## Abstract

We investigate the equation of the dual-phase-lag heat conduction proposed by Tzou. To describe this equation, we use the phase lag of the heat flux and the phase lag of the gradient of the temperature. We analyse the basic properties of the solutions of this problem. First, we prove that when both parameters are positive, the problem is well posed and the spatial decay of solutions is controlled by an exponential of the distance. When the phase lag of the gradient of the temperature is bigger than the phase lag of the heat flux, the problem is exponentially stable (which is a natural property to expect for a heat equation) and the spatial behaviour is controlled by an exponential of the square of the distance. Also, a uniqueness result for unbounded solutions is proved in this case.

## 1. Introduction

It is well known that the usual theory of heat conduction based on Fourier's law predicts an infinite heat propagation speed. Heat transmission at low temperature has been observed to propagate by means of waves. These aspects have caused intense activity in the field of heat propagation. Extensive reviews on the so-called second sound theories (hyperbolic heat conduction) are given by Chandrasekharaiah (1998) and in the books of Müller & Ruggeri (1998) and Jou *et al*. (1996). A theory of heat conduction in which the evolution equation contains a third-order derivative with respect to time was proposed by Ghaleb & El-Deen Mohamedein (1989). Several instability results have been obtained for the theory (e.g. Franchi & Straughan 1994; Quintanilla 1997) as well as the proof of the non-existence of global solutions in the nonlinear theory by Quintanilla & Straughan (2002). Tzou (1995*a*) proposed a theory of heat conduction in which the Fourier law is replaced by an approximation of the equation(1.1)where *τ*_{q} is the phase lag of the heat flux; and *τ*_{θ} is the phase lag of the gradient of the temperature. The relation (1.1) states that the gradient of the temperature at a point in the material at time *t*+*τ*_{θ} corresponds to the heat flux vector at the same point at time *t*+*τ*_{q}. The delay time *τ*_{θ} is caused by microstructural interactions, such as phonon scattering or phonon–electron interactions. The delay *τ*_{q} is interpreted as the relaxation time due to fast-transient effects of thermal inertia.

Instead of Fourier's law, being equivalent to assuming *τ*_{q}=*τ*_{θ}=0 and leading to the paradox of infinite propagation speed, we consider the model proposed by Tzou (1995*a*), where *τ*_{q}>0, *τ*_{θ}>0 are positive relaxation times, and where a second-order approximation for ** q** and

*θ*is used, turning (1.1) into(1.2)

Thus, in this paper, we consider the theory developed by taking a Taylor series expansion on both sides of (1.1) and retaining terms up to the second-order in *τ*_{q} and *τ*_{θ}. This equation has not received much attention in the literature (until now). However, we can recall that the spectrum was studied by Quintanilla & Racke (2006*b*).

A natural question is the determination of the time parameters *τ*_{q} and *τ*_{θ} (see Hetnarski & Ignaczak 1999), and our work is motivated by this question. One might expect that the mathematical analysis of existence, uniqueness and stability issues, for example, would furnish certain restrictions on the parameters. One property to hold for solutions of a heat equation should be exponential stability (or at least stability). In Quintanilla & Racke (2006*b*), stability (for the heat conduction) was established whenever(1.3)

Thus, under condition (1.3), one has a heat theory with a third-order derivative in time in the equation that predicts stability. This is of interest in the light of the results obtained in the theory proposed by Ghaleb & El-Deen Mohamedein (1989). By means of several exact solutions, instability of solutions was also established by Quintanilla & Racke (2006*b*), whenever condition (1.3) is violated. Thus, one may assume that condition (1.3) must be satisfied in order to use this model to describe heat transmission.

In this paper, we study four kinds of questions. First, to determine the suitable frame where the third-order problem of heat conduction is well posed. Second, to prove the exponential stability. Third, to determine the spatial behaviour of the solutions of the heat conduction in a semi-infinite cylinder in ^{3}. The last goal is to obtain a uniqueness result in the case of unbounded domains and when we assume that the solutions can be unbounded at infinity.

It is worth recalling that in the case that we only consider the development until the first order in *τ*_{θ}, we obtain a hyperbolic theory, which has been studied and analysed by Quintanilla (2002), Horgan & Quintanilla (2005) and Quintanilla & Racke (2006*a*). Our aim here is to continue the clarification of these theories and to see what happens in the new theory. We believe that the mathematical analysis will help us to understand the applicability of these theories.

This paper is organized as follows. In §2, we settle the field equations and the boundary and initial conditions of the problem we consider in this paper. A uniqueness and existence result is proved in §3. In §4, we prove the exponential stability. In §5, we obtain some results of Saint-Venant's type concerning the spatial behaviour of solutions in a semi-infinite cylinder and some improvements of them as obtained in §6. Section 7 is devoted to the uniqueness of solutions when the solutions can grow in a fast way at infinite.

We shall denote the three-dimensional semi-infinite cylinder by *R*, with cross-section *D*. The finite end face of the cylinder is in the plane *x*_{3}=0. The boundary ∂*D* is supposed regular enough to allow the use of the divergence theorem. We denote the set of points of the cylinder *R* by *R*(*z*), such that *x*_{3} is greater than *z*, and the cross-section of the points by *D*(*z*), such that *x*_{3}=*z*.

## 2. Preliminaries

In this paper, we study solutions *θ*=*θ*(** x**,

*t*) of the heat equation(2.1)We have used the notation(2.2)where

*τ*

_{θ}>0 and

*τ*

_{q}>0 are the dimensionless time lag parameters.

We study the qualitative behaviour of the classical solutions subject to the initial conditions(2.3)and the boundary conditions(2.4)(2.5)where the prescribed boundary data *g*(*x*_{α},*t*) on the end *x*_{3}=0 is such that *g*(*x*_{α},0)=*θ*^{0}(*x*_{α}) and *g* are assumed to vanish on ∂*D*(0)×[0,∞).

Observe that in the limit as *τ*_{θ} and *τ*_{q} tend to zero, we recover from (2.1) the usual heat equation. In this limit, the existence, stability and the spatial evolution of solutions has been studied in a variety of contexts. When *τ*_{q} and *τ*_{θ} are positive, the results to be described in the sequel will be seen to be similar to those obtained previously for such equations.

In some parts of the paper (see §§4, 6 and 7), we will assume that *τ*_{θ}>*τ*_{q}. We note that from the empiric point of view, this is a rational assumption, as it can be seen in Tzou (1995*a*,*b*). We note that our equation can be written as(2.6)where(2.7)As *τ*_{θ}>*τ*_{q}, the constants *B* and *C* are positive.

In the course of our calculations, we will use the knowledge of the eigenvalues of the real symmetric positive definite matrix(2.8)It is clear that one of the eigenvalues is *m* and the others are the eigenvalues of the matrix(2.9)The eigenvalues of this matrix are(2.10)so that the smallest eigenvalue is(2.11)When(2.12)the matrix(2.13)is defined positive and the minimum eigenvalue can be calculated by means of the arguments proposed previously. We shall denote the smaller eigenvalue by *λ*_{γ}.

## 3. Well-posedness

We shall formulate the problem for the semi-infinite cylinder *R* in three space dimensions, but the well-posedness holds for general domains.

The well-posedness result for this system—not being trivial for third-order in time systems, cf. p. 125 in Goldstein (1985)—can be achieved by an appropriately sophisticated choice of variables and spaces that reflect the special structure of the system.

We first transform systems (2.1)–(2.5) to zero boundary conditions on all of ∂*R* by defining(3.1)and using *θ* instead of *ψ* again, we obtain the initial boundary-value problem(3.2)(3.3)(3.4)where the given heat supply *p* arises from transformation (3.1) in terms of the boundary data *g*.

We remark that finding a solution *θ* to (3.2)–(3.4) allows to determine the desired solutions *θ* of the original system.

Defining ** V**≔(

*θ*,

*θ*

_{t},

*θ*

_{tt},)′ we obtain(3.5)with the (yet formal) differential operator

*A*given by the symbol(3.6)the right-hand side

**given by**

*F***≔(0, 0,**

*F**p*)′ and the initial value

*V*^{0}(

**)≔(**

*x**θ*,

*θ*

_{t},

*θ*

_{tt},)′(

**, 0), with its components being given in terms of the originally prescribed initial data in (3.3) by using the differential equations.**

*x*As underlying Hilbert space, we choose with inner productwhere 〈., .〉 denotes the usual *L*^{2}(*R*)-inner product. The operator *A* is now given aswith

*There exists a constant c*_{1}>0, *such that for all V*∈*D*(*A*)*holds*.

We have (without loss of generality for a real inner product)which was possible owing to the specific choice of the inner product. Sincewhere *ϵ* is an arbitrary positive constant and, if we takewe see thatwhich implies the assertion with ▪

As a consequence, we see that for *d*>*c*_{1}, the operator *A*−*d* is dissipative and invertible.

*For all d*>*c*_{1}, *we have that the range of A*−*d is all of* .

The solvability of (*A*−*d*)*V*=*F* is equivalent to solving(3.7)(3.8)(3.9)Eliminating *V*^{2} and *V*^{3}, we have to solve(3.10)First, let . Hence, we consider for *G*^{1}∈*L*^{2}(*R*) the equation(3.11)where *γ*_{1} and *δ*_{1} are positive. The existence of solving (3.11) is well known, then *V*^{2} and *V*^{3} both in can be determined from equations (3.7) and (3.8), respectively, and *V*∈*D*(*A*) will solve (*A*−*d*)*V*=*F*. By a density argument using the continuity of (*A*−*d*)^{−1}, we conclude the solvability for any *F*∈. ▪

Now we conclude from lemmas 3.1 and 3.2 that *A* generates a *C*_{0}-semigroup, and hence the initial (boundary) value problem (3.5) is uniquely solvable.

*For any F*∈*C*^{0}([0,∞), *D*(*A*)) *or F*∈*C*^{1}([0,∞), ) *and any V*^{0}∈*D*(*A*), *there is a unique solution V to* (3.5) *with V*∈*C*^{1}([0,∞), )∩*C*^{0}([0,∞), *D*(*A*)).

The well-posedness consideration in this section extends naturally to other domains *Ω*⊂^{n}, *n*=1, 2, 3, instead of the three-dimensional cylinder *R*.

## 4. Exponential stability

The aim of this section is to show that the solutions of the problem decay in an exponential way whenever we assume that *τ*_{θ}>*τ*_{q}. In this section we assume *g*=0.

From relation (2.6), we know that the following relation,(4.1)is satisfied for a solution.

As we assume that *τ*_{θ}>*τ*_{q}, we note that(4.2)and that the matrix(4.3)is positively definite. If we define the function(4.4)we see that(4.5)

It is clear from the positivity of *P* that we can obtain two positive constants *M* and *m*, such that(4.6)Poincaré's inequality implies that(4.7)where *C* is a positive constant, which depends on the cross-section of *R* and can be calculated. Thus, observing (4.2), we get the existence of a positive constant *N*, such that(4.8)It follows the existence of a positive constant *K*, such that(4.9)This inequality leads to the exponential decay estimate(4.10)Thus, observing (4.6), we have proved forthe following theorem.

*Let θ be a solution of the problem determined by equation* (2.1), *the initial conditions* (2.3) *and the boundary conditions* (3.4). *Then, the energy term E decays exponentially*,

## 5. Some spatial estimates

In this section we establish the results on the spatial evolution of solutions of (2.1), (2.2), (2.4) and (2.5), provided that the initial data (2.3) is assumed to be zero. To this end, we only need that the two parameters *τ*_{q} and *τ*_{θ} are positive. It is worth recalling that the state of the art of the spatial behaviour of several thermal problems can be found in the papers by Horgan & Knowles (1983), Horgan *et al*. (1984), Horgan (1989, 1996), Quintanilla (1997, 2002) and Horgan & Quintanilla (2005).

The analysis begins by considering the function(5.1)where the positive constant *γ* satisfies (2.12).

Using the differential equation (2.1) and the partial integration, we have(5.2)Thus,(5.3)Our next step is to establish an inequality between |*F*_{γ}(*z*,*t*)| and spatial derivatives of *F*_{γ}(*z*,*t*). Applying Schwarz's inequality in (5.1), we get(5.4)where *ϵ* is an arbitrary positive constant and *λ*_{γ} was defined in §2. If we takeit follows that(5.5)where *β*_{γ}=*ϵ*^{−1}. This inequality is well known in the study of spatial decay estimates. It implies that(5.6)From (5.6), we can obtain an alternative of Phragmen–Lindelöf type, which states (see Flavin *et al*. 1989) that the solutions either grow exponentially for *z* sufficiently large or solutions decay exponentially in the form(5.7)for all *z*≥0, where(5.8)

Thus, we have proved the following theorem.

*Let θ be a solution of the initial boundary-value problem* (2.1), (2.2), (2.4), (2.5) *and null initial conditions. Then either the solutions grow exponentially or the estimate* (5.7) *is satisfied, where E _{γ}=−F_{γ} is defined in* (5.8).

We note that this result gives an answer to the question proposed by Hetnarski & Ignaczak (1999, p. 474), a principle of Saint-Venant's type in this theory.

## 6. A faster spatial estimate

It is clear that when *τ*_{q}=*τ*_{θ}, our equation is very similar to the usual heat equation, cf. (3.6). For the classical heat equation, we know that the decay is of the type of a exponential of the square of the distance. This suggests that the spatial decay (5.7) could be improved in general. To point out this fact, we concentrate our attention to the case that *τ*_{θ}>*τ*_{q}. In this section we assume that *τ*_{θ}>*τ*_{q} and we will prove that the spatial decay can be controlled by an exponential of the square of the distance.

Let us define the function(6.1)We have(6.2)and(6.3)We get(6.4)where *λ*_{1} is the first eigenvalue of the cross-section and *d*=*k*^{−1}*A*.

If we define the (non-negative) functional(6.5)we have that *P*(*z*, *t*) satisfies the second-order differential inequality(6.6)(6.7)Now, we show that the function *P*(*z*, *t*) satisfying (6.6) can be bounded above by the solutions to a related initial boundary-value problem for the one-dimensional heat equation. The argument here follows that of Horgan *et al*. (1984). By virtue of its definition, *P*(*z*, *t*) satisfies the initial condition(6.8)and the boundary condition(6.9)where(6.10)As the aim of this section is to obtain an estimate for the spatial decay of the solutions, we start with assuming the following asymptotic behaviour for *P*(*z*, *t*):(6.11)Thus, the temperature field, satisfying (2.1), (2.4) and (2.5) and null initial conditions, is assumed to vanish in a weighted mean-square sense as the axial variable tends to infinity. Let(6.12)Then, it follows from (6.6), (6.9), (6.10) and (6.11) that *v*(*z*, *t*)≥0 satisfies(6.13)(6.14)(6.15)(6.16)An upper bound for *v*(*z*, *t*) in terms of the solution of an initial boundary-value problem for the one-dimensional *heat equation* now follows immediately from the maximum principle. Let *w*(*z*, *t*) be such that(6.17)(6.18)(6.19)(6.20)The maximum principle for the heat equation now yields(6.21)and hence, from (6.12), we find that(6.22)The representation for *w*(*z*, *t*) that is useful for our purposes (see Sokolnikoff & Redheffer 1966) is(6.23)where the non-negative function *G*(*z*, *t*) is given by(6.24)Here, the complementary error function erfc(*x*) is defined by(6.25)Thus, on using (6.22), we get the upper bound(6.26)On recalling the definition of *P*(*z*, *t*) in (6.5), the result (6.26) can be written directly in terms of the solution *θ*(** x**,

*t*) to the original problem as(6.27)where

*G*(

*z*,

*t*) is given in (6.24).

The result (6.27) provides a weighted *mean-square* estimate for the solution subject to the assumption that *τ*_{θ}>*τ*_{q}.

The arguments by Horgan *et al*. (1984) can be used to show that estimate (6.27) implies that the spatial decay of end effects in the transient problem is *faster* than that for the steady state. To see this, we use the fact that(6.28)and hence, since *G*(*z*, *t*) is monotonically increasing in *t*, we obtain(6.29)Thus, we have established that the *rate of spatial decay is at least as fast as*(6.30)This decay rate is (optimal) the one for the steady state for *θ*(** x**).

If we employ the inequality(6.31)(see Abramowitz & Stegun 1965, p. 298) in (6.27), then, for , (6.24) and (6.27) yield the estimate*This result shows that, for fixed t, the spatial decay is ultimately controlled by the factor*(6.32)*rather than the factor* *found in the steady-state case*.

A natural question is to relate the estimate for with *θ*. To this end, we recall that(6.33)It is worth noting that the expressionis positive in the sense that it is equivalent to the measure defined by .

If we assume null initial conditions, then we obtain the spatial estimate in the usual norm.

## 7. Uniqueness in exterior domains

The aim of this section is to prove a uniqueness result for the solutions of equation (2.1) whenever we assume that the solutions do not grow faster than the exponential of the square of the distance from the origin. Thus, we assume here that *R* is an unbounded domain and that the solutions satisfy the asymptotic conditions(7.1)where *k*^{**} is a positive constant and *r*^{2}=*x*_{i}*x*_{i}.

In this section we also assume that *τ*_{θ}>*τ*_{q}. To prove a uniqueness result, it is sufficient to show that the only solution of the null initial and boundary conditions is the null solution.We define the function(7.2)whereIn view of conditions (7.1) on the asymptotic behaviour, this function is well defined whenever *t* is sufficiently small. We have(7.3)where(7.4)and(7.5)Using (4.1) and the Dirichlet boundary condition, we have(7.6)where(7.7)and(7.8)This implies(7.9)and, by the choice of *β*, we conclude *F*′(*t*)≤0 for every *t*. After integration, we obtain *F*(*t*)≤*F*(0). As we assume null initial conditions, we obtain that in an interval with a non-zero measure. It is clear that the only solution to the problem determined by the equation with null initial conditions is the null solution and then we obtain that *θ*≡0 in an interval. We can repeat this argument to obtain that *θ*=0, for all *t*≥0, and the uniqueness result is proved.

## 8. Conclusions

In this paper, we have investigated the dual-phase-lag heat equation proposed by Tzou (1985*a*), which is described by (2.1) and (2.2). In the description of this equation, the parameters *τ*_{θ} and *τ*_{q} are fundamental, and they are related to the microstructural interactions. We have proved that whenever both parameters are positive, the problem is well posed and the spatial decay of solutions is controlled by a negative exponential of the distance. When *τ*_{θ}>*τ*_{q}, we have found that the problem is exponentially stable and that the spatial decay of solutions is controlled by an exponential of the square of the distance (see equation (6.32)). We finished the paper by proving the uniqueness of solutions for unbounded regions whenever the solutions are controlled by the exponential of the square of the distance (see equation (7.1)).

## Acknowledgements

R.Q. was supported by the project ‘Estudio Cualitativo de Problemas Termomecánicos’ (MTM2006-03706) of the MEC. R.R. was supported by the DFG-project ‘Hyperbolic Thermoelasticity’ (RA 504/3-1).

## Footnotes

- Received July 19, 2006.
- Accepted October 4, 2006.

- © 2006 The Royal Society