Research Article | Open Access
Moncef Aouadi, "Exponential Stability in Hyperbolic Thermoelastic Diffusion Problem with Second Sound", International Journal of Differential Equations, vol. 2011, Article ID 274843, 21 pages, 2011. https://doi.org/10.1155/2011/274843
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 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.
The classical model for the propagation of heat turns into the well-known Fourier’s law where is temperature (difference to a fixed constant reference temperature), is the heat conduction vector, and 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 . In this theory, a modified law of heat conduction, the Cattaneo's law, 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 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  developed the classic theory of thermoelastic diffusion under Fourier’s law. Sherief et al.  derived the theory of thermoelastic diffusion with second sound. Aouadi  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  and Slemrod  and it has been studied in the book of Jiang and Racke  and the contribution of Lebeau and Zuazua . One should mention a paper by Sherief , where the stability of the null solution also in higher dimension is proved. We mention also the work of Tarabek  who studied even one dimensional nonlinear systems and obtained the strong convergence of derivatives of solutions to zero. Racke  proved the exponential decay of linear and nonlinear thermoelastic systems with second sound in one dimension for various boundary conditions. Messaoudi and Said-Houari  proved the exponential stability in one-dimensional nonlinear thermoelasticity with second sound. Soufyane  established an exponential and polynomial decay results of porous thermoelasticity including a memory term.
Recently, Aouadi and Soufyane  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 or , are close to each other, in the sense of energy estimates, of order and .
2. Basic Equations and Preliminaries
The governing equations for an isotropic, homogenous thermoelastic diffusion solid are as follows (see ):(i)the equation of motion (ii)the stress-strain-temperature-diffusion relation (iii)the displacement-strain relation (iv)the energy equation (v)the Cattaneo's law for temperature (vi)the entropy-strain-temperature-diffusion relation (vii)the equation of conservation of mass (viii)the Cattaneo's law for chemical potential (ix)the chemical-strain-temperature-diffusion relation where and , , and are, respectively, the coefficients of linear thermal and diffusion expansion and and are Lamé’s constants. is small temperature increment, is the absolute temperature of the medium, and is the reference uniform temperature of the body chosen such that . is the heat conduction vector, is the coefficient of thermal conductivity, and is the specific heat at constant strain. are the components of the stress tensor, are the components of the displacement vector, are the components of the strain tensor, is the entropy per unit mass, is the chemical potential per unit mass, is the concentration of the diffusive material in the elastic body, is the diffusion coefficient, denotes the flow of the diffusing mass vector, “” is a measure of thermodiffusion effect, “” is a measure of diffusive effect, and is the mass density. is the thermal relaxation time, which will ensure that the heat conduction equation will predict finite speeds of heat propagation. 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 The alternative form can be written by substituting (2.10) into (2.1)–(2.8), where 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  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 , and we study the exponential stability in one-dimension space. If , , and describe the displacement, relative temperature and chemical potential, respectively, our equations take the form where . The system is subjected to the following initial conditions: and boundary conditions
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 , let be defined as
We consider the Hilbert space with inner product Let such that The domain of is that is,
On the other hand, if satisfies (2.24) for 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 as a generator of a -semigroup of contractions.
(i) is dense in , and the operator is dissipative.
(ii) is closed.
Proof. (i) The density of in is obvious, and we have
Then is dissipative.
(ii) Let , and , as . Then, Choosing successively(1), (2),(3), (4),(5), (6),we obtain(1) and (first component),(2) and ,(3) and ,(4) and ,(5) and ,(6) and .
(iii) Choosing appropriately as in the proof of (ii), the conclusion follows.
With the Hille-Yosida theorem (see ) -semigroups, we can state the following result.
(i) The operator is the infinitesimal generator of a -semigroup of linear contractions over the space for .
(ii) For any , there exists a unique solution to (2.24) given by .
(iii) If , , then and (2.24) yields higher regularity in .
Moreover, we will use the Young inequality
The differential of (2.15)2 and (2.15)4 together with boundary conditions (2.17) yields Then, and defined by satisfy with , and the same differential equations (2.15)1–(2.17) as , but additionally, we have the Poincaré inequality for as well as for , , or .
In the sequel, we will work with and but still write and for simplicity until we will have proved Theorem 3.2.
From (2.15)3 and (2.15)5, we conclude Finally, for the sake of simplicity, we will employ the same symbols for different constants, even in the same formula. In particular, we will denote by the same symbol different constants due to the use of Poincaré's inequality on the interval .
3. Exponential Stability
Let be a solution to problem (2.15)1–(2.17). Multiplying (2.15)1 by , (2.15)2 by , (2.15)3 by , (2.15)4 by , and (2.15)5 by and integrating from 0 to , we get where Differentiating (2.15) with respect to , we get in the same manner where
Let us define the functionals
Proof. We will only prove (3.6) and (3.7) can be obtained analogously. Multiplying (2.15)1 by and using the Young inequality, we get
Multiplying (2.15)2 by and using the Young inequality, we get Substituting (2.15)1 in the above equation, yields
Using the estimates
Combining (3.9) and (3.13), we get
Now, we conclude from (2.15) that whence
Combining (3.14) and (3.16), we get our conclusion follows.
Using the estimates
Choosing such as yields Now, we will show the main result of this section.
Theorem 3.2. Let be a solution to problem (2.15)1–(2.17). Then, the associated energy of first and second order
decays exponentially; that is,
Bounds for and can be given explicitly in terms of the coefficient , and .
Proof. Now, we define the desired Lyapunov functional . For , to be determined later on, let
Then, we conclude from (3.1)–(3.6) and (3.14) that
Using (2.34), we obtain
Using (2.13) and choosing such that all terms on the right-hand side of (3.27) become negative, Choosing as in (3.27), we obtain from where which implies with
On the other hand, we have where are determined as follows. Let then with
Choosing we have Moreover, from (3.30) and (3.32), we derive with hence Applying (3.32) again, we have proved with and it holds.
Successively we can approximately compute , and from the previous equations, getting finally 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 Although the estimate for and 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 do not decay. Instead, the decay rate of the thermodiffusion system provides a better rate; that is, with initial conditions and boundary conditions In this case, we have where Using the Poincaré inequality, we get with
4. The Limit Case
We will show that the energy of the difference of the solution to (2.15)1–(2.17) and the the solution to the corresponding system with (see ) vanishes of order as , provided the values at coincide. For this purpose, let denote the difference then satisfies