International Journal of Non-Linear Mechanics 46 (2011) 1252–1257

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International Journal of Non-Linear Mechanics

0020-74

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/nlm

On the spatial behavior of solutions for non-linear type II heat conduction

Antonio Magana �, Ramon Quintanilla

Dept. Matematica Aplicada 2, UPC C. Colon 11, 08222 Terrassa, Barcelona, Spain

a r t i c l e i n f o

Article history:

Received 1 February 2011

Received in revised form

27 May 2011

Accepted 7 June 2011Available online 3 July 2011

Keywords:

Type II thermoelasticity

Energy methods

62/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ijnonlinmec.2011.06.003

esponding author: Tel.: þ34 93 739 8206; fa

ail address: antonio.magana@upc.edu (A. Mag

a b s t r a c t

In this paper we study the spatial behavior of the solutions for a problem determined by the non-linear

version of the Green and Naghdi type II heat conduction theory. We obtain a spatial decay estimates for

the usual boundary-initial-value problem and also an upper bound for the amplitude term of the spatial

estimate. Finally, we analyze a non-standard initial value problem defined on a particular family of heat

conductors.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Maxwell [30] pointed out that the classical linear theory ofheat conduction based on Fourier’s law predicts that a thermalperturbation at some point in a material body will be feltinstantly at all other points of the body, however distant theyare. This is referred to as the paradox of heat conduction. It isphysically unrealistic since it implies that thermal signals propa-gate with infinite speed. Thus, given the classical theory’snon-causal nature, and causality’s fundamental role in modernphysics, different theories of heat conduction have been put forthover the course of the 20th century (see, for example, [13,14]and the references cited therein). In the book of Ignaczack andOstoja-Starzewski [20] several studies concerning applicability ofnon-classical thermoelastic theories are proposed. Green andNaghdi [9–12] proposed some of these thermomechanical theorieswhere the heat conduction does not agree with the usual one (seealso [36]). They considered three theories labelled as type I, II and III,respectively. These theories were based on an entropy balance lawrather than the usual entropy inequality. However, we want tomention the total compatibility of the entropy balance law with theentropy inequality. Their thermodynamics proposed the use of thethermal displacement

aðx,tÞ ¼

Z t

t0

yðx,sÞ dsþa0,

where y is the empirical temperature.The type I thermoelasticity coincides with the classical one; in

type II, the heat is allowed to propagate by means of thermal

ll rights reserved.

x: þ34 93 739 8101.

ana).

waves but without dissipating energy and for this reason it is alsoknown as thermoelasticity without energy dissipation; the type IIIthermoelasticity includes the two above-mentioned theories asparticular cases. All these theories have recently been the aim ofgreat amount of works (as a matter of illustration see[2–5,19,21–24,26,29,33–35]. In this paper we study the non-linearversion of the type II heat conduction.

The study of edge effects in several thermomechanical situationshas deserved much attention in the last three decades. They weremotivated by the desire of giving a precise mathematical version ofthe result known in elastostatics as Saint-Venant Principle. Thisinterest was later extended to other thermomechanical fields and,in general, to abstract partial differential equations. Several decayestimates for elliptic [7], parabolic [15,16], hyperbolic [17,32] and/orcombination of them [27] have been obtained. In these contributionsthe authors obtain growth/decay estimates for the solutions.

It is worth noting that there are few contributions about thespatial decay in non-linear dynamical problems. This fact appliesto parabolic problems, but it is more marked for hyperbolicproblems. Flavin et al. [8] provided a method for hyperbolicproblems that was later completed by Chirita and Quintanilla [6].The former contribution applied for non-linear problems, but thekind of non-linearity was very restrictive and, then, their techni-que was used later only for linear problems. General non-linearproblems add extra difficulties not easy to solve. For example, wedo not know any contributions of this type for non-linearelastodynamics. Our aim here is to find a class of non-linearequations (with a physical meaning) to which the methods ofFlavin et al. can be applied. The equations that we consider appearin a natural way in the analysis of the non-linear problem of theGreen and Naghdi heat conduction theory. Paying attention to theexamples proposed in this paper, the key point of our proposal isthe introduction of terms like ym, for m42, in the energy of the

A. Magana, R. Quintanilla / International Journal of Non-Linear Mechanics 46 (2011) 1252–1257 1253

system (theta is the temperature). This fact allows us to apply theabove cited methods.

As we mentioned above, there are neither physical normathematical studies to support type II theory. Therefore, at thisstage, we think that it is important to analyze different proposalswith their respective conclusions. Our work is addressed to thisobjective from a mathematical point of view. In order to establishthe more appropriate scenery for this theory, some initialattempts must be done.

The organization of this paper is as follows. In Section 2 we setthe equation we are going to deal with and the assumptions thatwe impose to it. We also show a couple of examples that satisfy ourassumptions. Section 3 is devoted to obtain a spatial decayestimates for the usual boundary-initial-value problem. As acorollary, we get a result of domain of influence type. Later, inSection 4, we obtain an upper bound for the amplitude term (of thespatial estimate) by means of the problem’s initial data. In Section5, we find spatial decay estimates for a non-standard initial valueproblem defined in a particular case following one of our examples.Finally, in the last section, some concluding remarks are stated.

2. Preliminaries

Let R be the semi-infinite cylinder ð0,1Þ � D, where D is a twodimensional bounded domain such that the boundary @D issmooth enough to apply the divergence theorem. The finite endface of the cylinder is in the plane x1 ¼ 0. Let DðzÞ be the cross-section of the points in R such that x1 ¼ z, and let RðzÞ be thepoints of the cylinder such that x14z. The equations we studyhere are determined on R.

In the following, we consider a non-linear type II heatconduction material. We denote by r the mass density, whichdepends on the material point, and by Z the entropy density.The evolution equation is given by

r _Z ¼Fi,i, ð2:1Þ

where

rZ¼� @C@y

and Fi ¼@C@a,i

: ð2:2Þ

Cðy,a,iÞ is the free energy and Fi is the entropy flux vector.If we substitute (2.2) into (2.1) we obtain

@

@t�@C@y

� �¼

@

@xi

@C@a,i

� �: ð2:3Þ

To have a well-posed problem in R we impose the following initialconditions:

aðx1,x2,x3,0Þ ¼ gðx1,x2,x3Þ, _aðx1,x2,x3,0Þ ¼ hðx1,x2,x3Þ ð2:4Þ

and also the boundary conditions given by

að0,x2,x3,tÞ ¼ f ðx2,x3,tÞ, 8ðx2,x3ÞAD, aðx1,x2,x3,tÞ ¼ 0 8ðx2,x3ÞA@D:

ð2:5Þ

In order to avoid difficulties concerning the smoothness of thesolutions, we assume that f ðx2,x3,tÞ ¼ 0 for ðx2,x3ÞAD.

In our study, it will be relevant to consider the internal energyfunction

S¼CþryZ:

Our aim is to obtain the spatial behavior for the solutions ofthe problem determined by (2.3) with initial conditions (2.4) andboundary conditions (2.5) whenever we assume thatZ

RS dV o1 ð2:6Þ

at time t¼ 0.

The class of functions we are going to work with is restrictedto satisfy the following assumptions:

A1:

jF1jrC1S1=2þC2Sp with C1,C240 and 1=2opo1.

A2:

SZC3y2þC4y

q with C3,C440 and pþ1=q¼ 1.

Let us show several examples that we had in mind and led usto propose the above assumptions.

Example 2.1. Let us consider the functions

CðmÞ ¼ �c1mjyjmþc2mjrajmþc3myjrajm�1þyðdima,iÞm�1 for mZ2,

where c1m and c2m are positive constants and c3m and dim arearbitrary constants. If we define

C¼Cð2Þ þCðmÞ, m42,

then we have

rZ¼ 2c12y�c32jrajþmc1mjyjm�2y�c3mjrajm�1�di2a,i�ðdima,iÞm�1

and

S¼ c12jyj2þc22jraj2þðm�1Þc1mjyjmþc2mjrajm: ð2:7Þ

In this case,

F1 ¼@C@a,1¼ 2c22a,1þc32y

a,1

jrajþmc2mjrajm�1 a,1

jraj

þðm�1Þc3myjrajm�2 a,1

jraj þyd12þðm�1Þyd1mðd1ma,1Þm�2:

Note that from (2.7) we obtain

ja,1jr jrajrK1S1=2,

jyjrK2S1=2,

jrajm�1 ¼ ðjrajmÞðm�1Þ=mrK3Sðm�1Þ=m,

jyjjrajm�2 ¼ ðjyjmÞ1mðjrajmÞðm�2Þ=mrK4S1=mSðm�2Þ=m

¼ K4Sðm�1Þ=m

and

jyjja,1r jyjjraj:

Therefore, jF1jrC1S1=2þC2Sp with C1,C240 and 1=2opo1.

Note also that SZC3y2þC4y

q for q¼m.

Example 2.2. We define GðmÞ ¼CðmÞ þbmpjyjpjrajm�p. And wetake

C¼Cð2Þ þGðmÞ, m42:

In this case,

rZ¼ 2c12y�c32jrajþmc1mjyjm�2y�c3mjrajm�1�pbmpyjyjp�2jrajm�p

�di2a,i�ðdima,iÞm�1:

Thus,

S¼ c12jyj2þc22jraj2þðm�1Þc1mjyjmþc2mjrajm

þð1�pÞbmpjyjpjrajm�p:

3. Spatial estimates

In this section we obtain our spatial decay estimate. Thedomain of influence type result for the solutions will be acorollary.

Our analysis starts by considering the following function:

Iðz,tÞ ¼�

Z t

0

ZDF1y dA ds:

A. Magana, R. Quintanilla / International Journal of Non-Linear Mechanics 46 (2011) 1252–12571254

Let us compute the divergence of the vector field Fiy:

½Fiy�,i ¼Fi,iyþFiy,i ¼@

@t�@C@y

� �yþ

@C@a,i

_a ,i:

Or, equivalently,

½Fiy�,i ¼�@

@t

@C@y

y� �

þ@C@y

_yþ@C@a,i

_a ,i ¼�@

@t

@C@y

y� �

þ@C@t:

Using Eq. (2.2) we obtain

½Fiy�,i ¼@

@t½ryZ�þ @C

@t¼ _S:

We consider the part of R between x1 ¼ zn and x1 ¼ z. Applyingthe divergence theorem, we obtain

Iðz,tÞ�Iðzn,tÞ ¼

Z t

0

Z z

zn

ZD

_S dA dx1 ds: ð3:1Þ

And integrating:

Iðz,tÞ�Iðzn,tÞ ¼�

Z z

zn

ZDS dVþ

Z z

zn

ZDSð0Þ dV ,

where dV stands for dA dx1 and Sð0Þ for function S evaluated att¼ 0. Note that the integral of Sð0Þ depends on the initial data ofthe problem.

It is clear that

@I

@t¼�

ZDF1y dA

and

@I

@z¼�

ZDS dAþ

ZDSð0Þ dA:

Applying assumption A1 we obtain

@I

@t

��������rZ

DjF1jjyj dArC1

ZDS1=2jyj dAþC2

ZDSpjyj dA:

And applying A2,

@I

@t

��������rC1C�1=2

3

ZDS1=2S1=2 dAþC2C�1=q

4

ZDSpS1=q dA

¼ ðC1C�1=23 þC2C�1=q

4 Þ

ZDS dA:

If we denote C5 ¼ C1C�1=23 þC2C�1=q

4 , then we have

@I

@t

��������rC5 �

@I

@zþ

ZDSð0Þ dA

� �:

Or, equivalently,

@I

@t

��������þC5

@I

@zrC5

ZDSð0Þ dA: ð3:2Þ

To simplify the notation a little bit, let us denoteE1ðzÞ ¼

RDSð0Þ dA. Then, inequality (3.2) implies that

@I

@tþC5

@I

@zrC5E1ðzÞ ð3:3Þ

and

@I

@t�C5

@I

@zZ�C5E1ðzÞ: ð3:4Þ

These inequalities have been studied in several references before(see, for instance, [8]).

Note that (3.3) and (3.4) can be rewritten, respectively, as

c@I

@tþ@I

@zrE1ðzÞ ð3:5Þ

and

c@I

@t�@I

@zZ�E1ðzÞ, ð3:6Þ

where c¼ 1=C5.Let us fix ðz0,t0ÞA ½0,L� � ½0,1Þ: If we set t¼ t0þcðz�z0Þ, then

the expression on the left hand side of inequality (3.5) can bethought of as

@

@zIðz,t0þcðz�z0ÞÞ: ð3:7Þ

Therefore, on integrating (3.5) we obtain

Iðz,t0þcðz�z0ÞÞ�Iðz0,t0ÞrZ z

z0

E1ðxÞ dx, ð3:8Þ

where zZz0.On the other hand, if we set now t¼ t1þcðz1�zÞ, and integrat-

ing (3.6) we obtain

Iðz,t1þcðz1�zÞÞ�Iðz1,t1ÞZ�

Z z1

zE1ðxÞ dx, ð3:9Þ

where zrz1.Taking t0 ¼ t1 ¼ 0, assuming that the initial data satisfy (2.6),

and making z-1, from inequalities (3.8) and (3.9) we obtainthat, for a finite time t,

limz-1

Iðz,tÞ ¼ 0:

Taking into account this result and relation (3.1) we get

Iðz,tÞ ¼

ZRðzÞ

SðtÞ dV�

ZRðzÞ

Sð0Þ dV :

Now inequality (3.3) implies that

Eðz,tÞrEðzn,0Þ, ð3:10Þ

where

Eðz,tÞ ¼

Z 1z

ZDSðtÞ dV ð3:11Þ

and z, zn and t are related by t¼ cðz�znÞ. In a similar way, we get

Eðz,tÞrEðznn,0Þ ð3:12Þ

for t¼ cðznn�zÞ. Finally, from inequalities (3.10) and (3.12), weconclude that

Eðz,tÞrEðzn,tnÞ ð3:13Þ

for jt�tnjrcðz�znÞ.Therefore, we have the following result.

Theorem 3.1. Let a be a solution of the initial-boundary-value

problem defined by (2.3)–(2.5). Then the energy function Eðz,tÞdefined in (3.11) satisfies inequality (3.13) whenever jt�tnjrcðz�znÞ, provided that the initial data satisfy (2.6).

If one defines the measure

Enðz,tÞ ¼

Z t

0Eðz,sÞ ds ð3:14Þ

the following inequalities can be obtained (see [16]):

Enðz,tÞrc

Z z

z�c�1tEðp,0Þ dp, c�1trz,

Enðz,tÞrc

Z z

0Eðp,0Þ dpþ 1�

cz

t

� �Enð0,tÞ, c�1tZz: ð3:15Þ

If g and h vanish, we have that Eðp,0Þ ¼ 0 for every pZ0. Thus,the last inequalities become

Enðz,tÞ ¼ 0, c�1trz,

A. Magana, R. Quintanilla / International Journal of Non-Linear Mechanics 46 (2011) 1252–1257 1255

Enðz,tÞr 1�cz

t

� �Enð0,tÞ, c�1tZz: ð3:16Þ

Note that the first equality implies that the solution vanishes fortrcz, which is a domain of influence kind of result.

Therefore, we have proved the following property.

Corollary 3.2. If homogeneous initial conditions are supposed, then

estimates (3.16) hold. In particular, a¼ 0 whenever c�1trz.

4. The amplitude term

To have a better description of our estimates, we need anupper bound for the amplitude term by means of the dataproblem. In this section we obtain a bound for Enð0,tÞ when theinitial data vanish.

In this section we will also assume that

A3:

jrZjrD1S1=2þD2Sp with D1,D240 and 1=2opo1.

A4:

jFijrD1S1=2þD2Sp.

It is worth noting that the families of functions proposed inExamples 2.1 and 2.2 satisfy these conditions.

It is clear thatZ t

0

ZRðr _Z�Fi,iÞy dV ds¼ 0:

Thus, making some calculations, we obtain

Eð0,tÞ ¼�

Z t

0

ZDð0Þ

F1y dA ds:

Therefore, for any arbitrary function Gðx,tÞ which decays in a(sufficiently) fast way as x1 goes to infinity and such that it agreeswith yðx,tÞ at Dð0Þ, we have

Enð0,tÞ ¼�

Z t

0

ZDð0Þðt�sÞF1G dA ds:

Using the divergence theorem we obtain that

Enð0,tÞ ¼

Z t

0

ZRðt�sÞr _ZG dV dsþ

Z t

0

ZRðt�sÞFiG,i dV ds:

We denote by I1 the first integral and by I2 the second one. I1 canbe rewritten as

I1 ¼

Z t

0

ZRrZG dV ds�

Z t

0

ZRðt�sÞrZ _G dV ds¼ I11�I12:

We now bound the absolute values of integrals I11, I12 and I2.Taking into account assumption A3 we get

jI11jrZ t

0

ZRðD1S1=2

þD2SpÞjGj dV ds:

Thus,

jI11jrD1

Z t

0

ZRS dV ds

� �1=2 Z t

0

ZRjGj2 dV ds

� �1=2

þD2

Z t

0

ZRS dV ds

� �p Z t

0

ZRjGj1=ð1�pÞ dV ds

� �1�p

:

Or, equivalently,

jI11jrD1Enð0,tÞ1=2Z t

0

ZRjGj2 dV ds

� �1=2

þD2Enð0,tÞpZ t

0

ZRjGj1=ð1�pÞ dV ds

� �1�p

:

Now we will use the inequality

a � brðeaÞa

a þðe�1bÞb

bwhen

1

a þ1

b¼ 1 for any e40:

In this way, we obtain

jI11jre1Enð0,tÞþN1

Z t

0

ZRjGj2 dV dsþe2Enð0,tÞ

þN2

Z t

0

ZRjGj1=ð1�pÞ dV ds:

In the above expression, e1 and e2 are two positive constants thatwe can select as small as we want, and N1 and N2 are twocalculable positive constants.

In a similar way we have

jI12jrZ t

0

ZRðt�sÞðD1S1=2

þD2SpÞ _G dV ds:

Using the same reasoning as before, we can obtain

jI12jre3Enð0,tÞþN3ðtÞ

Z t

0

ZRj _Gj2 dV dsþe4Enð0,tÞ

þN4ðtÞ

Z t

0

ZRj _Gj1=ð1�pÞ dV ds,

where e3 and e4 are positive but as small as we want, and N3ðtÞ

and N4ðtÞ can depend on the time.Using A4 and the fact that jG,ijr ðG,iG,iÞ

1=2, we obtain

jI2jr3

Z t

0

ZRðt�sÞðD1S1=2

þD2SpÞðG,iG,iÞ

1=2 dV ds:

Now we bound jI2j as before:

jI2jre5Enð0,tÞþN5ðtÞ

Z t

0

ZRðG,iG,iÞ dV dsþe6Enð0,tÞ

þN6ðtÞ

Z t

0

ZRðG,iG,iÞ

1=2ð1�pÞ dV ds,

where e5, e6, N5ðtÞ and N6ðtÞ are as above.In view of the previous estimates and selecting ei ¼ 1=12 for

i¼ 1, . . . ,6 we obtain

Enð0,tÞrHðtÞ

Z t

0

ZRðjGj2þj _Gj2þðG,iG,iÞþjGj1=ð1�pÞ þj _Gj1=ð1�pÞ

�

þðG,iG,iÞ1=2ð1�pÞ

Þ dV si, ð4:1Þ

where HðtÞ is a function calculable in terms of the constitutiveconstants and the time.

We now need to obtain an upper estimate for the integral onthe right hand side of (4.1) in terms of the boundary data. In orderto do so, we take

Gðx1,x2,x3,tÞ ¼ e�dx1 f ðx2,x3,tÞ,

where d is an arbitrary but positive constant and f ðx2,x3,tÞ is oneof our boundary conditions given at (2.5).

Note that

_Gðx1,x2,x3,tÞ ¼ e�dx1 _f ðx2,x3,tÞ,

and

G,iG,i ¼ d2e�2dx1 f 2ðx2,x3,tÞþe�2dx1 ðf 2,2ðx2,x3,tÞþ f 2

,3ðx2,x3,tÞÞ:

Therefore, each one of the integrals on the right hand side of(4.1) can be bounded. In fact, we haveZ

RjGjk dV ¼

ZD

Z 10

e�kdx1 jf kðx2,x3,tÞj dx1 dA¼1

kd

ZDjf kðx2,x3,tÞj dA,

for k¼ 2 or k¼ 1=ð1�pÞ:

ZRj _Gjk dV ¼

ZD

Z 10

e�kdx1 j_fkðx2,x3,tÞj dx1 dA¼

1

kd

ZDj_f

kðx2,x3,tÞj dA,

for k¼ 2 or k¼ 1=ð1�pÞ:

ZRðG,iG,iÞ

k dV ¼

ZD

Z 10

e�2kdx1 ½d2f 2ðx2,x3,tÞþ f 2,2ðx2,x3,tÞ

A. Magana, R. Quintanilla / International Journal of Non-Linear Mechanics 46 (2011) 1252–12571256

þ f 2,3ðx2,x3,tÞ�k dx1 dA

¼1

2kd

ZD½d2f 2ðx2,x3,tÞþ f 2

,2ðx2,x3,tÞþ f 2,3ðx2,x3,tÞ�k dA,

for k¼ 1 or k¼ 1=2ð1�pÞ:

And, then,

Enð0,tÞrHðtÞ

Z t

0

ZD

1

2dðð1þd2

Þjf j2þj_f j2þ f 2,2þ f 2

,3Þ

�

þ1�p

dðjf j1=ð1�pÞ þj_f j1=ð1�pÞ þðd2f 2þ f 2

,2þ f 2,3Þ

1=2ð1�pÞÞ dA ds

�:

ð4:2Þ

5. A non-standard problem for Eq. (2.3)

In this section we restrict our attention to the problemdetermined by the functions given at Example 2.1:

C¼Cð2Þ þCðmÞ, m42,

where

CðmÞ ¼ �c1mjyjmþc2mjrajmþc3myjrajm�1þyðdima,iÞm�1 for mZ2:

c1m and c2m are positive constants and c3m and dim are arbitraryconstants.

We briefly discuss the behavior of the solutions of (2.3) subjectto the boundary conditions (2.5) and the non-standard conditions

aðx,TÞ ¼ kaðx,0Þ, yðx,TÞ ¼ kyðx,0Þ, ð5:1Þ

where jkj41. Such non-standard conditions have been thesubject of recent attention. In 2002 Payne and Schaefer [31]proposed the study of a non-standard initial problem associatedwith elastodynamics. This kind of problems was motivated byregularization of ill-posed Cauchy problems for the backward intime classical heat equation [1].

The analysis begins by considering the function

FgðzÞ ¼�

Z T

0

ZDðzÞ

e�2gsF1y dA ds, ð5:2Þ

where, as established in [17], the positive constant g is given by

g¼ 1

Tln jkj:

It is clear, see (3.1), that

FgðzÞ�FgðznÞ ¼

Z T

0

Z z

zn

ZD

e�2gs _S dA dx1 ds,

for every zZznZ0.Or, equivalently,

FgðzÞ�FgðznÞ ¼

Z T

0

Z z

zn

ZD

d

ds½e�2gsS� dA dx1 ds

þ2gZ T

0

Z z

zn

ZD

e�2gsS dA dx1 ds:

If we denote by Rðz,znÞ the part of the cylinder from zn to z andwe integrate with respect to the time in the first integral above,we get

FgðzÞ�FgðznÞ ¼

ZRðz,znÞ½e�2gTSðTÞ�Sð0Þ� dVþ2g

Z T

0

ZRðz,znÞ

e�2gsS dV ds,

where function S is the one defined by (2.7).Hence, taking into account the non-standard conditions (5.1)

and the value of g (that gives k2e�2gT ¼ 1) we obtain

e�2gTSðTÞ�Sð0Þ ¼ ðkm�2�1Þ½ðm�1Þc1mjyð0Þjmþc2mjrað0Þjm�:

As a result,

FgðzÞ�FgðznÞ ¼

ZRðz,znÞðkm�2�1Þ½ðm�1Þc1mjyð0Þjmþc2mjrað0Þjm� dV

þ2gZ T

0

ZRðz,znÞ

e�2gsS dV ds:

And, computing now F 0gðzÞ we obtain

F 0gðzÞ ¼

ZDðzÞðkm�2�1Þ½ðm�1Þc1mjyð0Þjmþc2mjrað0Þjm� dA

þ2gZ T

0

ZDðzÞ

e�2gsS dA ds:

Being the first integral positive, it is clear that

F 0gðzÞZ2gZ T

0

ZDðzÞ

e�2gsS dA ds:

As we want to estimate the absolute value of FgðzÞ in terms ofits derivative, we remind here that functions F1 and y satisfyassumptions A1 and A2. Therefore

jFgðzÞjrZ T

0

ZDðzÞ

e�2gs½C1S1=2þC2Sp

�jyj dA ds

rZ T

0

ZDðzÞ

e�2gs½C1S1=2C�1=23 S1=2

þC2SpC�1=q4 S1=q

� dA ds:

ð5:3Þ

Finally,

jFgðzÞjrC1C�1=2

3 þC2C�1=q4

2gF 0gðzÞ ¼ CgF 0gðzÞ:

Hence, we can obtain an alternative of Phragmen–Lindeloftype which states (see Ref. [7]) that the solutions either growexponentially for z large enough or solutions decay exponentiallyin the form

EðzÞrEð0Þe�z=Cg ,

for all zZ0, where

EðzÞ ¼Z T

0

ZRðzÞ

2ge�2gsS dV dsþ

ZRðzÞðkm�2�1Þ½ðm�1Þc1mjyð0Þjm

þc2mjrað0Þjm� dV :

As a final comment, let us say that this result can be extendedfor jkjo1. The idea is to observe that if a is a solution of ourproblem, then it is also a solution of the backward in timeproblem determined by the same boundary conditions, but withthe non-standard initial conditions given by

k�1aðx,TÞ ¼ aðx,0Þ, k�1yðx,TÞ ¼ yðx,0Þ,

for function

C¼Cð2ÞnþCðmÞ

n, m42,

with

CðmÞn¼�c1mjyjmþc2mjrajm�c3myjrajm�1�yðdima,iÞ

m�1 for mZ2:

In this case, k�1 is now greater than one and the analysis proposedpreviously can be adapted to this new system.

6. Concluding remarks

This paper was focussed in the analysis of a mathematicalaspect that can appear in the non-linear theory of type II heatconduction. The mathematical structure of this theory is given bya hyperbolic differential equation while, in the classical theory,the structure is given by a parabolic equation. We have seen that,under some constitutive assumptions, the energy does not grow

A. Magana, R. Quintanilla / International Journal of Non-Linear Mechanics 46 (2011) 1252–1257 1257

along certain space–time directions. In particular, a result ofdomain of influence has been obtained. This kind of result cannotbe obtained for the classical theory of heat conduction due to thephenomenon of diffusion.

We think that, nowadays, it is impossible to decide which ofboth theories is appropriate to better describe real situations: itdepends on many different aspects. Surely, for certain scenariosone of these theories is more suitable, while for other situations,the other one is preferable. Let us say that the heat conductionphenomena at very low temperatures seem to be well describedby the theory of wave propagation at finite speed. For instance,Hwang et al. [18], Maıga et al. [28], Kim et al. [25] or Vadasz et al.[37] suggest that a mechanism for the increased heat transfercharacteristics of a nanofluid may be through a hyperbolicequation for the temperature field.

Acknowledgments

The authors want to thank two anonymous referees for theiruseful comments.

The investigation reported in this paper is supported byprojects ‘‘Ecuaciones en Derivadas Parciales en Termomecanica.Teorıa y aplicaciones’’ (MTM2009-08150) and ‘‘Teorıa de Juegos ysistemas de decision colectiva con aplicaciones’’ (MTM 2009-08037)of the Science and Innovation Spanish Ministry and the EuropeanRegional Development Fund.

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