of high technologies in the years before, during, and after the second world war pronouncedly affected the investigations in which the fields of temperature and diffusion in solids cannot be neglected. The problems connected with the diffusion of matter in thermoelastic bodies and the interaction of mechanodiffusion processes have become the subject of research by many authors. At elevated and low temperatures, the processes of heat and mass transfer play a decisive role in many satellite problems, returning space vehicles, and landing on water or land. These days, oil companies are interested in the process of thermodiffusion for more efficient extraction of oil from oil deposits.

Nowacki [2] developed the classic theory of thermoelastic diffusion under Fourier’s law. Sherief et al. [3] derived the theory of thermoelastic diffusion with second sound. Aouadi [4] derived the theory of micropolar thermoelastic diffusion under Cattaneo's law. Recently, Aouadi [5–7] derived the general equations of motion and constitutive equations of the linear thermoelastic diffusion theory in the context of different media, with uniqueness and existence theorems.

In recent years, a relevant task has been developed to obtain exponential stability of solutions in thermoelastic theories. The classical theory was first considered by Dafermos [8] and Slemrod [9] and it has been studied in the book of Jiang and Racke [10] and the contribution of Lebeau and Zuazua [11]. One should mention a paper by Sherief [12], where the stability of the null solution also in higher dimension is proved. We mention also the work of Tarabek [13] who studied even one dimensional nonlinear systems and obtained the strong convergence of derivatives of solutions to zero. Racke [14] proved the exponential decay of linear and nonlinear thermoelastic systems with second sound in one dimension for various boundary conditions. Messaoudi and Said-Houari [15] proved the exponential stability in one-dimensional nonlinear thermoelasticity with second sound. Soufyane [16] established an exponential and polynomial decay results of porous thermoelasticity including a memory term.

Recently, Aouadi and Soufyane [17] proved the polynomial and exponential stability for one-dimensional problem in thermoelastic diffusion theory under Fourier's law. To the author's knowledge, no work has been done regarding the exponential stability of the thermoelastic diffusion theory with second sound though similar research in thermoelasticity has been popular in recent years. This paper will devote to study the exponential stability of the solution of the one-dimensional thermoelastic diffusion theory under Cattaneo's law. The model that we consider is interesting not only because we take into account the thermal-diffusion effect, but also because Cattaneo's law is physically more realistic than Fourier’s law. In this case, the governing equations corresponds to the coupling of three hyperbolic equations. This question is new in thermoelastic theories and poses new analytical and mathematical difficulties. This kind of coupling has not been considered previously, and we have a few results concerning the existence, uniqueness, and exponential decay. For this reason, the exponential decay of the solution is very interesting and also very difficult. We obtain the exponential decay by the multiplier method and constructing generalized Lyapunov functional.

The remaining part of this paper is organized as follows: in Section 2, we give basic equations, and for completeness, we discuss the well-posedness of the initial boundary value problem in a semigroup setting. In Section 3, we derive the various energy estimates, and we state the exponential decay of the solution. In Section 4, we provide arguments for showing that the two systems, either 𝜏₀>0,𝜏>0 or 𝜏₀=𝜏=0, are close to each other, in the sense of energy estimates, of order 𝜏²₀ and 𝜏².

2. Basic Equations and Preliminaries

The governing equations for an isotropic, homogenous thermoelastic diffusion solid are as follows (see [3]):(i)the equation of motion 𝜎_{𝑖𝑗,𝑗}=𝜌𝑢̈_𝑖, (2.1)(ii)the stress-strain-temperature-diffusion relation 𝜎_𝑖𝑗=2𝜇𝑒_𝑖𝑗+𝛿_𝑖𝑗𝜆𝑒_𝑘𝑘−𝛽₁𝜃−𝛽₂𝐶, (2.2)(iii)the displacement-strain relation 𝑒_𝑖𝑗=12𝑢_𝑖,𝑗+𝑢_𝑗,𝑖, (2.3)(iv)the energy equation 𝑞_𝑖,𝑖=−𝜌𝑇₀𝑆,̇ (2.4)(v)the Cattaneo's law for temperature −𝑘𝜃_,𝑖=𝑞_𝑖+𝜏₀𝑞̇_𝑖, (2.5)(vi)the entropy-strain-temperature-diffusion relation 𝜌𝑇₀𝑆=𝛽₁𝑇₀𝑒_𝑘𝑘+𝜌𝑐_𝐸𝜃+𝑎𝑇₀𝐶, (2.6)(vii)the equation of conservation of mass 𝜂_𝑖,𝑖=−𝐶̇, (2.7)(viii)the Cattaneo's law for chemical potential −ℏ𝑃_,𝑖=𝜂_𝑖+𝜏𝜂̇_𝑖, (2.8)(ix)the chemical-strain-temperature-diffusion relation 𝑃=−𝛽₂𝑒_𝑘𝑘−𝑎𝜃+𝑏𝐶, (2.9) where 𝛽₁=(3𝜆+2𝜇)𝛼_𝑡 and 𝛽₂=(3𝜆+2𝜇)𝛼_𝑐, 𝛼_𝑡, and 𝛼_𝑐 are, respectively, the coefficients of linear thermal and diffusion expansion and 𝜆 and 𝜇 are Lamé’s constants. 𝜃=𝑇−𝑇₀ is small temperature increment, 𝑇 is the absolute temperature of the medium, and 𝑇₀ is the reference uniform temperature of the body chosen such that |𝜃/𝑇₀|≪1. 𝑞_𝑖 is the heat conduction vector, 𝑘 is the coefficient of thermal conductivity, and 𝑐_𝐸 is the specific heat at constant strain. 𝜎_𝑖𝑗 are the components of the stress tensor, 𝑢_𝑖 are the components of the displacement vector, 𝑒_𝑖𝑗 are the components of the strain tensor, 𝑆 is the entropy per unit mass, 𝑃 is the chemical potential per unit mass, 𝐶 is the concentration of the diffusive material in the elastic body, ℏ is the diffusion coefficient, 𝜂_𝑖 denotes the flow of the diffusing mass vector, “𝑎” is a measure of thermodiffusion effect, “𝑏” is a measure of diffusive effect, and 𝜌 is the mass density. 𝜏₀ is the thermal relaxation time, which will ensure that the heat conduction equation will predict finite speeds of heat propagation. 𝜏 is the diffusion relaxation time, which will ensure that the equation satisfied by the concentration will also predict finite speeds of propagation of matter from one medium to the other.

We will now formulate a different alternative form that will be useful in proving uniqueness in the next section. In this new formulation, we will use the chemical potential as a state variable instead of the concentration. From (2.9), we obtain 𝐶=𝛾₂𝑒_𝑘𝑘+𝑛𝑃+𝑑𝜃. (2.10) The alternative form can be written by substituting (2.10) into (2.1)–(2.8), 𝜎_{𝑖𝑗,𝑗}=𝜌𝑢̈_𝑖,𝜎_𝑖𝑗=2𝜇𝑒_𝑖𝑗+𝛿_𝑖𝑗𝜆₀𝑒_𝑘𝑘−𝛾₁𝜃−𝛾₂𝑃,𝑞_𝑖,𝑖=−𝜌𝑇₀𝑆,̇−𝑘𝜃_,𝑖=𝑞_𝑖+𝜏₀𝑞̇_𝑖,𝜌𝑆=𝑐𝜃+𝛾₁𝑒_𝑘𝑘𝜂+𝑑𝑃,_𝑖,𝑖=−𝐶̇,−ℏ𝑃_,𝑖=𝜂_𝑖+𝜏𝜂̇_𝑖, (2.11) where 𝛾₁=𝛽₁+𝑎𝛽₂𝑏,𝛾₂=𝛽₂𝑏,𝜆₀𝛽=𝜆−²²𝑏,𝑐=𝜌𝑐_𝐸𝑇₀+𝑎²𝑏𝑎,𝑑=𝑏1,𝑛=𝑏 (2.12) are physical positive constants satisfying the following condition: 𝑐𝑛−𝑑²>0. (2.13) Note that this condition implies that 𝑐𝜃²+2𝑑𝜃𝑃+𝑛𝑃²>0. (2.14) Condition (2.13) is needed to stabilize the thermoelastic diffusion system (see [18] for more information on this).

We assume throughout this paper that the condition (2.13) is satisfied.

For the sake of simplicity, we assume that 𝜌=1, and we study the exponential stability in one-dimension space. If 𝑢=𝑢(𝑥,𝑡), 𝜃=𝜃(𝑥,𝑡), and 𝑃=𝑃(𝑥,𝑡) describe the displacement, relative temperature and chemical potential, respectively, our equations take the form 𝑢_𝑡𝑡−𝛼𝑢_𝑥𝑥+𝛾₁𝜃_𝑥+𝛾₂𝑃_𝑥=0,in]0,ℓ[×ℝ⁺,𝑐𝜃_𝑡+𝑑𝑃_𝑡+𝑞_𝑥+𝛾₁𝑢_𝑥𝑡=0,in]0,ℓ[×ℝ⁺,𝜏₀𝑞_𝑡+𝑞+𝑘𝜃_𝑥=0,in]0,ℓ[×ℝ⁺,𝑑𝜃_𝑡+𝑛𝑃_𝑡+𝜂_𝑥+𝛾₂𝑢_𝑥𝑡=0,in]0,ℓ[×ℝ⁺,𝜏𝜂_𝑡+𝜂+ℏ𝜂_𝑥=0,in]0,ℓ[×ℝ⁺, (2.15) where 𝛼=𝜆₀+2𝜇>0. The system is subjected to the following initial conditions: 𝑢(𝑥,0)=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥)][,𝜃(),𝑥∈0,ℓ𝑥,0=𝜃₀(𝑥),𝑞(𝑥,0)=𝑞₀(𝑥)][,𝑃(),𝑥∈0,ℓ𝑥,0=𝑃₀(𝑥),𝜂(𝑥,0)=𝜂₀(𝑥),𝑥∈]0,ℓ[, (2.16) and boundary conditions 𝑢(0,𝑡)=𝑢(ℓ,𝑡)=0,𝑞(0,𝑡)=𝑞(ℓ,𝑡)=0,𝜂(0,𝑡)=𝜂(ℓ,𝑡)=0,𝑡≥0. (2.17)

For the sake of simplicity, we present a short direct discussion of the the well-posedness for the linear initial boundary value (2.15)₁–(2.17). We transform the system (2.15)₁–(2.17) into a first-order system of evolution type, finally applying semigroup theory. For a solution (𝑢,𝜃,𝑞,𝑃,𝜂), let 𝒰 be defined as 𝒰=𝛼𝑢_𝑥𝑢_𝑡𝜃𝑞𝑃𝜂,𝒰(0,⋅)=𝒰₀=𝛼𝑢_0,𝑥𝑢₁𝜃₀𝑞₀𝑃₀𝜂₀. (2.18)

The initial-boundary value problem (2.15)₁–(2.17) is equivalent to problem 𝑑𝒰𝑑𝑡+𝒬⁻¹ℳ𝒰=0,𝒰(0)=𝒰₀, (2.19) where 1𝒬=𝛼𝜏0000001000000𝑐0𝑑0000⁰𝑘𝜏0000𝑑0𝑛000000ℏ,ℳ=0−𝜕_𝑥0000−𝜕_𝑥0𝛾₁𝜕_𝑥0𝛾₂𝜕_𝑥00𝛾₁𝜕_𝑥0𝜕_𝑥0000𝜕_𝑥1𝑘000𝛾₂𝜕_𝑥000𝜕_𝑥0000𝜕_𝑥1ℏ. (2.20)

We consider the Hilbert space £=𝒰∈(𝐿²(0,ℓ))⁶ with inner product ⟨𝑈,𝑊⟩_£=⟨𝑈,𝑄𝑊⟩_𝐿². (2.21) Let 𝒜∶𝒟(𝒜)⊂£→£ such that 𝒜𝒰=𝒬⁻¹ℳ𝒰. (2.22) The domain of 𝒜 is 𝒟(𝒜)=𝒰=𝑈¹,𝑈²,𝑈³,𝑈⁴,𝑈⁵,𝑈^6𝑇∈£/𝑈²,𝑈⁴,𝑈⁶∈𝐻¹₀(0,ℓ),𝑈¹,𝑈³,𝑈⁵∈𝐻¹(0,ℓ), (2.23) that is, 𝑑𝒰(0)𝑑𝑡+𝒜𝒰=0,𝒰=𝒰₀∈𝒟(𝒜). (2.24)

On the other hand, if 𝒰 satisfies (2.24) for 𝒰₀ defined in (2.18), then 𝑢(⋅,𝑡)=𝑢₀(⋅)+^𝑡₀𝑈²(⋅,𝑠)𝑑𝑠,𝜃=𝑈³,𝑞=𝑈⁴,𝑃=𝑈⁵,𝜂=𝑈⁶ (2.25) satisfy (2.15)₁–(2.17); that is, (2.24) and (2.15)₁–(2.17) are equivalent (in the chosen spaces). The well-posedness is now a corollary of the following lemma characterizing 𝒜 as a generator of a 𝐶₀-semigroup of contractions.

Lemma 2.1. (i)𝒟(𝒜) is dense in £, and the operator −𝒜 is dissipative.
(ii)𝒜 is closed.
(iii)𝒟𝒜^∗=𝒟(𝒜),𝒜^∗𝑊=𝒬⁻¹0𝜕_𝑥𝜕0000_𝑥0−𝛾₁𝜕_𝑥0−𝛾₂𝜕_𝑥00−𝛾₁𝜕_𝑥0−𝜕_𝑥0000−𝜕_𝑥1𝑘000−𝛾₂𝜕_𝑥000−𝜕_𝑥0000−𝜕_𝑥1ℏ𝑊. (2.26)

Proof. (i) The density of 𝒟(𝒜) in £ is obvious, and we have 𝑅e⟨−𝐴𝑈,𝑈⟩_£1=−𝑘^ℓ₀𝑞²1𝑑𝑥−ℏ^ℓ₀𝜂²𝑑𝑥≤0. (2.27) Then −𝒜 is dissipative.
(ii) Let (𝒰_𝑛)_𝑛⊂𝒟(𝒜),𝒰_𝑛→𝒰∈£, and 𝒜𝒰_𝑛→ℱ∈£, as 𝑛→∞. Then, ∀Φ∈£∶𝐴𝒰_𝑛,Φ_£⟶⟨𝐹,Φ⟩_£. (2.28) Choosing successively(1)Φ=(Φ¹,0,0,0,0,0)^𝑇,Φ¹∈𝐻¹(0,ℓ), (2)Φ=(0,0,Φ³,0,0,0)^𝑇,Φ³∈𝐻¹(0,ℓ),(3)Φ=(0,0,0,0,Φ⁵,0)^𝑇,Φ⁵∈𝐻¹(0,ℓ), (4)Φ=(0,0,0,Φ⁴,0,0)^𝑇,Φ⁴∈𝐶^∞₀(0,ℓ),(5)Φ=(0,0,0,0,0,Φ⁶)^𝑇,Φ⁶∈𝐶^∞₀(0,ℓ), (6)Φ=(0,Φ²,0,0,0)^𝑇,Φ²∈𝐶^∞₀(0,ℓ),we obtain(1)𝑈²∈𝐻¹₀(0,ℓ) and −𝜕_𝑥𝑈²=[𝒬ℱ]¹ (first component),(2)𝑈⁴∈𝐻¹₀(0,ℓ) and 𝛾₁𝜕_𝑥𝑈²+𝜕_𝑥𝑈⁴=[𝒬ℱ]³,(3)𝑈⁶∈𝐻¹₀(0,ℓ) and 𝛾₂𝜕_𝑥𝑈²+𝜕_𝑥𝑈⁶=[𝒬ℱ]⁵,(4)𝑈³∈𝐻¹(0,ℓ) and 𝜕_𝑥𝑈³+1/𝑘𝑈⁴=[𝒬ℱ]⁴,(5)𝑈⁵∈𝐻¹(0,ℓ) and 𝜕_𝑥𝑈⁵+1/ℏ𝑈⁶=[𝒬ℱ]⁶,(6)𝑈¹∈𝐻¹(0,ℓ) and −𝜕_𝑥𝑈¹+𝛾₁𝜕_𝑥𝑈³+𝛾₁𝜕_𝑥𝑈⁵=[𝒬ℱ]².
(iii) 𝒲∈𝒟𝒜^∗⟺∃ℱ∈£,∀Φ∈𝒟(𝒜)∶⟨𝐴Φ,𝒲⟩_£⟶⟨Φ,ℱ⟩_£. (2.29) Choosing Φ appropriately as in the proof of (ii), the conclusion follows.

With the Hille-Yosida theorem (see [19]) 𝐶₀-semigroups, we can state the following result.

Theorem 2.2. (i) The operator −𝒜 is the infinitesimal generator of a 𝐶₀-semigroup of linear contractions 𝑇(𝑡)=𝑒^−𝑡𝒜 over the space £ for 𝑡≥0.
(ii) For any 𝒰₀∈𝒟(𝒜), there exists a unique solution 𝒰(𝑡)∈𝐶¹([0,∞);£)∩𝐶⁰([0,∞);𝒟(𝒜))to (2.24) given by 𝒰(𝑡)=𝑒^−𝑡𝒜𝒰₀.
(iii) If 𝒰₀∈𝒟(𝒜^𝑛), 𝑛∈ℕ, then 𝒰(𝑡)∈𝐶⁰([0,∞);𝒟(𝒜^𝑛)) and (2.24) yields higher regularity in 𝑡.

Moreover, we will use the Young inequality 𝑎±𝑎𝑏≤²𝛿+𝛿4𝑏²,∀𝑎,𝑏∈ℝ,𝛿>0. (2.30)

The differential of (2.15)₂ and (2.15)₄ together with boundary conditions (2.17) yields ^ℓ₀𝜃(𝑥,𝑡)𝑑𝑥=^ℓ₀𝜃₀(𝑥)𝑑𝑥,^ℓ₀𝑃(𝑥,𝑡)𝑑𝑥=^ℓ₀𝑃₀(𝑥)𝑑𝑥,𝑡≥0. (2.31) Then, 𝜃 and 𝑃 defined by 𝜃()()−1𝑥,𝑡∶=𝜃𝑥,𝑡ℓ^ℓ₀𝜃₀(𝑥)𝑑𝑥,𝑃()()−1𝑥,𝑡∶=𝑃𝑥,𝑡ℓ^ℓ₀𝑃₀(𝑥)𝑑𝑥 (2.32) satisfy with 𝑢,𝑞, and 𝜂 the same differential equations (2.15)₁–(2.17) as (𝑢,𝜃,𝑞,𝑃,𝜂), but additionally, we have the Poincaré inequality ^ℓ₀𝑣²ℓ𝑑𝑥≤²𝜋^2ℓ₀𝑣²_𝑥𝑑𝑥, (2.33) for 𝑣=𝜃(⋅,𝑡) as well as for 𝑣=𝑃, 𝑣=𝑢, 𝑣=𝑞 or 𝑣=𝜂.

In the sequel, we will work with 𝜃 and 𝑃 but still write 𝜃 and 𝑃 for simplicity until we will have proved Theorem 3.2.

From (2.15)₃ and (2.15)₅, we conclude ^ℓ₀𝜃²_𝑥𝑑𝑥≤2𝜏²₀𝑘^2ℓ0𝑞²_𝑡2𝑑𝑥+𝑘^2ℓ0𝑞²𝑑𝑥,_ℓ0𝑃²_𝑥𝑑𝑥≤2𝜏²ℏ^2ℓ0𝜂²_𝑡2𝑑𝑥+ℏ^2ℓ0𝜂²𝑑𝑥. (2.34) Finally, for the sake of simplicity, we will employ the same symbols 𝐶 for different constants, even in the same formula. In particular, we will denote by the same symbol 𝐶_𝑖 different constants due to the use of Poincaré's inequality on the interval [0,ℓ].

3. Exponential Stability

Let (𝑢,𝜃,𝑞,𝑃,𝜂) be a solution to problem (2.15)₁–(2.17). Multiplying (2.15)₁ by 𝑢_𝑡, (2.15)₂ by 𝜃, (2.15)₃ by 𝑞, (2.15)₄ by 𝑃, and (2.15)₅ by 𝜂 and integrating from 0 to ℓ, we get 𝑑ℰ𝑑𝑡¹(𝑡)1=−𝑘^ℓ₀𝑞²1𝑑𝑥−ℏ^ℓ₀𝜂²𝑑𝑥, (3.1) where ℰ₁(𝑡)=12^ℓ₀𝑢²_𝑡+𝛼𝑢²_𝑥+𝑐𝜃²+𝑛𝑃²𝜏+2𝑑𝜃𝑃+⁰𝑘𝑞²+𝜏ℏ𝜂²𝑑𝑥. (3.2) Differentiating (2.15) with respect to 𝑡, we get in the same manner 𝑑ℰ𝑑𝑡²(𝑡)1=−𝑘^ℓ₀𝑞²_𝑡1𝑑𝑥−ℏ^ℓ₀𝜂²_𝑡𝑑𝑥, (3.3) where ℰ₂(𝑡)=12^ℓ₀𝑢²_𝑡𝑡+𝛼𝑢²_𝑥𝑡+𝑐𝜃²_𝑡+𝑛𝑃²_𝑡+2𝑑𝜃_𝑡𝑃_𝑡+𝜏₀𝑘𝑞²_𝑡+𝜏ℏ𝜂²_𝑡𝑑𝑥. (3.4)

Let us define the functionals ℱ(𝑡)=−^ℓ₀1𝛼𝑢_𝑥𝑡𝑢_𝑥+3𝛼²𝛾₁𝑞𝑢_𝑡𝑡+3𝛼𝑞𝜃_𝑥+3𝛾₂𝛼𝛾₁𝑞𝑃_𝑥+3𝑐𝛼𝛾₁𝜃_𝑥𝑢_𝑡+3𝑑𝛼𝛾₁𝑃_𝑥𝑢_𝑡𝒢(𝑡)𝑑𝑥,=−^ℓ₀1𝛼𝑢_𝑥𝑡𝑢_𝑥+3𝛼²𝛾₂𝜂𝑢_𝑡𝑡+3𝛼𝜂𝑃_𝑥+3𝛾₁𝛼𝛾₂𝜂𝜃_𝑥+3𝑑𝛼𝛾₂𝑃_𝑥𝑢_𝑡+3𝑛𝛼𝛾₂𝜃_𝑥𝑢_𝑡𝑑𝑥. (3.5)

Lemma 3.1. Let (𝑢,𝜃,𝑞,𝑃,𝜂) be a solution to problem (2.15)₁–(2.17). Then, one has 𝑑ℱ(𝑡)≤𝑑𝑡−136𝛼^2ℓ₀𝑢²_𝑡𝑡+𝛼𝜋²ℓ²𝑢²_𝑡+𝜃²+𝑃²1𝑑𝑥−𝛼^ℓ₀𝑢²_𝑥𝑡𝑑𝑥+27𝛾²₁𝛼^2ℓ₀𝑞²_𝑡+𝑑𝑥27𝑐²𝛾²₁+3𝑐𝛾₂2𝛼𝛾₁+5𝛾²₁6𝛼²+ℓ²36𝛼²𝜋²+3𝑐2𝛼^ℓ₀𝜃²_𝑥+𝑑𝑥27𝑑²𝛾²₂+3𝑐𝛾₂2𝛼𝛾₁+5𝛾²₂6𝛼²+ℓ²36𝛼²𝜋²+3𝑐2𝛼^ℓ₀𝑃²_𝑥𝑑𝑥, (3.6)𝑑𝒢(𝑡)≤𝑑𝑡−136𝛼^2ℓ₀𝑢²_𝑡𝑡+𝛼𝜋²ℓ²𝑢²_𝑡+𝜃²+𝑃²1𝑑𝑥−𝛼^ℓ₀𝑢²_𝑥𝑡𝑑𝑥+27𝛾²₂𝛼^2ℓ₀𝜂²_𝑡+𝑑𝑥27𝑛²𝛾²₂+3𝑛𝛾₁2𝛼𝛾₂+5𝛾²₂6𝛼²+ℓ²36𝛼²𝜋²+3𝑛2𝛼^ℓ₀𝑃²_𝑥+𝑑𝑥27𝑑²𝛾²₁+3𝑛𝛾₁2𝛼𝛾₂+5𝛾²₁6𝛼²+ℓ²36𝛼²𝜋²+3𝑛2𝛼^ℓ₀𝜃²_𝑥𝑑𝑥. (3.7)

Proof. We will only prove (3.6) and (3.7) can be obtained analogously. Multiplying (2.15)₁ by 𝑢_𝑥𝑥/𝛼 and using the Young inequality, we get ^ℓ₀𝑢²_𝑥𝑥1𝑑𝑥=−𝛼^ℓ₀𝑢_𝑡𝑡𝑥𝑢_𝑥𝛾𝑑𝑥+¹𝛼^ℓ₀𝜃_𝑥𝑢_𝑥𝑥𝛾𝑑𝑥+²𝛼^ℓ₀𝑃_𝑥𝑢_𝑥𝑥1𝑑𝑥≤−𝛼𝑑𝑑𝑡^ℓ₀𝑢_𝑥𝑡𝑢_𝑥1𝑑𝑥+𝛼^ℓ₀𝑢²_𝑥𝑡𝑑𝑥+3𝛾²₁4𝛼^2ℓ₀𝜃²_𝑥𝑑𝑥+3𝛾²₂4𝛼^2ℓ₀𝑃²_𝑥2𝑑𝑥+3^ℓ₀𝑢²_𝑥𝑥𝑑𝑥, (3.8) which implies 1𝛼𝑑𝑑𝑡^ℓ₀𝑢_𝑥𝑡𝑢_𝑥1𝑑𝑥≤−3^ℓ₀𝑢²_𝑥𝑥1𝑑𝑥+𝛼^ℓ₀𝑢²_𝑥𝑡𝑑𝑥+3𝛾²₁4𝛼^2ℓ₀𝜃²_𝑥𝑑𝑥+3𝛾²₂4𝛼^2ℓ₀𝑃²_𝑥𝑑𝑥. (3.9)
Multiplying (2.15)₂ by 3𝑢_𝑥𝑡/(𝛼𝛾₁) and using the Young inequality, we get 3𝛼^ℓ₀𝑢²_𝑥𝑡3𝑑𝑥=𝛼𝛾₁^ℓ₀𝑞𝑢_𝑥𝑥𝑡𝑑𝑥+3𝑐𝛼𝛾₁^ℓ₀𝜃_𝑥𝑡𝑢_𝑡𝑑𝑥+3𝑑𝛼𝛾₁^ℓ₀𝑃_𝑥𝑡𝑢_𝑡=3𝑑𝑥𝛼𝛾₁𝑑𝑑𝑡^ℓ₀𝑞𝑢_𝑥𝑥3𝑑𝑥−𝛼𝛾₁^ℓ₀𝑞_𝑡𝑢_𝑥𝑥𝑑𝑥+3𝑐𝛼𝛾₁𝑑𝑑𝑡^ℓ₀𝜃_𝑥𝑢_𝑡𝑑𝑥−3𝑐𝛼𝛾₁^ℓ₀𝜃_𝑥𝑢_𝑡𝑡+𝑑𝑥3𝑑𝛼𝛾₁𝑑𝑑𝑡^ℓ₀𝑃_𝑥𝑢_𝑡𝑑𝑥−3𝑑𝛼𝛾₁^ℓ₀𝑃_𝑥𝑢_𝑡𝑡𝑑𝑥. (3.10) Substituting (2.15)₁ in the above equation, yields 3𝛼^ℓ₀𝑢²_𝑥𝑡3𝑑𝑥=𝛼²𝛾₁𝑑𝑑𝑡^ℓ₀𝑞𝑢_𝑡𝑡3𝑑𝑥+𝛼𝑑𝑑𝑡^ℓ₀𝑞𝜃_𝑥𝑑𝑥+3𝛾₂𝛼𝛾₁𝑑𝑑𝑡^ℓ₀𝑞𝑃_𝑥3𝑑𝑥−𝛼𝛾₁^ℓ₀𝑞_𝑡𝑢_𝑥𝑥+𝑑𝑥3𝑐𝛼𝛾₁𝑑𝑑𝑡^ℓ₀𝜃_𝑥𝑢_𝑡𝑑𝑥−3𝑐𝛾₁^ℓ₀𝜃_𝑥𝑢_𝑥𝑥𝑑𝑥+3𝑐𝛼^ℓ₀𝜃²_𝑥𝑑𝑥+3𝑐𝛾₂𝛼𝛾₁^ℓ₀𝜃_𝑥𝑃_𝑥+𝑑𝑥3𝑑𝛼𝛾₁𝑑𝑑𝑡^ℓ₀𝑃_𝑥𝑢_𝑡𝑑𝑥−3𝑑𝛾₁^ℓ₀𝑃_𝑥𝑢_𝑥𝑥𝑑𝑥+3𝑐𝛼^ℓ₀𝜃_𝑥𝑃_𝑥𝑑𝑥+3𝑐𝛾₂𝛼𝛾₁^ℓ₀𝑃²_𝑥𝑑𝑥. (3.11)
Using the estimates −3𝛼𝛾₁^ℓ₀𝑞_𝑡𝑢_𝑥𝑥1𝑑𝑥≤12^ℓ₀𝑢²_𝑥𝑥𝑑𝑥+27𝛾²₁𝛼^2ℓ₀𝑞²_𝑡−𝑑𝑥,3𝑐𝛾₁^ℓ₀𝜃_𝑥𝑢_𝑥𝑥1𝑑𝑥≤12^ℓ₀𝑢²_𝑥𝑥𝑑𝑥+27𝑐²𝛾²₁^ℓ₀𝜃²_𝑥𝑑𝑥,3𝑐𝛾₂𝛼𝛾₁^ℓ₀𝜃_𝑥𝑃_𝑥𝑑𝑥≤3𝑐𝛾₂2𝛼𝛾₁^ℓ₀𝜃²_𝑥𝑑𝑥+3𝑐𝛾₂2𝛼𝛾₁^ℓ₀𝑃²_𝑥−𝑑𝑥,3𝑑𝛾₂^ℓ₀𝑃_𝑥𝑢_𝑥𝑥1𝑑𝑥≤12^ℓ₀𝑢²_𝑥𝑥𝑑𝑥+27𝑑²𝛾²₂^ℓ₀𝑃²_𝑥𝑑𝑥,3𝑐𝛼^ℓ₀𝜃_𝑥𝑃_𝑥𝑑𝑥≤3𝑐2𝛼^ℓ₀𝜃²_𝑥𝑑𝑥+3𝑐2𝛼^ℓ₀𝑃²_𝑥𝑑𝑥, (3.12)3𝛼^ℓ₀𝑢²_𝑥𝑡𝑑𝑑𝑥≤𝑑𝑡^ℓ₀3𝛼²𝛾₁𝑞𝑢_𝑡𝑡+3𝛼𝑞𝜃_𝑥+3𝛾₂𝛼𝛾₁𝑞𝑃_𝑥+3𝑐𝛼𝛾₁𝜃_𝑥𝑢_𝑡+3𝑑𝛼𝛾₁𝑃_𝑥𝑢_𝑡1𝑑𝑥+4^ℓ₀𝑢²_𝑥𝑥+𝑑𝑥27𝛾²₁𝛼^2ℓ₀𝑞²_𝑡𝑑𝑥+3𝑐𝛼9𝑐𝛼𝛾²₁+𝛾₂2𝛾₁+12^ℓ₀𝜃²_𝑥𝑑𝑥+3𝑛𝛼9𝑐𝛼𝛾²₂+𝛾₁2𝛾₂+12^ℓ₀𝜃²_𝑥𝑑𝑥. (3.13)
Combining (3.9) and (3.13), we get 𝑑ℱ(𝑡)1𝑑𝑡≤−12^ℓ₀𝑢²_𝑥𝑥2𝑑𝑥−𝛼^ℓ₀𝑢²_𝑥𝑡𝑑𝑥+27𝛾²₁𝛼^2ℓ₀𝑞²_𝑡+𝑑𝑥3𝑐𝛼9𝑐𝛼𝛾²₁+𝛾₂2𝛾₁+𝛾²₁+14𝑐𝛼2^ℓ₀𝜃²_𝑥+𝑑𝑥3𝑐𝛼9𝑑²𝛼𝛾²₂𝑐+𝛾₂2𝛾₁+𝛾²₂+14𝑐𝛼2^ℓ₀𝑃²_𝑥𝑑𝑥. (3.14)
Now, we conclude from (2.15) that ^ℓ₀𝑢²_𝑡𝑡+𝛼𝜋²ℓ²𝑢²_𝑡+𝜃²+𝑃²=𝑑𝑥_ℓ0𝛼²𝑢²_𝑥𝑥+𝛾²₁𝜃²_𝑥+𝛾²₂𝑃²_𝑥−2𝛼𝛾₁𝑢_𝑥𝑥𝜃_𝑥−2𝛼𝛾₂𝑢_𝑥𝑥𝑃_𝑥+2𝛾₁𝛾₂𝜃_𝑥𝑃_𝑥+𝛼𝜋²ℓ²𝑢²_𝑡+𝜃²+𝑃²𝑑𝑥≤3𝛼^2ℓ₀𝑢²_𝑥𝑥𝑑𝑥+3𝛾²₁+ℓ²𝜋^2ℓ₀𝜃²_𝑥𝑑𝑥+3𝛾²₂+ℓ²𝜋^2ℓ₀𝑃²_𝑥𝑑𝑥+𝛼^ℓ₀𝑢²_𝑥𝑡𝑑𝑥 (3.15) whence −^ℓ₀𝑢²_𝑥𝑥𝑑𝑥≤−13𝛼^2ℓ₀𝑢²_𝑡𝑡+𝛼²𝜋²ℓ²𝑢²_𝑡+𝜃²+𝑃²+1𝑑𝑥𝛼²𝛾²₁+ℓ²3𝜋^2ℓ₀𝜃²_𝑥1𝑑𝑥+𝛼²𝛾²₂+ℓ²3𝜋^2ℓ₀𝑃²_𝑥1𝑑𝑥+3𝛼^ℓ₀𝑢²_𝑥𝑡𝑑𝑥. (3.16)
Combining (3.14) and (3.16), we get our conclusion follows.

Multiplying(2.15)₂ by 𝜃_𝑡 and (2.15)₄ by 𝑃_𝑡, and summing the results, yields −𝑑𝑑𝑡^ℓ₀𝑞𝜃_𝑥+𝜂𝑃_𝑥𝑑𝑥=−^ℓ₀𝑐𝜃²_𝑡+𝑛𝑃²_𝑡+2𝑑𝜃_𝑡𝑃_𝑡𝑑𝑥−^ℓ₀𝑞_𝑡𝜃_𝑥𝑑𝑥−𝛾₁^ℓ₀𝑢_𝑥𝑡𝜃_𝑡𝑑𝑥−^ℓ₀𝜂_𝑡𝑃_𝑥𝑑𝑥−𝛾₂^ℓ₀𝑢_𝑥𝑡𝑃_𝑡𝑑𝑥. (3.17)

Using the estimates 𝛾₁^ℓ₀𝑢_𝑥𝑡𝜃_𝑡𝑐𝑑𝑥≤4^ℓ₀𝜃²_𝑡𝛾𝑑𝑥+²¹𝑐^ℓ₀𝑢²_𝑥𝑡𝛾𝑑𝑥,_2ℓ0𝑢_𝑥𝑡𝑃_𝑡𝛿𝑑𝑥≤4^ℓ₀𝑃²_𝑡𝛾𝑑𝑥+²²𝛿^ℓ₀𝑢²_𝑥𝑡𝑑𝑥,−2𝑑^ℓ₀𝜃_𝑡𝑃_𝑡𝑐𝑑𝑥≤4^ℓ₀𝜃²_𝑡𝑑𝑥+4𝑑²𝑐^ℓ₀𝑃²_𝑡−𝑑𝑑𝑥,𝑑𝑡^ℓ₀𝑞𝜃_𝑥+𝜂𝑃_𝑥𝑐𝑑𝑥≤−2^ℓ₀𝜃²_𝑡𝛿𝑑𝑥−𝑛−4−4𝑑²𝑐^ℓ₀𝑃²_𝑡+1𝑑𝑥2^ℓ₀𝑞²_𝑡1𝑑𝑥+2^ℓ₀𝜂²_𝑡1𝑑𝑥+2^ℓ₀𝜃²_𝑥+1𝑑𝑥2^ℓ₀𝑃²_𝑥𝛾𝑑𝑥+²¹𝑐+𝛾²₂𝛿^ℓ₀𝑢²_𝑥𝑡𝑑𝑥. (3.18)

Choosing 𝛿 such as 𝛿𝑛−4−4𝑑²𝑐>𝑛2 (3.19) yields −𝑑𝑑𝑡^ℓ₀𝑞𝜃_𝑥+𝜂𝑃_𝑥𝑐𝑑𝑥≤−2^ℓ₀𝜃²_𝑡𝑛𝑑𝑥−2^ℓ₀𝑃²_𝑡1𝑑𝑥+2^ℓ₀𝑞²_𝑡1𝑑𝑥+2^ℓ₀𝜂²_𝑡+1𝑑𝑥2^ℓ₀𝜃²_𝑥1𝑑𝑥+2^ℓ₀𝑃²_𝑥𝛾𝑑𝑥+²¹𝑐+𝛾²₂𝛿^ℓ₀𝑢²_𝑥𝑡𝑑𝑥. (3.20) Now, we will show the main result of this section.

Theorem 3.2. Let (𝑢,𝜃,𝑞,𝑃,𝜂) be a solution to problem (2.15)₁–(2.17). Then, the associated energy of first and second order ℰ(𝑡)=ℰ₁(𝑡)+ℰ₂(𝑡)=12²_𝑗=1^ℓ₀𝜕_𝑡^𝑗−1𝑢_𝑡²+𝛼𝜕_𝑡^𝑗−1𝑢_𝑥²+𝑐𝜕_𝑡^𝑗−1𝜃²+𝑛𝜕_𝑡^𝑗−1𝑃²+2𝑑𝜕_𝑡^𝑗−1𝜃𝜕_𝑡^𝑗−1𝜏𝑃+⁰𝑘𝜕_𝑡^𝑗−1𝑞²+𝜏ℏ𝜕_𝑡^𝑗−1𝜂²(𝑥,𝑡)𝑑𝑥 (3.21) decays exponentially; that is, ∃𝑐₀>0,∃𝐶₀>0,∀𝑡≥0ℰ(𝑡)≤𝐶₀ℰ(0)𝑒^−𝑐0𝑡. (3.22)
Bounds for 𝑐₀ and 𝐶₀ can be given explicitly in terms of the coefficient 𝛼,𝛾₁,𝛾₂,𝑐,𝑛,𝑑,𝑘,ℏ,𝜏₀,𝜏, and ℓ.

Proof. Now, we define the desired Lyapunov functional 𝒩(𝑡). For 𝜀>0, to be determined later on, let 𝒩(𝑡)=1𝜀ℰ(𝑡)+ℱ(𝑡)+𝒢(𝑡)−𝜀^ℓ₀𝑞𝜃_𝑥+𝜂𝑃_𝑥𝑑𝑥. (3.23) Then, we conclude from (3.1)–(3.6) and (3.14) that 𝑑𝒩(𝑡)1𝑑𝑡≤−𝜀𝑘^ℓ₀𝑞²1𝑑𝑥−𝜀ℏ^ℓ₀𝜂²1𝑑𝑥−18𝛼^2ℓ₀𝑢²_𝑡𝑡+𝛼𝜋²ℓ²𝑢²_𝑡+𝜃²+𝑃²−2𝑑𝑥𝛼−𝜀𝛾²₂𝑛−𝜀𝛾²₁𝛿^ℓ₀𝑢²_𝑥𝑡1𝑑𝑥−−𝜀𝑘27𝛾²₁𝛼²−𝜀2^ℓ₀𝑞²_𝑡−1𝑑𝑥−𝜀ℏ27𝛾²₂𝛼²−𝜀2^ℓ₀𝜂²_𝑡𝜀𝑑𝑥−2^ℓ₀𝑐𝜃²_𝑡+𝑛𝑃²_𝑡+𝜀𝑑𝑥𝜛+2^ℓ₀𝜃²_𝑥𝜀𝑑𝑥+𝜚+2^ℓ₀𝑃²_𝑥𝑑𝑥, (3.24) where 𝑐𝜛=27²+𝑑²𝛾²₁+3𝑐𝛾₂2𝛼𝛾₁+3𝑛𝛾₁2𝛼𝛾₂+5𝛾²₁3𝛼²+ℓ²18𝛼²𝜋²+3(),𝑛2𝛼𝑐+𝑛𝜚=27²+𝑑²𝛾²₁+3𝑐𝛾₂2𝛼𝛾₁+3𝑛𝛾₁2𝛼𝛾₂+5𝛾²₂3𝛼²+ℓ²18𝛼²𝜋²+3().2𝛼𝑐+𝑛 (3.25)
Using (2.34), we obtain 𝑑𝒩(𝑡)1𝑑𝑡≤−𝑘1𝜀−2𝑘𝜀𝜛+2^ℓ₀𝑞²1𝑑𝑥−ℏ1𝜀−2ℏ𝜀𝜚+2^ℓ₀𝜂²−1𝑑𝑥18𝛼^2ℓ₀𝑢²_𝑡𝑡+𝛼𝜋²ℓ²𝑢²_𝑡+𝜃²+𝑃²2𝑑𝑥−𝛼−𝜀𝛾²₂𝑛−𝜀𝛾²₁𝛿^ℓ₀𝑢²_𝑥𝑡−1𝑑𝑥−𝜀𝑘27𝛾²₁𝛼²−𝜀2−2𝜏²₀𝑘²𝜀𝜛+2^ℓ₀𝑞²_𝑡−1𝑑𝑥−𝜀ℏ27𝛾²₂𝛼²−𝜀2−𝜀2−2𝜏²₀ℏ²𝜀𝜚+2^ℓ₀𝜂²_𝑡𝜀𝑑𝑥−2^ℓ₀𝑐𝜃²_𝑡+𝑛𝑃²_𝑡𝑑𝑥. (3.26)
Using (2.13) and choosing 0<𝜀<1 such that all terms on the right-hand side of (3.27) become negative, 2𝛼𝛾²₁/𝑐+𝛾²₂𝑘/𝛿<𝜀<min,ℏ1+2𝜛,11+2𝜚𝑘(1/2)+𝜏²₀/𝑘²(1+2𝜛)+27/𝛾²₁𝛼²,1ℏ(1/2)+𝜏²/ℏ²1+2𝜚+27/𝛾²₂𝛼². (3.27) Choosing 𝜀 as in (3.27), we obtain from 𝑑𝒩(𝑡)𝑑𝑡≤−𝑐₁^ℓ₀𝑢²_𝑡𝑡+𝑢²_𝑡+𝑢²_𝑥𝑡+𝜃²+𝑃²+𝜃²_𝑡+𝑃²_𝑡+𝑞²_𝑡+𝜂²_𝑡+𝑞²+𝜂²𝑑𝑥, (3.28) where 𝑐₁=121min,1𝜀𝑘,1𝜀ℏ9𝛼²,2𝛼,𝜀, (3.29) which implies 𝑑𝒩(𝑡)𝑑𝑡≤−𝑐₂ℰ(𝑡), (3.30) with 𝑐₂=𝑐₁2𝜏min1,𝛼,𝑐,𝑑,𝑛,⁰𝑘,𝜏ℏ. (3.31)
On the other hand, we have ∃𝜀₂>0,∃𝐶₁,𝐶₂>0,∀𝜀≤𝜀₂,∀𝑡>0∶𝐶₁ℰ(𝑡)≤𝒩(𝑡)≤𝐶₂ℰ(𝑡), (3.32) where 𝐶₁,𝐶₂ are determined as follows. Let ℋ(𝑡)(𝑡)−1=𝒩𝜀ℰ(𝑡), (3.33) then |ℋ(𝑡)|≤𝐶₁ℰ(𝑡), (3.34) with 𝐶₁3()=max𝑐+𝑑𝛼𝛾₁+3(𝑛+𝑑)𝛼𝛾₂,3𝛼²1𝛾₁+1𝛾₂,1𝛼²,3𝑘𝜏₀1𝛼²𝛾₁+1𝛼+𝛾₂𝛼𝛾₁+2𝛼𝑘²𝑐1+𝛾₁,3ℏ𝜏1𝛼²𝛾₂+1𝛼+𝛾₁𝛼𝛾₂+2𝛼ℏ²𝑛1+𝛾₂,6𝜏₀𝑐𝛼𝑘1+𝛾₁,6𝜏𝑛𝛼ℏ1+𝛾₂. (3.35)
Choosing 𝜀≤𝜀₂=12𝐶₁, (3.36) we have 𝐶₂=1𝜀+𝐶₁,𝜀=min𝜀₁,𝜀₂. (3.37) Moreover, from (3.30) and (3.32), we derive 𝑑𝒩(𝑡)𝑑𝑡≤−𝑐₀𝒩(𝑡), (3.38) with 𝑐₀=𝑐₂𝐶₂, (3.39) hence 𝒩(𝑡)≤𝑒^−𝑐0𝑡𝒩(0). (3.40) Applying (3.32) again, we have proved ℰ(𝑡)≤𝐶₀ℰ(0)𝑒^−𝑐0𝑡, (3.41) with 𝐶₀=𝐶₂𝐶₁, (3.42) and it holds.

The copper material was chosen for purposes of numerical evaluations. The physical constants given by Table 1 are found in [20].

Table 1 

Values of the constants.

Successively we can approximately compute 𝜀₁,𝐶₁,𝜀₂,𝜀,𝑐₁,𝑐₂,𝐶₀, and 𝑐₀ from the previous equations, getting finally 𝑐₀≈2.68×10⁻⁵⁶, (3.43) which indicates a slow decay of the energy in the beginning but does not mean that solutions do not decay.

Remark 3.3. In particular, we can get 𝑑₀=𝑂𝜏₀,𝜏as𝜏₀,𝜏⟶(0,0). (3.44) Although the estimate for 𝜏₀ and 𝜏 are very coarse and might be far from being sharp, it indicates a slow decay of the energy in usually measured time periods. The above relation of course does not imply that solutions to the limiting case (𝜏₀,𝜏)=(0,0) do not decay. Instead, the decay rate of the thermodiffusion system provides a better rate; that is, 𝑐𝜃_𝑡+𝑑𝑃_𝑡−𝑘𝜃_𝑥𝑥=0,in]0,ℓ[×ℝ⁺,𝑑𝜃_𝑡+𝑛𝑃_𝑡−ℏ𝑃_𝑥𝑥=0,in]0,ℓ[×ℝ⁺, (3.45) with initial conditions 𝜃(𝑥,0)=𝜃₀(𝑥),𝑃(𝑥,0)=𝑃₀(𝑥),𝑥∈]0,ℓ[, (3.46) and boundary conditions 𝜃(0,𝑡)=𝜃(ℓ,𝑡)=0,𝑃(0,𝑡)=𝑃(ℓ,𝑡)=0,𝑡≥0. (3.47) In this case, we have 𝑑𝐸(𝑡)𝑑𝑡=−𝑘^ℓ₀𝜃²𝑑𝑥−ℏ^ℓ₀𝑃²𝑑𝑥, (3.48) where 𝐸(𝑡)=12^ℓ₀𝑐𝜃²+𝑛𝑃²𝜏+2𝑑𝜃𝑃+⁰𝑘𝑞²+𝜏ℏ𝜂²𝑑𝑥. (3.49) Using the Poincaré inequality, we get 𝐸(𝑡)≤𝑒^−𝜈𝑡𝐸(0), (3.50) with 𝜋𝜈=2²ℓ²𝑘𝑐+ℏ𝑛≈2.43×10⁻³. (3.51)

4. The Limit Case (𝜏₀,𝜏)→(0,0)

We will show that the energy of the difference of the solution (𝑢,𝜃,𝑃,𝑞,𝜂) to (2.15)₁–(2.17) and the the solution (𝑢,𝜃,𝑃,𝑞,𝜂) to the corresponding system with (𝜏₀,𝜏)=(0,0) (see [17]) vanishes of order 𝜏²₀+𝜏² as (𝜏₀,𝜏)→(0,0), provided the values at 𝑡=0 coincide. For this purpose, let (𝑈,Θ,𝜙,Φ,𝜓) denote the difference 𝑈=𝑢−𝑢,Θ=𝜃−𝜃,𝜙=𝑞−𝑞,Φ=𝑃−𝑃,𝜓=𝜂−𝜂, (4.1) then (𝑈,Θ,𝜙,Φ,𝜓) satisfies 𝑈_𝑡𝑡−𝛼𝑈_𝑥𝑥+𝛾₁Θ_𝑥+𝛾₂Φ_𝑥=0,𝑐Θ_𝑡+𝑑Φ_𝑡+𝜙_𝑥+𝛾₁𝑈_𝑥𝑡𝜏=0,₀𝜙_𝑡+𝜙+𝑘Θ_𝑥=𝜏₀𝑘𝜃_𝑥𝑡,𝑑Θ_𝑡+𝑛Φ_𝑡+𝜓_𝑥+𝛾₂𝑈_𝑥𝑡=0,𝜏𝜓_𝑡+𝜂+ℏ𝜓_𝑥=𝜏ℏ𝑃_𝑥𝑡, (4.2)𝑈(𝑥,0)=0,𝑈_𝑡()()()𝜙()()][,𝑥,0=0,Θ𝑥,0=0,Φ𝑥,0=0,𝑥,0=0,𝜓𝑥,0=0,𝑥∈0,ℓ (4.3)𝑈(0,𝑡)=𝑈(ℓ,𝑡)=0,Θ(0,𝑡)=Θ(ℓ,𝑡)=0,Φ(0,𝑡)=Φ(ℓ,𝑡)=0. (4.4) Here, we assumed the compatibility conditions 𝑞₀=−𝑘𝜃_0,𝑥,𝜂₀=−ℏ𝑃_0,𝑥. (4.5) If 𝐸(𝑡) denotes the energy of first order for (𝑈,Θ,𝜙,Φ,𝜓); that is, 𝑑𝐸(𝑡)1𝑑𝑡=−𝑘^ℓ₀𝜙²𝑑𝑥+𝜏₀^ℓ₀𝜃_𝑥𝑡1𝜙𝑑𝑥−ℏ^ℓ₀𝜓²𝑑𝑥+𝜏^ℓ₀𝑃_𝑥𝑡𝜓𝑑𝑥, (4.6) where 𝐸(𝑡)=12^ℓ₀𝑈²_𝑡+𝛼𝑈²_𝑥+𝑐Θ²+𝑛Φ²𝜏+2𝑑ΘΦ+⁰𝑘𝜙²+𝜏ℏ𝜓²𝑑𝑥. (4.7) Using the Young inequality, we obtain 𝑑𝐸(𝑡)1𝑑𝑡≤−2𝑘^ℓ₀𝜙²𝜏𝑑𝑥+²⁰𝑘2^ℓ₀𝜃²_𝑥𝑡1𝑑𝑥−2ℏ^ℓ₀𝜓²𝜏𝑑𝑥+²ℏ2^ℓ₀𝑃²_𝑥𝑡𝑑𝑥. (4.8)

Using initial condition (4.3) yields 𝐸(𝑡)≤𝜏²₀𝑘2^𝑡₀^ℓ₀𝜃²_𝑥𝑡()𝜏𝑥,𝑠𝑑𝑥𝑑𝑠+²ℏ2^𝑡₀^ℓ₀𝑃²_𝑥𝑡(𝑥,𝑠)𝑑𝑥𝑑𝑠, (4.9) from where we get for 𝑇>0 fixed, 𝑡∈[0,𝑇]𝐸(𝑡)≤𝜏²₀𝑘2^𝑇₀^ℓ₀𝜃²_𝑥𝑡()𝜏𝑥,𝑠𝑑𝑥𝑑𝑠+²ℏ2^𝑇₀^ℓ₀𝑃²_𝑥𝑡(𝑥,𝑠)𝑑𝑥𝑑𝑠. (4.10) Moreover, since ^∞₀^ℓ₀𝜃²_𝑥𝑡(𝑥,𝑠)𝑑𝑥𝑑𝑠<∞,^∞₀^ℓ₀𝑃²_𝑥𝑡(𝑥,𝑠)𝑑𝑥𝑑𝑠<∞, (4.11) because of the exponential decay of the solution corresponding to the problem when (𝜏₀,𝜏)=(0,0) (see [17]), we obtain a uniform bound on the right-hand side, ∃𝐶>0,∀𝑡≥0∶𝐸(𝑡)≤𝐶𝜏²₀+𝜏², (4.12) where 1𝐶=2min𝑘^𝑇₀^ℓ₀𝜃²_𝑥𝑡(𝑥,𝑠)𝑑𝑥𝑑𝑠,ℏ^𝑇₀^ℓ₀𝑃²_𝑥𝑡(𝑥,𝑠)𝑑𝑥𝑑𝑠. (4.13) Then we have 𝐸(𝑡)≤𝒪𝜏²₀+𝜏²as𝜏₀,𝜏⟶(0,0), (4.14) also 𝐸(𝑡)𝜏²₀+𝜏²⟶0as𝑡⟶0. (4.15)

5. Concluding Remarks

(1)By comparison of the approximate value of 𝑐₀≈2.68×10⁻⁵⁶ with the value 𝑐₀≈1.75×10⁻¹³ of the problem corresponding to (𝜏₀,𝜏)=(0,0) computed in [17], we remark the second value is significantly larger than the first. This confirms that thermoelastic models with second sound are physically more realistic than those given in the classic context.(2)By comparison of the approximate value of 𝑐₀≈2.68×10⁻⁵⁶ with the value 𝜈≈2.43×10⁻³ of the thermodiffusion problem, we conclude that the slow decay of the elastic part is responsible for the low bounds on the decay rates obtained in this paper and in [17].(3)Finally, we remark that in [17], the exponential decay of the solution was proved by means of the first energy only, while in our case, it is necessary to use second-order derivatives because of the more complicated system with second sound.