## Abstract

In the middle of the 1990s, Green & Naghdi proposed three theories of thermoelasticity that they labelled as types I, II and II. The type II theory, which is also called thermoelasticity without energy dissipation, is conservative and the solutions cannot decay with respect to time. It is well known that, in general, in the linear theories of thermoelasticity of types I and III, the solutions decay with respect to time. In many situations this decay is at least exponential. In this paper we study whether this decay can be fast enough to guarantee the solutions to be zero in a finite time. We investigate the impossibility of the localization in time of the solutions of linear thermoelasticity for the theories of Green & Naghdi. This means that the only solution that vanishes after a finite time is the null solution. The main idea is to show the uniqueness of solutions for the backward in time problem. To be precise, for type III thermoelasticity we will prove the impossibility of localization of solutions in the case of bounded domains, and for the type I thermoelasticity in the case of exterior domains, even when the solutions can be unbounded, whether the spatial variable goes to infinity.

## 1. Introduction

In the middle of the 1990s, Green & Naghdi proposed several thermomechanical theories (see Green & Naghdi 1992, 1993, 1995*a*,*b*, 1996) where the heat conduction was different from the usual one. In particular, they considered three theories of thermoelasticity which they labelled as types I, II and III. These theories were based on an entropy balance law rather than the usual entropy inequality. Their thermodynamics makes use of the thermal displacement *τ* that satisfies where *θ* is the temperature. It is worth noting that the field equations of the linear version of the type I theory agree with the equations of classical thermoelasticity. Type II thermoelasticity, also known as *thermoelasticity without energy dissipation*, is a limiting case of the type III theory and satisfies the conservation of energy. These theories have deserved much attention in recent years, and much research has been developed to understand them (see Quintanilla 1999, 2002; Quintanilla & Straughan 2000, 2004, 2005*a*,*b*; Quintanilla & Racke 2003).

Generally speaking, for types I and III the energy decays with respect to time (Racke 1987, 1990; Lebeau & Zuazua 1999; Quintanilla & Racke 2003; Zhang & Zuazua 2003; Reissig & Wang 2005; Yang & Wang 2006). That is, for a generic class of domains, we can expect that the solutions tend to zero when the time tends to infinity. Moreover, it is known that in many situations the decay can be exponential. Thus, it is natural to enquire whether the decay can be so fast that the solution becomes zero in a finite time. In this paper we prove the impossibility of localization of solutions in a finite time for the type III thermoelasticity in the case of bounded domains. The same is proved for type I thermoelasticity in the case of exterior domains for solutions which can be unbounded whether the spatial variable goes to infinity. Results of this kind are a good complement to the decay (with respect to time) results. Previous contributions give upper bounds for the temporal decay of solutions; here we are going to give information on the lower bounds of the decay of solutions. In fact, we shall prove that the solutions cannot vanish after a finite interval of time. Note that the main tool to prove the impossibility of localization will be to show the uniqueness of solutions for the backward in time problem. In this sense, it is worth recalling the contribution of Ciarletta (2003), in which he proved the uniqueness of solutions for the backward in time problem of the usual thermoelasticity and then the impossibility of localization for classical thermoelasticity. Thus, our contribution can be seen as extensions of this result along two trends: the case of the type III thermoelasticity which contains the type I as a limiting case and the case where the domain and the solutions can be unbounded when the spatial variable tends to infinity. The author has recently proved (Quintanilla in press) the impossibility of localization of solutions in the case of thermoelasticity with voids. We also recall the contribution of Lasiecka *et al*. (2001) concerning uniqueness of solutions for the backward in time problem for several thermoelastic situations. However, this contribution proposes an alternative approach for strongly continuous semigroups. In fact, it is based on the assumption that the resolvent operator of the semigroup is bounded on suitable rays of the complex plane.

As in the contribution of Ciarletta, our approach is based on the Lagrange identities (see Brun 1969) and energy arguments. However, we note that our arguments do not agree with those proposed by Ciarletta. This is because we successfully treat the heat capacity as being *strictly positive* and introduce more sophisticated energy functions. Then, it is possible to apply the arguments to the type III thermoelasticity or in the case of unbounded domains. Our approach needs several new arguments, different from the usual ones used by Ciarletta. Thus, this the paper contains some new ideas in the study of uniqueness results in thermoelasticity. For instance we point out the use of the function (*t*) proposed in (3.22) (where *E*_{0} is defined at (3.19)) for type III thermoelasticity, and the use of the weight functions in the way proposed by Galdi *et al*. (1986) for type I thermoelasticity. Nevertheless, our arguments are closely related to those proposed by Ciarletta.

In §2 we recall the field equations and the basic assumptions in the theories of Green & Naghdi. In §3 we give the proof of the impossibility of localization of solutions in type III thermoelasticity. Section 4 proves the impossibility of solutions in the classical theory in the case of unbounded domains and solutions.

## 2. Basic equations

The aim of this section is to state the basic equations and assumptions we are going to work with in this paper.

In §3, we consider a body that at some instant occupies a bounded region *B* of the Euclidean space, with a smooth boundary. But in §4 we will consider an unbounded region that we shall also denote by *B*.

We recall that in the absence of supply terms the field equations that govern the problem of the thermoelasticity for the Green & Naghdi theories are(2.1)(2.2)Here *ρ* is the mass density; *c* is the heat capacity; *C*_{ijkl} is the elasticity tensor; *b*_{ij} is the heat conductivity tensor; *β*_{ij} is the dilatation tensor; and *k*_{ij} is a tensor which is typical in the Green & Naghdi theories. As usual *u*_{i} means the displacement vector and *τ* is the thermal displacement that satisfies , with *θ* being the temperature.

It is worth recalling that this system of equations is the general type III thermoelasticity system. Two limiting cases are also relevant. When *k*_{ij}=0, we obtain the type I thermoelasticity that agrees with the system of equations of the classical linear thermoelasticity. In this case equation (2.2) becomes(2.3)

The other limiting case proposed by Green & Naghdi is when *b*_{ij}=0. In this case, we have(2.4)The system of equations (2.1) and (2.4) defines the theory called type II thermoelasticity. It is also known as the system of *thermoelasticity without energy dissipation*. We point out that in this theory the impossibility of localization is well known, as is shown by Quintanilla & Straughan (2000). Thus, from now on, we restrict our attention to the type I and III theories.

Here, we set down the basic assumptions we are going to work with.

All the constitutive tensors are bounded above.

The mass density

*ρ*and the heat capacity*c*are strictly positive. That is,(2.5)The symmetry relations(2.6)are satisfied.

The elasticity tensor

*C*_{ijkl}and the heat conductivity tensor*b*_{ij}are strictly positive. That is, the inequalities(2.7)are satisfied for every tensor*ξ*_{ij}and*ξ*_{i}, where*C*_{1}and*C*_{2}are strictly positive.

The assumptions above are in agreement with the physical experience. The thermomechanical interpretation of conditions (ii) is obvious. Assumption (iv)_{2} is related to the defining property of a definite mechanical heat conductor and assumption (iv)_{1} is related to the positivity of the internal energy and may be interpreted with the help of the theory of mechanical stability. The symmetries (iii) are natural in the context of the thermomechanical theories.

Note that we do not impose any positivity assumption on the tensor *k*_{ij}. In particular, our assumptions are adequate in the case of type I thermoelasticity, because it corresponds to the case *k*_{ij}=0.

When we work with type III thermoelasticity, we impose initial conditions for and *θ*. However, we do not impose any initial conditions on *τ* for the type I.

## 3. Thermoelasticity of type III

In this section, we prove the impossibility of localization in time of the solutions of type III thermoelasticity when *B* is a bounded domain.

To define our problem, we impose the boundary conditions(3.1)

To prove the impossibility of localization, we will show the uniqueness of solutions for the backward in time problem. The system of equations which governs the backward in time problem is given by(3.2)

(3.3)

To prove the uniqueness it is sufficient to show that the only solution for the homogeneous initial conditions,(3.4)is the null solution.

To reach the main aim of this section, it will be useful to state and prove the following.

*Let* (*u*_{i}, *τ*) *be a solution of the problem determined by system* *(3.2)*, *(3.3)*, *initial conditions* *(3.4)* *and boundary conditions* *(3.1)*. *Then* (*u*_{i}, *τ*)=(0, 0) *for every t*≥0.

First of all we define several functions and compute their time derivatives. We define(3.5)Using the evolution equations, the boundary conditions and the divergence theorem, we find(3.6)Considering now(3.7)we obtain(3.8)We also need a well-known identity in type III thermoelasticity. To be self-contained we will derive it. However, we point out that it can also be obtained in Quintanilla & Straughan (2000). For a fixed *t*∈(0,*T*), we form the identities(3.9)(3.10)(3.11)(3.12)Now, we form the combination (3.9)+(3.11)−(3.10)−(3.12). After some integration and the use of the initial conditions, we find(3.13)From this relation, we see that(3.14)

Now, we multiply equation (3.3) by *τ* and equation (3.2) by −*u*_{i}. Integrating we obtain(3.15)and(3.16)After addition and using the relation (3.13), we see that(3.17)Now, we form several relations. Let 0<*ϵ*<1 be a constant; we have(3.18)Now, let *λ* be a positive constant. We form(3.19)where(3.20)We note that(3.21)We point out that we can select *λ* large enough to guarantee thatis a positive definite function.

We consider the function(3.22)We have that(3.23)As we assume that *τ*(0)=0, we can use the Poincaré inequalityThen(3.24)Thus, whenever we assume that *t*≤*t*_{0}, where *t*_{0} is sufficiently small, we can guarantee the existence of a positive constant *K* such that(3.25)

Here *λ*^{*} is a positive constant which can be selected to guarantee that all the integrands have the same dimension. We point out that *K*>0 depends on the constitutive coefficients, *λ*, *λ*^{*}, *ϵ* and *t*_{0}, but as we do not need the exact expression to prove our result we skip this tedious and cumbersome calculation.

Using the null initial condition assumptions, we have(3.26)

From the A–G mean inequality, we can obtain the following expressions:Here *K*_{1}, *K*_{2} are two positive constants that can be easily calculated.

From these relations, we can obtain the existence of a positive constant *K*^{*}=*K*^{−1}(*K*_{1}+*K*_{2})>0 such that(3.27)for every 0≤*t*≤*t*_{0}. After a quadrature, it follows that:(3.28)From the definition of (*t*), we have (0)=0 and then (*t*)=0 for every 0≤*t*≤*t*_{0}. It follows that for every 0≤*t*≤*t*_{0}. We can prove that for every 0≤*t*≤2*t*_{0}. It is clear that this process can be used to prove that for every 0≤*t*<∞. Therefore, the lemma is proved. ▪

From this lemma we obtain the following theorem.

*Let* (*u*_{i}, *τ*) *be a solution to the problem determined by system* *(2.1), (2.2)*, *boundary conditions* *(3.1)* *and initial conditions*(3.29)*that vanishes after a finite time. Then*, (*u*_{i}, *τ*) *is the null solution for every t*≥0.

## 4. Thermoelasticity of type I: exterior domain

In this section we prove that the only solution to the system of classical thermoelasticity (2.1), (2.3), which vanishes after a finite time in the case of unbounded domains, is the null solution. In this section we assume that *B* is an exterior domain and that the solutions satisfy the spatial asymptotic conditionsfor every 0<*α*≤*α*_{0} where *α*_{0} is a positive constant, *r*^{2}=*X*_{i}*X*_{i}, and(4.1)It is worth noting that this kind of condition has been considered by Galdi *et al*. in several works (see Galdi *et al*. 1986). They gave sufficient conditions to guarantee it. For instance, they showed that whenever is asymptotically as *r*^{−1/2−δ}, where *δ* is a positive constant arbitrarily small, then the limit (4.1) holds. To guarantee the uniqueness of solutions it is natural to impose spatial asymptotic conditions on the class of functions we work with. For instance, the uniqueness of solutions fails for the usual heat conduction equation if we relax the asymptotic spatial behaviour (see John 1982, pp. 210, 211). In fact, the assumption we impose here is that the growth when the spatial variable is unbounded must be controlled by every positive exponential. Thus, *α*_{0} does not play any relevant role. However, we point out that the functions which tend (in the spatial variable) to infinity as a polynomial satisfy the required conditions.

Again, we note that to show the impossibility of localization is equivalent to proving the uniqueness of solutions for the system(4.2)(4.3)To make the calculations easier we assume that the constitutive coefficients are independent of the material point. However, the analysis could be extended to the case of non-homogeneous materials.

*Let* (*u*_{i}, *θ*) *be a solution of the problem determined by system* *(4.2), (4.3)*, *initial conditions* *(3.4)*, *boundary conditions* *(3.1)* *and asymptotic conditions* *(4.1)*. *Then* (*u*_{i}, *θ*)=(0, 0) *for every t*≥0.

We consider the function(4.4)We have(4.5)

Now, we define the function(4.6)We obtain(4.7)We also need several relations. For a fixed *t*∈(0, *T*), we form the identities(4.8)(4.9)(4.10)(4.11)From the previous relations, we obtain(4.12)where *C*_{3} can be calculated. Here, it is relevant to note that *C*_{3} depends on the constitutive coefficients and *α*. This constant is bounded above as *α* tends to zero. Let us consider *ϵ* a positive constant, but strictly less than one. We form the function(4.13)It is clear that there exists a positive constant *C* such that(4.14)In view of the null initial conditions, we have(4.15)At this point it is suitable to point out that we can obtain a positive constant *K*_{1} such thatIn view of (4.4) and (4.7), we can obtain a positive constant *K*_{2} such thatwhere *K*_{2} is bounded above when *α* tends to zero.

From the previous relations and inequality (4.12), we can obtain the estimate(4.16)where *K*^{*}=*C*^{−1}(*K*_{2}+*C*_{3}) is bounded as *α* tends to zero and(4.17)As in §3 we do not calculate a bound for *K*^{*}. This is because the calculation would be very cumbersome and tedious, and it would not be relevant to the proof of the result.

After a quadrature we have the estimate(4.18)If we select *t* small enough to guarantee that 1−2*tK*^{*}>0, we see(4.19)For every *R*>0 we have(4.20)where *B*(*R*) denotes the set of points in *B* that are at a distance less than or equal to *R* from the origin. In view of condition (4.1), it follows that:(4.21)for every *R*>0. It shows that *u*_{i}=*θ*=0 for every * X*∈

*B*(

*R*). Taking the limit as

*R*goes to infinity, we deduce that the solution is zero in

*B*for every

*t*small enough. Thus, we have proved the following theorem. ▪

*Let* (*u*_{i}, *θ*) *be a solution of the problem determined by the system* *(2.1), (2.3)*, *boundary conditions* *(3.1)*, *the asymptotic conditions* *(4.1)* *and the initial conditions*(4.22)*that vanishes after a finite time. Then*, (*u*_{i}, *θ*) *is the null solution for every t*≥0.

We believe that the analysis used in this section can be adapted to the study of type III thermoelasticity.

## Acknowledgments

This work is partially supported by project ‘Qualitative study of thermomechanical problems’ (MTM2006-03706) supported by the Education and Science Spanish Ministry and the European Regional Development Fund.

## Footnotes

- Received June 14, 2007.
- Accepted September 7, 2007.

- © 2007 The Royal Society