Exponential Stability in Hyperbolic Thermoelastic Diffusion Problem with Second Sound

We consider a thermoelastic diffusion problem in one space dimension with second sound. The thermal and diffusion disturbances aremodeled 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.


Introduction
The classical model for the propagation of heat turns into the well-known Fourier's law q k∇θ 0, 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 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 τ 0 > 0, τ > 0 or τ 0 τ 0, are close to each other, in the sense of energy estimates, of order τ 2 0 and τ 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 ii the stress-strain-temperature-diffusion relation iii the displacement-strain relation e ij 1 2 u i,j u j,i , 2.3 iv the energy equation v the Cattaneo's law for temperature −kθ ,i q i τ 0 qi , 2.5 vi the entropy-strain-temperature-diffusion relation vii the equation of conservation of mass International Journal of Differential Equations viii the Cattaneo's law for chemical potential ix the chemical-strain-temperature-diffusion relation where β 1 3λ 2μ α t and β 2 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 − T 0 is small temperature increment, T is the absolute temperature of the medium, and T 0 is the reference uniform temperature of the body chosen such that |θ/T 0 | 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, 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.τ 0 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 γ 2 e kk nP dθ.

2.10
The alternative form can be written by substituting 2.10 into 2.1 -2.8 , where 12 are physical positive constants satisfying the following condition: Note that this condition implies that 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 γ 1 θ x γ 2 P x 0, in 0, × R , cθ t dP t q x γ 1 u xt 0, in 0, × R , τ 0 q t q kθ x 0, in 0, × R , dθ t nP t η x γ 2 u xt 0, in 0, × R , τη t η η x 0, in 0, × R ,

2.17
For the sake of simplicity, we present a short direct discussion of the the wellposedness 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 The initial-boundary value problem 2.15 1 -2.17 is equivalent to problem where

2.20
We consider the Hilbert space £ U ∈ L 2 0, 6 with inner product

2.22
The domain of A is On the other hand, if U satisfies 2.24 for U 0 defined in 2.18 , then 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 C 0 -semigroup of contractions.

2.26
Proof.i The density of D A in £ is obvious, and we have

2.29
Choosing Φ appropriately as in the proof of ii , the conclusion follows.
With the Hille-Yosida theorem see 19 C 0 -semigroups, we can state the following result.

Theorem 2.2. i
The operator −A is the infinitesimal generator of a C 0 -semigroup of linear contractions T t e −tA over the space £ for t ≥ 0. ii For any U 0 ∈ D A , there exists a unique solution U t ∈ C 1 0, ∞ ; £ ∩ C 0 0, ∞ ; D A to 2.24 given by U t e −tA U 0 .iii If U 0 ∈ D A n , n ∈ N, then U t ∈ C 0 0, ∞ ; D A n and 2.24 yields higher regularity in t.
Moreover, we will use the Young inequality 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 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.

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, .

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 , we get where Differentiating 2.15 with respect to t, we get in the same manner where

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 0

3.11
Using the estimates

3.14
Now, we conclude from 2.15 that

3.17
Using the estimates

3.18
Choosing δ such as

3.20
Now, we will show the main result of this section.

14
International Journal of Differential Equations 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 decays exponentially; that is,
Proof.Now, we define the desired Lyapunov functional N t .For ε > 0, to be determined later on, let

3.23
Then, we conclude from 3.1 -3.6 and 3.14 that

3.25
International Journal of Differential Equations 15 Using 2.34 , we obtain

3.26
Using 2.13 and choosing 0 < ε < 1 such that all terms on the right-hand side of 3.27 become negative, On the other hand, we have where C 1 , C 2 are determined as follows.Let with

3.37
Moreover, from 3.30 and 3.32 , we derive Although the estimate for τ 0 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 τ 0 , τ 0, 0 do not decay.Instead, the decay rate of the thermodiffusion system provides a better rate; that is, 4. The Limit Case τ 0 , 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 τ 0 , τ 0, 0 see 17 vanishes of order τ 2 0 τ 2 as τ 0 , τ → 0, 0 , provided the values at t 0 coincide.For this purpose, let U, Θ, φ, Φ, ψ denote the difference

Concluding Remarks
1 By comparison of the approximate value of c 0 ≈ 2.68 × 10 −56 with the value c 0 ≈ 1.75 × 10 −13 of the problem corresponding to τ 0 , 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 c 0 ≈ 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 .
The copper material was chosen for purposes of numerical evaluations.The physical constants given by Table1are found in 20 .Successively we can approximately compute ε 1 , C 1 , ε 2 , ε, c 1 , c 2 , C 0 , and c 0 from the previous equations, getting finally c 0 ≈ 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

Table 1 :
Values of the constants.
If E t denotes the energy of first order for U, Θ, φ, Φ, ψ ; that is, exponential decay of the solution corresponding to the problem when τ 0 , τ 0, 0 see 17 , we obtain a uniform bound on the right-hand side, ∃C > 0, ∀t ≥ 0 : E t ≤ C τ 2