THE DECAY PROPERTY OF REGULARITY-LOSS
TYPE OF SOLUTIONS OF SOME HYPERBOLIC
AND HYPERBOLIC/PARABOLIC SYSTEMS
ﻲﻤﻠﻌﻟا ﺚﺤﺒﻟاو ﱄﺎﻌﻟا ﻢﻴﻠﻌﺘﻟا ةرازو
رﺎـﺘﺨﻣ ﻲـﺟﺎﺑ ﺔـﻌﻣﺎﺟ
ﺔـﺑﺎـﻨﻋ
Université Badji Mokhtar
Annaba
Badji Mokhtar University -
Annaba
Faculté des Sciences
Département de Mathématiques
THÈSE
Année : 2017/2018
Présentée en vue de l’obtention du diplôme de
Doctorat en Mathématiques
Option : Equations aux dérivées partielles
Présentée Par:
LEILA DJOUAMAI
DIRECTEUR DE THÈSE: BELKACEM SAID- HOUARI M.C.A. Univ. Alhosn U.A.E
CO-DIRECTEUR DE THÈSE: FAOUZIA REBBANI PROF. E.S.T.I Annaba
Devant le jury
PRESIDENT :
SAID MAZOUZI
Professeur
U.B.M. ANNABA
EXAMINATEURS :
LAHCEN CHORFI Professeur
U.B.M. ANNABA
MOHAMED SAID
MOULAY
Mathematical Department
The decay property of regularity-loss type of solutions of
some hyperbolic and hyperbolic
/parabolic systems
by
Leila Djouamai
Under the Supervision of
Dr. Belkacem Said-Houari
Pr. Faouzia Rebbani
This thesis is submitted in order to obtain the
degree of Doctor of Sciences
(Cauchy) (Fourier) (Cauchy)
-. (Cauchy) (Bresse-Cattaneo)Cette th`ese est consacr´ee `a l’´etude de la stabilit´e et taux de d´ecroissance des solutions pour certains syst`emes hyperboliques et hyperboliques-paraboliques. Le premier tra-vail, est consacr´e `a un probl`eme de Cauchy pour un syst`eme de thermo-´elasticit´e avec un retard `a double phase. En utilisant la m´ethode de l’´energie dans l’espace de Fourier, on montre que la solution a un lent taux de d´ecroissance et il est de type de perte de r´egularit´e. Le deuxi`eme probl`eme est li´e au probl`eme de Cauchy de la th´eorie de la con-duction thermique avec un retard triphas´e. Nous d´eterminons le taux de d´ecroissance optimale de la norme L2 des solutions. Le troisi`eme probl`eme est consacr´e `a l’´etude
de la stabilit´e de la th´eorie des mat´eriaux ´elastiques avec des vides, nous consid´erons un probl`eme de Cauchy pour un solide ´elastique avec des vides. D’abord nous mon-trons que la dissipation lin´eaire poreuse conduit le syst`eme `a un taux de d´ecroissance de type perte de r´egularit´e. Ensuite, nous d´emontrons que l’ajout d’un amortisse-ment visco-´elastique donne un gain de la r´egularit´e de la solution et on obtiendra un taux de d´ecroissance optimale. Le quatri`eme probl`eme, est consacr´e au syst`eme de Bresse-Cattaneo avec un amortissement de frottement, nous ´etudions les propri´et´es de d´ecroissance du syst`eme Bresse-Cattaneo dans un espace de dimension une. Nous mon-trons qu’il existe un nouveau param`etre de stabilit´e qui contrˆole le taux de d´ecroissance de la solution.
This thesis is devoted to the study of the stability and the decay rates of solutions for some hyperbolic and hyperbolic-parabolic system.
The first work is concerned with a Cauchy problem for a system of dual-phase-lag ther-moelasticity. Using the energy method in the Fourier space, we show that the solution has a slow decay rate and it is of regularity-loss type.
The second problem is related to the Cauchy problem of the theory of heat conduction with three-phase-lag. We show the optimal decay rate of the L2-energy norm of the solutions.
The third problem is concerned with a Cauchy problem in elastic solids with voids where a porous dissipation is present. We show the effect of the porous dissipation in the stability of the system and we prove that the solution has a slow decay rate and it is of regularity-loss type. Second, we show that by adding a viscoelastic damping term, then we gain the regularity of the solution and obtain the optimal decay.
The last problem is devoted to the study of the Bresse-Cattaneo system with a fric-tional damping, we investigate the decay properties of the Bresse-Cattaneo system in the whole space. We show that there is a completely new stability number that controls the decay rate of the solution.
Abstract ii
Acknowledgements v
Introduction 1
1 Preliminaries 9
1.1 The Fourier Space . . . 9
1.2 The Sobolev Space . . . 10
1.3 Some Inequalities . . . 11
1.4 Notations and Definitions . . . 13
2 Decay property of regularity-loss type for solutions in dual-phase-lag ther-moelasticity 14 2.1 Introduction . . . 14
2.2 Energy method . . . 17
2.3 Decay estimates . . . 28
3 Decay property for solutions in the three-phase-lag heat conduction 31 3.1 Introduction . . . 31
3.2 The third order model . . . 32
3.2.1 The energy method in the Fourier space . . . 32
3.2.2 Decay estimates . . . 35
3.3 The Fourth order model . . . 37
4 Decay property of regularity-loss type for solutions in elastic solids with voids 44 4.1 Introduction . . . 44
4.2 Porous-elastic model . . . 45
4.2.1 The energy method in the Fourier space . . . 46
4.2.2 Decay estimates of regularity-loss type . . . 53
4.3 Porosity and viscoelasticity . . . 55
5 A new stability number of the Bresse-Cattaneo system 61 5.1 Introduction . . . 61
5.2 Main results . . . 64
5.3 Energy method in the Fourier space . . . 66
5.4 Eigenvalues expansion . . . 81
5.4.1 Eigenvalues expansion for low frequencies . . . 81
5.4.2 Eigenvalues expansion for high frequencies . . . 85
Firstly, I would like to express my sincere gratitude to my advisor Dr. B. SAID-HOUARI for the continuous support of my Ph.D study and related research, for his patience, motivation, and large knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D study.
I am very grateful to Prof. F. REBBANI who was always there to give me her academic administrative support to UBM Annaba.
I am very grateful to the president of the jury, Prof. Said Mazouzi and examiners: Prof. Lahcen Chorfi and Prof. Moulay Mohamed Said for agreeing to report on this thesis and for their valuable comments and suggestions.
I must thank the members of the Mathematics Department of Badji Mokhtar Annaba University (Algeria) including the Applied Math Lab of Badji Mokhtar Annaba Univer-sity for their support over the years.
Of course, this thesis is the accumulation of many years of study. I am indebted to all those who guided me on the road of mathematics.
I thank from my heart my teachers whose encouraged me during my studies .
Finally, I want to dedicate this work to my parents, my children and my husband, for their love and encouragement during the past years.
Leila
The aim of this thesis is to investigate the stability of some elastic and thermoelastic evolution problems, such as the Bresse systems, the theory of heat conduction and the elastic solids with voids. We consider different types of dissipation mechanisms and show their effects on the stability of such systems.
The theory of heat conduction
In 1807, the French mathematical physicist Joseph Fourier proposed a constitutive rela-tion of the heat flux of the form
q(x, t)= −κ∇θ(x, t), (1)
where x stands for the material point, t is the time, q is the heat flux, θ is the temperature, ∇ is the gradient operator and κ is the thermal conductivity of the material, which is a thermodynamic state property. Using the Fourier law (1), the behavior of an elastic heat body can be described by a coupled system of hyperbolic−parabolic type. This hyperbolic−parabolic system is interesting due to its large applications in mechanics, physics and engineering problems.
Over the past two decades, there has been a lot of work on local existence, global exis-tence, well-posedeness, and asymptotic behavior of solutions to some initial-boundary value problems as well as to the Cauchy problems in both one-dimensional and multi-dimensional thermoelasticity. See for instance [31, 32, 47, 53, 60] and references therein.
The one dimentional Cauchy problem, in which a thermoelastic body occupies the entire real line, was investigated by Kawashima & Okada [23], Zheng & Shen [61], and Hrusa & Tarabek [18] proving the global existence in time. In particular, Hrusa & Tarabek [18] combined certain estimates of Slemrod [53] that remain valid on unbounded in-tervals with some additional ones which exploited some relations associated with the
second law of thermodynamics, to obtain the energy estimate for lowest order terms, an estimate that cannot be obtained using Poincar´e’s inequality as in Slemrod’s paper.
The Fourier law of heat conduction is an early empirical law. It assumes that q and ∇θ appear at the same time instant t and as in (1) consequently equation (1) represents an instantaneous response to changes in the gradients of the temperature visible in the heat flux and implies that thermal signals propagate with an infinite speed. That is, any thermal disturbance at a single point has an instantaneous effect everywhere in the medium. More precisely, the thermoelastic theory based on the Fourier law has some disadvantages such as:
• infinite velocity of thermoelastic disturbances;
• unsatisfactory thermoelastic response of a solid to short laser pulses;
• poor description of thermoelastic behavior at low temperature.
With the development of science and technology such as the application of ultra-fast pulse-laser heating on metal films, heat conduction appears in the range of high heat flux and high unsteadiness. The drawback of infinite heat propagation speed in the Fourier law becomes unacceptable. This has inspired the work of searching for new constitutive relations. Consequently, a number of modifications of the basic assumption on the relation between the heat flux and the temperature have been made, such as: Cattaneo’s law, Gurtin & Pipkin’s theory, Jeffreys’ law, Green & Naghdi’s theory and others. The common feature of these theories is that all lead to hyperbolic differential equation and permit transmission of heat flow as thermal waves at finite speed. See [4,22] for more details.
The Cattaneo law (it is also known as Maxwell−Cattaneo or Maxwell−Cattaneo−Vernotte or Lord−Shulman model)
τqqt + q + κ∇θ = 0, (τq > 0, relatively small) (2)
was proposed by Cattaneo in [7]. It is perhaps the most obvious, the most widely accepted and simplest generalization of Fourier’s law that gives rise to a finite speed of propagation of heat.
When the Fourier law (1) is replaced by the Cattaneo law (2) for the heat conduction, the equations of thermoelasticity become purely hyperbolic. Indeed, from the energy
balance law
ρθt + %divq = 0 (3)
and (2), we obtain the telegraph equation ρθtt− %κ τq ∆θ + τρ q θt = 0, (4)
which is a hyperbolic equation and predicts a finite signal speed limited by%κ/(ρτq)
1/2 .
Concerning the thermoelasticity of second sound, the interested reader is refereed to Tarabek [56], Hrusa & Tarabek [18], Hrusa & Messaoudi [17], Racke [41–43], Mes-saoudi & Said-Houari [29,30] and references therein.
Note that the Cattaneo constitutive relation (2) can be seen as a first-order approximation of a more general constitutive relation (single−phase−lagging model; Tzou [57]),
q(x, t+ τq)= −κ∇θ(x, t). (5)
The relation (5) states that the temperature gradient established at a point x at time t gives rise to a heat flux vector at x at a later time t+ τq. The delay time τqis interpreted
as the relaxation time due to the fast-transient effects of thermal inertia (or small-scale effects of heat transport in time) and is called the phase−lag of the heat flux. It has been confirmed by many experiments that the Cattaneo law generates a more accurate prediction than the classical Fourier law. However, some studies show that the Catta-neo constitutive relation has only taken account of the fast-transient effects, but not the micro-structural interactions. See Tzou [58] for more details.
In [58], Tzou proposed a new theory of heat conduction which describes the interactions between phonons and electrons on the microscopic level as retarding sources causing a delayed response on the macroscopic scale. The physical meanings and the applicability of the dual−phase−lag model have been supported by the experimental results [59]. In this theory the Fourier law is replaced by an approximation of the equation
q(x, t+ τq)= −κ∇θ(x, t + τθ), τq> 0, τθ > 0, (6)
where τqis the phase lag of the heat flux and τθ is the phase lag of the gradient of the
temperature. According to the relation (6), the temperature gradient at a point x of the material at time t+ τθcorresponds to the heat flux density vector at x at time t+ τq. The
as phonon-electron interaction or phonon scattering, and is called the phase−lag of the temperature gradient (Tzou [58]).
If the two phase lags are equal, that is τq = τθ, then, the relation (6) is identical with
the classical Fourier law (1). While in the absence of the phase lag of the temperature gradient, τθ = 0 and by taking the first-order approximation for q, then equation (6)
reduces to the Cattaneo law (2).
A combination of the constitutive equation (6) with the classical energy equation, leads to an ill-posed problem (see [10]). However, if we replace the delay expressions in (6) by their Taylor expansions at different orders, we obtain several heat conduction theories. Indeed, in the case that we only consider the development until the first order in τθand a second order in τq, we obtain a hyperbolic theory, which has been studied and
analysed by Horgan & Quintanilla [16] and Quintanilla & Racke [39]. In particular, the authors in [39] analyzed the dual−phase−lag thermoelasticity, where the heat condition is given by (6) with second order approximation for q and first order approximation for θ were used. They showed that under the condition
τθ > τq/2, (7)
then solutions of the problem are generated by a semigroup of quasi-contractions. In addition, they showed that solutions of the one-dimensional problem are exponentially stable.
Choudhuri [5] proposed a three−phase− lag heat conduction model
q(x, t+ τq)= −[k∇θ(x, t + τθ)+ k∗∇ν(x, t + τν)], (8)
where νt = θ and v is the thermal displacement gradient, k and k∗ are two positive
constants.
Equation (8) states that the temperature gradient and the thermal displacement gradient established across a material volume located at a position x at time t+τθand t+τνresult
in heat flux to flow at a different instant of time t + τq. The third delay time t+ τν may
be interpreted, as the phase−lag of the thermal displacement gradient.
The case τθ = τq = τν corresponds to the Green & Naghdi type III model. See [13,14]
The elastic with voids
A large area of research is devoted to deduce qualitative properties of solutions for the equations of elastic solids with voids when they are complemented with initial and boundary conditions and certain restraints are placed on the coefficient of the system. Goodman and Cowin [12] first introduced the concept of a continuum theory granular materials with intersticial voids. It is considered to be a simple extension of the classical elasticity theory to porous media, where, in addition to the elastic effects, these materials (with voids) possess a microstructure with the property that the mass at each point is obtained as the product of the mass density of the material matrix by the volume fraction. Nunziato and Cowin [35] have presented a nonlinear theory of elastic materials with voids. The basic feature of this theory is the concept of a material for which the bulk density is written as the product of two fields, the matrix material density field and the volume fraction field. The linear theory of elastic materials with voids has been established by Cowin and Nunziato [6].
The one-dimensional porous-elastic model has the form
ρutt = µuxx+ bϕx ρκϕtt = αϕxx− bux−τϕt− aϕ (9)
where u is the longitudinal displacement, ϕ is the volume fraction, ρ > 0 is the mass density, κ > 0 is the equilibrated inertia, and µ, α, τ, a are the constitutive constants which are positive.
Quintanilla [38] considered (9) in a bounded domain with initial conditions and mixed boundary conditions and showed that the damping in the porous equation (−τϕt) is not
strong enough to obtain an exponential decay. Only the slow decay has been proved. Subsequently, many contributions have been made where the decay of solutions to the problems in elasticity with voids have been treated. See for instance [2, 3, 25,28, 37] and the references therein. In [33] the authors investigated the problem (9) with τ = 0 and a viscoelastic damping term of the form γutxx acting on the right hand side of the
first equation. They proved that the decay rate of the solution is polynomial and cannot be exponential.
The coupling of the system (9) with different laws of heat conduction has been also considered by many authors. The interested reader is refereed to [2,15,26,28,34,36, 54] and references therein.
The Bresse system
The stability of elastic beams, thermoelastic and viscoelastic is a highly active field of reaserch. Various types of beams has been considered, such as: Timoshenko, Bresse, Rayleign and Euler-Bernouli.
In their study on networks of flexible beams, Lagnese, Leugering and Schmidt [24] derived a general model of nonlinear elastic beam of three dimensions.
A particular case of this model is a linear model coupling three wave equations. It describes the moment of an elastic beam planar under the effect of small deformations. This is the Bresse system which is without feedback given by
ρ1ϕtt− Gh(ϕx+ ψ + lw)x − lEh(wx− lϕ) = 0, ρ2ψtt− EIψxx+ Gh (ϕx+ ψ + lw) = 0, ρ1wtt− Eh(wx− lϕ)x− lGh(ϕx+ ψ + lw) = 0. (10)
where (x, t) ∈ R × R+, the functions ϕ, ψ and w denote the vertical displacements of the beam, the rotation angle of the linear filaments material and longitudinal displacements respectively, and ρ1, ρ2, l, I, G, E and h are positive constants. Several types of
dissi-pation mechanisms have been previously introduced of frictional type, of viscoelastic type, and of thermal nature.
The system with frictional dissipative mechanism
ϕtt−(ϕx−ψ − lw)x− k20l(wx− lϕ) = 0, ψtt− a2ψxx−(ϕx−ψ − lw) = 0, wtt− k20(wx− lϕ)x− l(ϕx−ψ − lw) + γwt = 0, (11)
with the initial data
(ϕ, ϕt, ψ, ψt, w, wt) (x, 0) = (ϕ0, ϕ1, ψ0, ψ1, w0, w1). (12)
has beed first investigated by Soufyane and Said-Houari in [55] and several decay rates have been shown under different assumptions on the wave speeds of the three wave equations in (11). For the thermoelastic Bresse system, the initial boundary value prob-lem was first investigated by Liu and Rao [27], where they considered the Bresse system with two different thermal dissipative mechanisms. The authors showed that the energy
decays exponentially when the wave speeds of the three wave equations are equal, that is to say when a= k0= 1. Otherwise, if two of the wave speeds are different the authors
found polynomial decay rates depending on the boundary conditions.
For l= 0, the Bresse system reduces to the Timoshenko system ϕtt− (ϕx−ψ)x = 0, ψtt− a2ψxx− (ϕx−ψ) = 0, (13)
Several decay result have been shown for the solution of the Cauchy problem associated to (13) under different type of damping mechanisms. The interested reader is referred to the papers [19,20,49,50].
The main results of this thesis
This thesis contains four chapters.
Chapter 1. In this chapter, we considered the Cauchy problem of two models of the theory of heat conduction with three−phase−lag. Under appropriate assumptions on the material parameters, we showed the optimal decay rate of the L2-norm of solutions. More precisely, we proved that in each model the L2-norm of the solution is decaying with the rate (1+ t)−1/4 for initial data in L1(R). This decay rate is similar to the one of the heat kernel. Some faster decay rates have been also given for some weighted initial data in L1(R).
Chapter 2. This chapter is devoted to the study of the Cauchy problem for a system of dual-phase-lag thermoelasticity. We investigated the decay properties of the linear model in the whole space. To prove our result, we apply the energy method in the Fourier space to construct the appropriate Lyapunov functional. Unlike the classical system in thermoelasticity, and the thermoelasticity with second sound, we showed that the solution of the dual−phase−lag thermoelasticity enjoys the decay property of regularity-loss type.
Chapter 3. The goal of this chapter is to investigated the decay property of the one di-mensional linear Cauchy problem for a system of elastic solids with voids. First, we showed that a linear porous dissipation leads to decay rates of regularity-loss type of the solution. We show some decay estimates for initial data in Hs
(R) ∩ L1(R). Furthermore, we proved that by restricting the initial data to
be in Hs
solution. Second, we showed that by adding a viscoelstic damping term, then we gain the regularity of the solution and obtain the optimal decay rate.
Chapter 4. In this chapter, we studied the Bresse–Cattaneo system with a frictional damping term and prove some optimal decay results for the L2-norm of the solu-tion and its higher order derivatives. In fact, we show that there is a completely new stability number δ that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This results improves the one in [48].
Preliminaries
We recall some notations and we review some basic definitions will be used throughout this thesis.
1.1
The Fourier Space
- By ˆf we denote the Fourier transform of f :
ˆ f(ξ) = ˆ R f (x) e−iξxdx, f(x)= 1 (2π) ˆ R ˆ f (ξ) eiξxdξ.
Theorem 1.1. [11] Let f and g be locally integrable and have Fourier transforms f and ˆf , ˆg respectively. If α is a real number and a is a real or complex constant, then the Fourier transform of 1) f + g is fˆ+ ˆg, 2) a f is a ˆf, 3) f (x) cos αx is fˆ(ξ − α) + ˆf(ξ + α) 2 , 4) f (x) sin αx is fˆ(ξ − α) − ˆf(ξ + α) 2i , 5) f (x −α) is e−iξα fˆ(ξ) , and 6) f (αx) is 1 |α|fˆ ξ α if α , 0. 9
Theorem 1.2. [11] Let f : R → R be continuous and integrable on R and such that f(x) → 0 as x →+− ∞. If f0 is piecewise continuous and integrable on R, then
ˆ
f0(ξ) = iξ ˆf(ξ) .
Corollary 1.3. [11] If f, f0, f00,..., f(n−1)satisfy the same hypotheses that f satisfies in the theorem, then
c
f(n)(ξ) = (iξ)n fˆ(ξ) .
Theorem 1.4. Plancherel0s Theorem Assume d ≥ 1. The Fourier transformation ˆf of L2 = L2
Rd
is a unitary transformation of L2 . Namely we have the equality
fˆ = k f k . for an arbitrary f ∈ L2. Here k.k denotes the L2−norm.
1.2
The Sobolev Space
- The norm k.kLq and k.kHl stand for the Lq(R)-norm (2 ≤ q ≤ ∞) and the Hl(R)-norm.
Definition 1.5. [1] Let p ∈ R with 1 < p < ∞; we set
L∞(Ω) = { f : Ω → R; f is mesurable and | f | ∈ L1(Ω)}, with || f ||Lp = || f ||p = "ˆ Ω| f (x)| pdµ #1/p . Definition 1.6. [1] We set
L∞(Ω) = { f : Ω → R, f is measurable and there is a constant C, such that | f (x) ≤ C | a.e.on Ω}, with
|| f ||L∞ = || f ||∞ = inf {C; | f (x)| ≤ C a.e. on Ω} .
Definition 1.7. [1] Let Ω = (a, b) be an open interval, possibly unbounded, and let p ∈ R with 1 ≤ p ≤ ∞. The Sobolev space W1,p(Ω) is defined to be
W1,p(Ω) = ( u ∈ Lp(Ω) ; ∃g ∈ Lp(Ω) such that ˆ Ω uϕ0 = − ˆ Ω gϕ ∀ϕ ∈ C1 c(Ω) ) .
We set
H1(Ω) = W1,2(Ω) .
Definition 1.8. [1] Given an integer m ≥ 2 and a real number 1 ≤ p ≤ ∞ we define by induction the space
W1,p(Ω) =nu ∈ W m−1,p(Ω) ; u0 ∈ Wm−1,p(Ω)o . We also set
Hm(Ω) = Wm,2(Ω) .
Definition 1.9. For γ ∈ [0,+∞), we define the weighted function space L1,γ(R), as
follows: u ∈ L1,γ(R) iff u ∈ L1(R) and
kuk1,γ =
ˆ
R
(1+ |x|)γ|u(x)|dx <+∞.
1.3
Some Inequalities
- Let 1 ≤ p ≤ ∞; we denote by p0 the conjugate exponent, 1
p + 1 p0 = 1.
Theorem 1.10. [1] (Holder’s Inequality). Assume that f ∈ Lp and g ∈ Lp0with 1 ≤ p ≤ ∞. Then f g ∈ L1and
ˆ
| f g| ≤ k f kpkgkp0.
Theorem 1.11. [11] Minkowski0s Inequality. Assume that f ∈ Lp
and g ∈ Lp0with 1 ≤ p ≤ ∞. Then
k f + gkp≤ k f kp+ kgkp0 .
Theorem 1.12. (Young0s Inequality) For a, b > 0 and ε > 0, we have ab ≤ ε pa p+ 1 qεp/qb q. Si p= q = 2, on a ab ≤ ε 2a 2+ 1 2εb 2.
Lemma 1.13. For all k ≥ 0, c ≥ 0, there exists a constant C > 0 such that for all t ≥ 0 the following estimate holds :
ˆ
|ξ|≤1|ξ| σ
e−c|ξ|2tdξ ≤ C (1 + t)−(σ+1)/2. (1.1)
Proof. First, observe that ˆ |ξ|≤1 |ξ|σ e−c|ξ|2tdξ = 2 ˆ 1 0 rσe−cr2t dr. (1.2) Thus, it is enough to prove that for given c > 0 and σ ≥ 0, we have
ˆ 1 0
rσe−cr2tdr ≤ C(1+ t)−(σ+1)/2, (1.3) for all t ≥ 0, where C is a positive constant independent of t. To see this, observe first that, for 0 ≤ t ≤ 1, the (1.3) is obvious. On the other hand, for t ≥ 1, we have
(1+ t) ≤ 2t. (1.4)
Now, using (2.42), we have c(σ+1)/2(1+ t)(σ+1)/2 ˆ 1 0 rσe−cr2t dr ≤ c(σ+1)/2(2t)(σ+1)/2 ˆ 1 0 rσe−cr2tdr, 2−(σ+1)/2c(σ+1)/2(1+ t)(σ+1)/2 ˆ 1 0 rσe−cr2t dr ≤ ˆ 1 0 ctr2σ/2(ct)1/2e−cr2t dr. (1.5) Using the change of variables z= ctr2, we get
ˆ 1 0 ctr2σ/2(ct)1/2e−cr2t dr = 1 2 ˆ ct 0 zσ/2z−1/2e−zdr, ≤ 1 2 ˆ ∞ 0 z(σ+12 )−1e−zdr, = 1 2Γ σ + 1 2 ! < ∞, (1.6)
whereΓ is the Gamma function.
Inserting the above estimates into (1.5), we obtain ˆ 1 0 rσe−cr2tdr ≤ 1 2Γ k+ 1 2 ! 2(σ+1)/2c−(σ+1)/2(1+ t)−(σ+1)/2, ≤ C(1+ t)−(σ+1)/2.
Then (1.1) is fulfilled. Thus, the proof of Lemma 1 is finished.
1.4
Notations and Definitions
Definition 1.14. (Well-posed problems (Hadamard))
A problem is well-posed if the following three properties hold.
1) Existence : For all suitable data, a solution exists.
2) Uniqueness : For all suitable data, the solution is unique.
3) Stability : The solution depends continuously on the data.
Definition 1.15. (Ill-posed problems)
A problem that violates any of the three properties of well-posedness is called an ill-posed problem.
- Unless otherwise noted, C, C(), c, c()... are all positive constants which may vary from line to line and even sometimes in the same line.
Decay property of regularity-loss type
for solutions in dual-phase-lag
thermoelasticity
2.1
Introduction
In the classical theory of thermoelasticity, the behavior of an elastic heat body can be described by a coupled system of hyperbolic-parabolic type, where the classical Fourier model of heat conduction is used. This law assumes the flux q to be proportional to the gradient of the temperature θ at the same time t,
q(x, t)+ κ∇θ(x, t) = 0, (2.1)
where κ > 0 is the thermal conductivity depends on the properties of the material. This hyperbolic-parabolic system is interesting due to its large applications in mechanics, physics and engineering problems. Over the past two decades, there has been a lot of work on local existence, global existence, well-posedeness, and asymptotic behavior of solutions to some initial-boundary value problems as well as to Cauchy problems in both one-dimensional and multi-dimensional thermoelasticity. See for instance [31,32, 47,53,60], the book by Jiang & Racke [21] and the recent survey paper of Racke [43]. For some situations involving very low temperatures near absolute zero, for a heat source such as a laser or microwave of extremely short duration or very high frequency, very high temperature gradient and extremely short times, heat is found to propagate as
thermal waves with a finite speed. To account for the phenomena, the classical Fourier heat flux model (2.1) has been modified by Cattaneo [7], where he suggested, instead of (2.1), the new law of heat conduction:
τqqt+ q + κ∇θ = 0, (τq > 0, relatively small). (2.2)
Equation (2.2) together with the energy conservation law
ρcθt = −divq + g (2.3)
where ρ is the density, g is the heat generation per unit volume and c is the specific heat, lead to a hyperbolic equation which predicts a finite speed of heat propagation.
Note that the Cattaneo constitutive relation (2.2) can be seen as a first-order approxima-tion of a more general constitutive relaapproxima-tion (single-phase-lagging model; Tzou [57]),
q(x, t+ τq)= −κ∇θ(x, t). (2.4)
The relation (2.4) states that the temperature gradient established at a point x at time t gives rise to a heat flux vector at x at a later time t+ τq. The delay time τqis interpreted
as the relaxation time due to the fast-transient effects of thermal inertia (or small-scale effects of heat transport in time) and is called the phase-lag of the heat flux. It has been confirmed by many experiments that the Cattaneo law generates a more accurate prediction than the classical Fourier law. However, some studies show that the Catta-neo constitutive relation has only taken account of the fast-transient effects, but not the micro-structural interactions. See Tzou [58] for more details.
In [58], Tzou proposed a new theory of heat conduction which describes the interactions between phonons and electrons on the microscopic level as retarding sources causing a delayed response on the macroscopic scale. The physical meanings and the applicability of the dual-phase-lag model have been supported by the experimental results [59]. In this theory the Fourier law is replaced by an approximation of the equation
q(x, t+ τq)= −κ∇θ(x, t + τθ), τq > 0, τθ > 0, (2.5)
where τqis the phase lag of the heat flux and τθ is the phase lag of the gradient of the
temperature. According to the relation (2.5), the temperature gradient at a point x of the material at time t+ τθcorresponds to the heat flux density vector at x at time t+ τq. The
as phonon-electron interaction or phonon scattering, and is called the phase-lag of the temperature gradient (Tzou [58]).
A combination of the constitutive equation (2.5) with the classical energy equation, leads to an ill-posed problem (see [10]). However, if we replace the delay expressions in (2.5) by their Taylor expansions up to some orders, we obtain several heat conduction theories. Indeed, in the case that we only consider the development until the first order in τθand a second order in τq, we obtain a hyperbolic theory, which has been studied and
analysed by Horgan & Quintanilla [16] and Quintanilla & Racke [39]. In particular, the authors in [39] analyzed the dual-phase-lag thermoelasticity, where the heat condition is given by (2.5) with second order approximation for q and first order approximation for θ were used. They showed that under the condition
τθ > τq/2, (2.6)
then solutions of the problem are generated by a semigroup of quasi-contractions. In addition, they showed that solutions of the one-dimensional problem are exponentially stable.
In this chapter, we consider the one-dimensional system of dual-phase-lag thermoelas-ticity. Namely, our system looks like:
utt−αuxx+ τ2 qm 2 θttx+ τqmθtx+ mθx = 0, θttt+ 2 τq θtt+ 2 τ2 q θt + 2mθ0 τ2 q utx− 2τθk τ2 q θtxx− 2k τ2 q θxx = 0, x ∈ R, t > 0, (2.7)
where α, m, θ0, k, τθ and τqare positive constants and satisfy (2.6).
System (2.7) is subjected to the following initial condition u(x, 0) = u0(x), ut(x, 0) = u1(x), θ (x, 0) = ϑ0(x), θt(x, 0) = ϑ1(x), θtt(x, 0) = φ0(x). (2.8)
Expanding both sides of equation (2.5) by using the Taylor series and retaining only the first-order term of τθ and the second order term for τq, we obtain the following
constitutive relation that is valid at point x and time t:
q(x, t)+ τqqt(x, t)+
τ2 q
The relation (2.9) together with the energy equation of the heat conduction yields the equation θttt + 2 τq θtt+ 2 τ2 q θt− 2τθk τ2 q ∆θt − 2k τ2 q ∆θ = 0, (2.10)
which is the second equation in (2.7) (written in one-space dimension and without the coupling with u).
In this chapter, we study the Cauchy problem (2.7)-(2.8) and we show the following decay rate: ∂ k xV(t) L2 ≤ C(1+ t) −1/4−k/2 kV0kL1 + C(1 + t)−l ∂ k+l x V0 L2, (2.11)
for V(x, t) = (ut, ux, θtt+ θt + θ, θx, θtx)(x, t), V0(x) = V(x, 0), k is a nonnegative integer
and C is a positive constant. Obviously, the estimate (2.11) shows that the solution enjoys the decay property of regularity-loss type. To show the main result, we apply the energy method in the Fourier space and build the appropriate Lyapunov functional. This functional gives us the decay rate of the Fourier image of the solution (see (2.34)). Using this decay rate of the Fourier image together with Plancherel theorem, then the estimate (2.11) is proved.
2.2
Energy method
In this section, we prove certain decay estimates of the total energy of the system (2.7 )-(2.8). We employ the technique of converting the original system to a first order system with subsequent Fourier transform in the spatial variable and estimate the solutions of the transformed system. The new system is constructed as follows: setting
then, we get from (2.7) the following system yt− vx = 0, vt−αyx + τ2 qm 2 wx+ τqmz+ mη = 0, ηt− z= 0, x ∈ R, t > 0, ψt− w= 0, zt− wx = 0, wt+ 2 τq w+ 2 τ2 q ψ + 2mθ0 τ2 q vx− 2kτθ τ2 q zx− 2k τ2 q ηx = 0, (2.13)
with the initial data
(y, v, η, ψ, z, w) (x, 0) = (y0, v0, η0, ψ0, z0, w0) (x) . (2.14)
Thus, the system (2.13)-(2.14) can be rewritten as: Ut+ AUx+ LU = 0, U(x, 0) = U0. (2.15)
where U = (y, v, η, ψ, z, w, )T and A, L are matrices defined as
A= 0 −1 0 0 0 0 −α 0 0 0 0 τ 2 qm 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 2mθ0 τ2 q −2kτ2 q 0 − 2τθk τ2 q 0 , L= 0 0 0 0 0 0 0 0 m 0 τqm 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 τ22 q 0 2 τq .
As it has been shown in [39], the matrix A has real eigenvalues and thus, system (2.15) is hyperbolic.
Remark2.1. It is well known (see for instance [44, Theorem 5.1]) that for U0 ∈ Hs(R), s ∈
N and s ≥ 2, then problem (2.15) has a unique solution U such that U ∈ C0([0, ∞), Hs(R)) ∩ C1([0, ∞), Hs−1(R)).
By taking the Fourier transform of (2.13), we obtain ˆyt− iξˆv = 0, ˆvt−αiξˆy + τ2 qm 2 iξ ˆw + τqmˆz+ mˆη = 0, ˆηt− ˆz= 0, ξ ∈ R, t > 0, ˆ ψt− ˆw= 0, ˆzt− iξ ˆw = 0, ˆ wt+ 2 τq ˆ w+ 2 τ2 q ˆ ψ + 2mθ0 τ2 q iξˆv − 2kτθ τ2 q iξˆz − 2k τ2 q iξ ˆη = 0. (2.16)
The initial data (2.14) takes form
(ˆy, ˆv, ˆη, ˆψ, ˆz, ˆw) (ξ, 0) = (ˆy0, ˆv0, ˆη0, ˆψ0, ˆz0, ˆw0) (ξ) . (2.17)
Let us now define the first energy functional
ˆ E1(ξ, t) = 1 2 ( 4αθ0 τ4 q |ˆy|2+ 4θ0 τ4 q |ˆv|2+ 4k τ2 q Reˆη¯ˆz + 2kτθ τ2 q |ˆz|2+ 2 τ2 q ψˆ 2 + |ˆη|2+ | ˆw|2 ) . (2.18)
Lemma 2.2. Let ˆU(ξ, t) = (ˆy, ˆv, ˆη, ˆψ, ˆz, ˆw)T(ξ, t) be the solution of (
2.16)-(2.17). Then for any t ≥0 and ε1 > 0, the estimate
d dtEˆ1(ξ, t) ≤ − 2 τq | ˆw|2+ c (ε1) |ˆz|2+ 2ε1|ˆv|2+ c (ε1) | ˆη|2, (2.19) holds true.
Proof. Multiplying the first equation in (2.16) by 4αθ0
τ4
q ¯ˆy, the second equation by
4θ0
τ4 q ¯ˆv,
the third equation by ¯ˆη the fourth equation by τ22
q ¯ˆψ, the fifth equation by
2kτθ
τ2
q ¯ˆz, the sixth
equation by ¯ˆw, adding the results and taking the real part, we get 1 2 d dt ( 4αθ0 τ4 q |ˆy|2+ 4θ0 τ4 q |ˆv|2+ |ˆη|2+ 2 τ2 q ψˆ 2 + | ˆw|2+ 2kτθ τ2 q |ˆz|2 ) +τ2 q | ˆw|2+ Re ( τqm 4θ0 τ4 q ¯ˆvˆz+ 4mθ0 τ4 q ¯ˆvˆη − ¯ˆηˆz − 2k τ2 q iξ ¯ˆwˆη ) = 0. (2.20)
On the other hand, we have Re 2k τ2 q iξ ¯ˆwˆη ! = d dtRe 2k τ2 q ˆη¯ˆz ! − 2k τ2 q |ˆz|2. (2.21)
Inserting (2.21) into (2.20), to get d dtEˆ1(ξ, t) = − 2 τ2 q | ˆw|2+ 2k τ2 q |ˆz|2+ Re ( ˆz ¯ˆη − 4mθ0 τ3 q ˆz¯ˆv − 4mθ0 τ4 q ˆη¯ˆv ) . (2.22)
A simple application of Young’s inequality yields (2.19). This finishes the proof of
Lemma2.2.
In order to define another energy term, and following [39], we construct a new system as follows: setting
v= ut, y= ux, θ =˜
τ2 q
2θtt+ τqθt+ θ, θ∗= θ + τθθt, (2.23) then, we get from (2.7) the following system:
yt− vx = 0, vt−αyx+ m˜θx = 0, x ∈ R, t > 0, ˜ θt + mθ0vx− kθ∗xx = 0, (2.24)
with the initial data
(y, v, ˜θ) (x, 0)= (y0, v0, ˜θ0) (x) . (2.25)
By taking the Fourier transform of (2.24), we obtain ˆyt− iξˆv = 0, ˆvt− iξαˆy + iξm ˆ˜θ = 0, ξ ∈ R, t > 0, ˆ˜θt+ iξmθ0ˆv+ ξ2k ˆθ∗= 0. (2.26)
The initial data (2.25) takes form
Let us now define another energy term by ˆ E2(ξ, t) = 1 2 αθ0|ˆy|2+ θ0|ˆv|2+ |ˆ˜θ|2+ kτq+ τθ ξ2| ˆθ|2+ kτθτ2q 2 ξ 2| ˆθ t|2+ kτ2qReξ 2¯ˆθ tθˆ . (2.28)
Now, we show a dissipativity identity which states that the energy ˆE2(ξ, t) is a
non-increasing function, that is we have the following:
Lemma 2.3. Let ˆU(ξ, t) = (ˆy, ˆv, ˆ˜θ)T(ξ, t) be the solution of (2.26)-(2.27). Then for any t ≥0, the identity d dtEˆ2(t)= −kτqξ 2τ θ− τq 2 | ˆθt|2− kξ2| ˆθ|2, (2.29)
holds. In addition, assume that(2.6) is satisfied. Then, there exist two positive constants c1and c2such that
c1| ˆV(ξ, t)|2≤ ˆE2(ξ, t) ≤ c2| ˆV(ξ, t)|2, ∀t ≥ 0, (2.30)
where
| ˆV(ξ, t)|2 = |ˆy|2+ |ˆv|2+ |ˆ˜θ|2+ ξ2| ˆθ|2+ ξ2| ˆθt|2. (2.31)
Proof. Multiplying the first equation in (2.26) by αθ0¯ˆy, the second equation by θ0¯ˆv, the
third equation by ¯ˆ˜θ, adding the results and taking the real part to get 1 2 d dt nαθ0|ˆy| 2+ θ 0|ˆv|2+ |ˆ˜θ|2o + kReξ2¯ˆ˜θˆθ∗ = 0. (2.32)
On the other hand, we have
Reξ2 k¯ˆ˜θˆθ∗ = ξ2 kRe τ 2 q 2 ¯ˆθtt+ τq¯ˆθt+ ¯ˆθ ˆθ + τθθˆt = ξ2 kRe τ2 q 2 ¯ˆθttθ + τˆ q¯ˆθtθ + |ˆθ|ˆ 2+ τθτ 2 q 2 ¯ˆθttθˆt+ τqτθ| ˆθt| 2+ τ θθˆt¯ˆθ = ξ2kd dt τ q 2 + τθ 2 | ˆθ|2+ τθτ 2 q 4 | ˆθt| 2 + ξ2k| ˆθ|2+ ξ2kτ qτθ| ˆθt|2+ ξ2kRe τ2 q 2 ¯ˆθttθˆ . Also, we have kτ2 q 2 Reξ 2¯ˆθ ttθˆ = kτ2 q 2 d dtReξ 2¯ˆθ tθˆ − kτ 2 q 2 ξ 2| ˆθ t|2.
Consequently, we get from above Re ξ2k¯ˆ˜θˆθ ∗ = d dt ξ2kτq 2 + τθ 2 | ˆθ|2+ ξ2kτθτ 2 q 4 | ˆθt| 2+ kτ 2 q 2 Reξ 2¯ˆθ tθˆ +ξ2 k| ˆθ|2+ ξ2kτq τθ− τq 2 | ˆθt|2.
Inserting the previous identity into (2.32), we obtain 1 2 d dt αθ0|ˆy|2+ θ0|ˆv|2+ |ˆ˜θ|2+ kτ2qReξ2¯ˆθtθˆ + k τθ + τq ξ2| ˆθ|2+ kτθτ2q 2 ξ 2| ˆθ t|2 = −ξ2k τqτθ− τ2 q 2 | ˆθt| 2− kξ2| ˆθ|2. This yields (2.29).
Now, to prove (2.30), we apply Young’s inequality to the last term in (2.28), we obtain for any ε > 0, kτ2 qReξ 2¯ˆθ tθˆ ≤ kτ2 qεξ 2| ˆθ t|2+ kτ2q 1 4εξ 2| ˆθ|2. (2.33) Plugging the inequality (2.33) into (2.28), we get (2.30), where
c1 = 12min n 2k(τq+ τθ) − kτ2 q 2ε , αθ0, θ0, kτθτ2q− 2kτ2qε, 1o, c2 = 12max n kτθτ2q+ 2kτ2qε , αθ0, θ0, 2k(τq+ τθ)+ kτ2 q 2ε) , 1o. Now, to ensure that the constant c1is positive, we have to fix ε > 0, such that
τq
4 < ε < τθ
2.
This choice is possible since τθ > τ2q. Thus the proof of Lemma2.3is finished.
The following proposition will play a key role in the proof of our main result.
Proposition 2.4. Assume that(2.6) holds. Then for any t ≥ 0 and ξ ∈ R, we have the following pointwise estimate
Vˆ (ξ, t) 2 ≤ Ce−cρ(ξ)t Vˆ (ξ, 0) 2 , (2.34) where ρ (ξ) = ξ2 1+ ξ22 (2.35) and Vˆ (ξ, t) 2
Proof. The proof of the above proposition is based on the energy method in the Fourier space and will be done through several steps.
Multiplying the second equation in (2.16) by iξ ¯ˆy and the first equation by −iξ ¯ˆv, adding the resulting equalities and taking the real part, we have
d dtF(ξ, t) − ξ 2|ˆv|2+ αξ2|ˆy|2+ Re τqmiξ¯ˆyˆz + miξ¯ˆyˆη − ξ2 τ2 qm 2 w¯ˆˆy = 0, (2.36) where F(ξ, t) = Reiξ¯ˆyˆv .
Multiplying the sixth equation in (2.16) by −iξ ¯ˆv and the second equation by iξ ¯ˆw, adding the resulting equalities and taking the real part, we have
d dtRe −iξ¯ˆv ˆw + ξ22mθ0 τ2 q |ˆv|2−ξ2τ 2 qm 2 | ˆw| 2− Re( 2 τq iξ¯ˆv ˆw + 2 τ2 q iξ¯ˆv ˆψ + ξ22kτθ τ2 q ¯ˆvˆz ) +Re ( ξ2αˆy ¯ˆw − ξ22k τ2 q ¯ˆvˆη+ τqmiξ ¯ˆwˆz + miξ ¯ˆwˆη ) = 0 (2.37) Multiplying the third equation in (2.16) by −τ22
q¯ˆv and the second equation by −
2 τ2
q¯ˆη,
adding the resulting equalities and taking the real part, we have
d dtRe − 2 τ2 q ¯ˆvˆη ! − 2m τ2 q | ˆη|2+ Re( 2 τ2 q ˆz¯ˆv+ α2 τ2 q
iξ ¯ˆηˆy − miξ ˆw¯ˆη − 2 τq
mˆz ¯ˆη )
= 0. (2.38) Multiplying the fifth equation in (2.16) by −τ2
q¯ˆv and the second equation by −
2 τq¯ˆz, we get as above d dtRe − 2 τq ¯ˆvˆz ! − 2m |ˆz|2+ Re( 2 τq iξ ˆw¯ˆv +2α τq iξ¯ˆzˆy − τqmiξ ˆw¯ˆz − 2m τq ˆη¯ˆz ) = 0. (2.39) Adding (2.37), (2.38), (2.39), and using the fact that ˆz= iξ ˆψ, we obtain
d dtG(ξ, t) + ξ 22mθ0 τ2 q |ˆv|2−ξ2τ 2 qm 2 | ˆw| 2− 2m τ2 q | ˆη|2− 2m |ˆz|2 +Re ( ξ2αˆy ¯ˆw − ξ22kτθ τ2 q ¯ˆvˆz − ξ22k τ2 q ¯ˆvˆη+ τqmiξ ¯ˆwˆz + miξ ¯ˆwˆη + α 2 τ2 q iξ ¯ˆηˆy ) (2.40) +Re( 2ατ q iξ¯ˆzˆy − miξ ˆw¯ˆη − 2 τq mˆz ¯ˆη − τqmiξ ˆw¯ˆz − 2m τq ˆη¯ˆz ) = 0,
where G(ξ, t) = −Reiξ¯ˆv ˆw + 2 τq ¯ˆvˆz+ τ22 q ¯ˆvˆη. Computing τθtimes (2.40)+ mθτ0 q times (2.36), we obtain d dtΦ (ξ, t) + ξ 22mθ0 τ2 q τθ− τq 2 |ˆv|2+ ξ2mαθ0 τq |ˆy|2− 2mτθ|ˆz|2−ξ2 τ2 qmτθ 2 | ˆw| 2− 2mτθ τ2 q | ˆη|2 +Re ( τqτθmiξ ¯ˆwˆz − ξ2 2kτ2θ τ2 q ¯ˆvˆz − ξ22kτθ τ2 q ¯ˆvˆη+ τθmiξ ¯ˆwˆη + α2ττ2θ q iξ ¯ˆηˆy ) (2.41) +Re( 2αττ θ q iξ¯ˆzˆy − mτθiξ ˆw¯ˆη − 2ττθ q mˆz ¯ˆη − τqmτθiξ ˆw¯ˆz − 2mτθ τq ˆη¯ˆz+ m2θ0iξ¯ˆyˆz ) +Re( m2θ0 τq iξ¯ˆyˆη + ξ2 ατθ− τqθ0m2 2 ! ˆy ¯ˆw ) , where the functionalΦ is defined by
Φ (ξ, t) := τθG(ξ, t)+ mθτ 0 q F(ξ, t) = Re −τθiξ¯ˆv ˆw − 2ττθ q ¯ˆvˆz − 2τθ τ2 q ¯ˆvˆη+ mθ0 τq iξ¯ˆyˆv ! .
Multiplying the first equation in (2.16) by iξ¯ˆz and the fifth equation by −iξ ¯ˆy, adding the resulting equalities and taking the real part, we have
d dtRe n iξ¯ˆzˆyo + ξ2 Ren ¯ˆzˆv − ¯ˆy ˆwo = 0. (2.42) Computing (2.41)+ατθ −τqθ0m2/2 times (2.42), we get d dtΓ (ξ, t) + ξ 22mθ0 τ2 q τθ− τq 2 |ˆv|2+ ξ2mαθ0 τq |ˆy|2−ξ2τ 2 qmτθ 2 | ˆw| 2− 2mτ θ|ˆz|2− 2mττ2 θ q | ˆη|2 +Re ( τqτθmiξ ¯ˆwˆz − ξ2 2kτ2 θ τ2 q ¯ˆvˆz − ξ22kτθ τ2 q ¯ˆvˆη+ τθmiξ ¯ˆwˆη + α2ττ2θ q
iξ ¯ˆηˆy − mτθiξ ˆw¯ˆη
) +Re( 2αττ θ q iξ¯ˆzˆy −2τθ τq mˆz ¯ˆη − τqmτθiξ ˆw¯ˆz − 2mτθ τq ˆη¯ˆz ) +Re ( m2θ0iξ¯ˆyˆz + m2θ 0 τq iξ¯ˆyˆη + ξ2 ατ θ− τqθ0m2 2 ! ¯ˆzˆv ) = 0, (2.43) where Γ (ξ, t) = Φ (ξ, t) + Re ( ατθ− τqθ0m2 2 iξ¯ˆzˆy ) .
Multiplying the fifth equation in (2.16) by iξ ¯ˆwand the sixth equation by −iξ¯ˆz, adding the resulting equalities and taking the real part, we have
d dtRe iξˆz ¯ˆw −ξ22kτθ τ2 q |ˆz|2+ ξ2| ˆw|2+ Re ( ξ22mθ0 τ2 q ˆv¯ˆz − 2 τq iξ¯ˆz ˆw − ξ22k τ2 q ˆη¯ˆz − 2 τ2 q iξ¯ˆz ˆψ ) = 0. (2.44) Multiplying the third equation in (2.26) by iξ2
τ2
q ¯ˆψ and the fourth equation by −iξ
2 τ2
q¯ˆη, we
have, by the same method
d dtRe iξ 2 τ2 q ¯ˆψˆη!+ Re ( iξ2 τ2 q ¯ˆη ˆw − iξ 2 τ2 q ¯ˆψˆz) = 0. (2.45) Summing up (2.44) and (2.45), we get
d dtΨ (ξ, t) + ξ 2| ˆw|2−ξ22kτθ τ2 q |ˆz|2+ Re ( iξ 2 τ2 q ¯ˆη ˆw − 2 τq iξ¯ˆz ˆw + ξ22mθ0 τ2 q ¯ˆzˆv − ξ22k τ2 q ˆη¯ˆz ) = 0 (2.46) where Ψ (ξ, t) = Re iξˆz ¯ˆw + Re iξ2 τ2 q ¯ˆψˆη!. Computing τ2 θτqmtimes (2.46)+(2.43), we obtain d dtΘ (ξ, t) + ξ 22mθ0 τ2 q τθ− τq 2 |ˆv|2+ ξ2τθτqm τθ− τq 2 | ˆw|2 +ξ2mαθ0 τq |ˆy|2− 2mτθ τ2 q | ˆη|2−ξ22kτ 3 θm τq |ˆz|2− 2mτθ|ˆz|2 +Re ( τqτθmiξ ¯ˆwˆz − ξ2 2kτ2 θ τ2 q ¯ˆvˆz − ξ22kτθ τ2 q ¯ˆvˆη+ τθmiξ ¯ˆwˆη + α2ττ2θ q
iξ ¯ˆηˆy − mτθiξ ˆw¯ˆη
) +Re( 2αττ θ q iξ¯ˆzˆy −2τθ τq mˆz ¯ˆη − τqmτθiξ ˆw¯ˆz − 2mτθ τq ˆη¯ˆz ) +Re ( m2θ0iξ¯ˆyˆz + m2θ 0 τq iξ¯ˆyˆη + ξ2 ατθ − τqθ0m2 2 ! ¯ˆzˆv − 2τ2 θmiξ¯ˆz ˆw ) +Re ( iξ2τ 2 θm τq ¯ˆη ˆw − ξ22τ 2 θm2θ0 τq ¯ˆzˆv+ ξ22τ 2 θmk τq ˆη¯ˆz ) = 0, where Θ (ξ, t) = Γ (ξ, t) + τ2 θτqmΨ(ξ, t). (2.47)
Appling Young’s inequality, we obtain, for any ε2 > 0, d dtΘ (ξ, t) + ξ 22mθ0 τ2 q τθ− τq 2 −ε2 |ˆv|2+ ξ2 αmθ0 τq −ε2 ! |ˆy|2 +ξ2τ qm τθ− τq 2 −ε2 | ˆw|2 (2.48) ≤ c(ε2) 1+ ξ2|ˆz|2+ c (ε2) 1+ ξ2| ˆη|2. Now, we define the Lyapunov functional L(ξ, t) as:
L(ξ, t) := N1+ ξ2 ˆE2(ξ, t) + ξ2Eˆ1(ξ, t) + Θ (ξ, t) , (2.49)
where N is a large positive constant that have to be chosen later. Taking the derivative of L(ξ, t) with respect to t, and exploiting (2.19), (2.29) and (2.48), we obtain
d dtL(ξ, t) + ξ 2( 2mθ0 τ2 q τθ− τq 2 −ε2 − 2ε1 ) |ˆv|2+ ξ2 αmθ0 τq −ε2 ! |ˆy|2 +ξ2( 2 τ2 q + τqm τθ− τq 2 −ε2 ) | ˆw|2+1+ ξ2 (Nkτq τθ− τq 2 − c(ε1, ε2) |ˆz|2 + 1+ ξ2(kN − c (ε1, ε2)) | ˆη|2 ≤ 0, (2.50) where we have used the fact that ξ2|θ|2 = |η|2and ξ2|θ
t|2= |z|2. Our goal now is to choose
the constants ε1, ε2and N in (2.50) as follows:
First, we fix ε2 small enough such that
ε2< min ( τθ− τq 2 ,αmθ0 τq ) . Once ε2is fixed, we choose ε1 small enough such that
ε1 < 2mθ0 τ2 q τθ − τq 2 −ε2 /2. Once ε1and ε2are fixed, we choose N large enough such that
N > max c(ε1, ε2)/kτq τθ− τq 2 , c (ε1, ε2)/k . Consequently, the estimate (2.50) takes the form
d
where δ is a positive constant and
W(ξ, t) = ξ2|ˆy|2+ |ˆv|2+ | ˆw|2+ |ˆz|2+ |ˆη|2 + |ˆz|2+ |ˆη|2. (2.52) On the other hand, it is not hard to see that for N large enough there exist two positive constants β1and β2such that for all t ≥ 0, we have
β1
1+ ξ2 ˆE2(ξ, t) ≤ L (ξ, t) ≤ β2
1+ ξ2 ˆE2(ξ, t) . (2.53)
To prove the estimates in (2.53), we have from (2.49) that L(ξ, t) − N 1+ ξ2 ˆE2(ξ, t) = ξ 2Eˆ 1(ξ, t) + Θ (ξ, t) . (2.54)
Exploiting (2.18), (2.30) and (2.47), and using the fact that ˆ w= 2 τ2 q ˆ˜θ − 2 τq ˆ θt− 2 τ2 q ˆ θ, and ξ2 ψˆ 2 = |ˆz|2, ξ2 θˆt 2 = |ˆz|2, ξ2 θˆ 2 = |ˆη|2,
then, the right-hand side of (2.54) can be estimated as ξ 2ˆ E1(ξ, t) + Θ (ξ, t) ≤ C˜1nξ 2| ˆ˜θ|2+ 1+ ξ2|ˆv|2+1+ ξ2| ˆη|2+1+ ξ2|ˆz|2+ |ˆy|2o , ≤ C˜1 1+ ξ2 |ˆy|2+ |ˆv|2+ |ˆ˜θ|2+ |ˆz|2+ |ˆη|2 . (2.55) From (2.30) and (2.55), we deduce that
ξ 2Eˆ 1(ξ, t) + Θ (ξ, t) ≤ ˜ C1 c1 1+ ξ2 ˆE2(ξ, t) ≤ C˜1+ ξ2 ˆE2(ξ, t) .
where ˜C1and ˜C are two positive constants. Consequently, (2.54) leads to
N − ˜C 1+ ξ2 ˆE2(ξ, t) ≤ L (ξ, t) ≤
N+ ˜C 1+ ξ2 ˆE2(ξ, t) .
From this last inequality, it is obvious that (2.53) holds for β1= N − ˜C and β2 = N + ˜C.
On the other hand, from (2.52) and (2.30), we deduce that W(ξ, t) ≥ δ1 ξ2 1+ ξ2 |ˆy|2+ |ˆv|2+ |ˆ˜θ|2+ |ˆz|2+ |ˆη|2 (2.56) ≥ δ1 c2 ξ2 1+ ξ2Eˆ2(ξ, t) , ∀t ≥ 0,
where δ1is a positive constant. Now, combining (2.51), (2.53) and (2.56), we have
d dtL(ξ, t) ≤ − δδ1 β2c2 ξ2 1+ ξ22L(ξ, t) , ∀t ≥ 0. (2.57) Integrating (2.57) with respect to t and exploiting once again (2.53), we get (2.34). This
completes the proof of Proposition2.4.
2.3
Decay estimates
In this section, we prove a decay estimate for an energy term associated to the solution of (2.16). Our mean result reads as follows:
Theorem 2.5. Let s be a nonnegative integer and let V(x, t) = (y, v, ˜θ, θx, θtx)(x, t).
As-sume that V0 = V(x, 0) ∈ Hs(R) ∩ L1(R) . Suppose that (2.6) holds. Then V(x, t)
satisfies the following decay estimates:
∂ k xV(t) L2 ≤ C(1+ t) −1/4−k/2 kV0kL1 + C(1 + t)−l ∂ k+l x V0 L2, (2.58)
Proof. The proof is essentially based on the pointwise estimate in Proposition2.4. In-deed, applying the Plancherel theorem, we may write
∂ k xV(t) 2 L2 = ˆ R |ξ|2k Vˆ(ξ, t) 2 dξ, ≤ C ˆ R |ξ|2k V(ξ, t)ˆ 2 dξ, ≤ C ˆ R |ξ|2k e−ρ(ξ)t V(ξ, 0)ˆ 2 dξ, ≤ C ˆ |ξ|≤1 |ξ|2ke−ρ(ξ)t V(ξ, 0)ˆ 2 dξ + C ˆ |ξ|≥1 |ξ|2ke−ρ(ξ)t V(ξ, 0)ˆ 2 dξ, = L1+ L2. (2.59)
It is known that the decay of the solution depends on the low frequency part L1and the
regularity of the solution depends the high frequency part L2. It is not hard to see that
ρ1(ξ) ≥ cξ2for |ξ| ≤ 1. Thus, we have
L1 = C ˆ |ξ|≤1 |ξ|2ke−ρ(ξ)t V(ξ, 0)ˆ 2 dξ ≤ C sup |ξ|≤1 Vˆ(ξ, 0) 2 ˆ |ξ|≤1 |ξ|2ke−ctξ2dξ. (2.60) By using the inequality
1
ˆ
0
|ξ|σ
e−ctξ2dξ ≤ C(1 + t)−(σ+1)/2, we deduce from (2.60) that
L1≤ C(1+ t)−
1 2−k||V
On the other hand, using the fact since ρ1(ξ) ≥ cξ−2, then we get L2 = C ˆ |ξ|≥1 |ξ|2k e−ρ(ξ)t V(ξ, 0)ˆ 2 dξ ≤ C sup|ξ|−2le−cξ−2t ˆ |ξ|≥1 |ξ|2(k+l) V(ξ, 0)ˆ 2 dξ ≤ C(1+ t)−l ∂ k+l 0 V0 L2. (2.62)
Combining (2.59), (2.60) and (2.62), we obtain the estimate (2.58). This completes the
Decay property for solutions in the
three-phase-lag heat conduction
3.1
Introduction
In this chapter we consider two models that can be obtained by taking the Taylor series expansion of (8) up to the first or second-order terms in τθ, τq and τν. These models
have been recently investigated in [40] where an exponential stability has been shown in bounded domains.
Here we consider the Cauchy problem of these models and by using the energy method in the Fourier space, we build the appropriate Lyapunov functional for each model and show the optimal (compared to the heat kernel) decay rate of solutions provided that the coefficients satisfy appropriate assumptions. This chapter is organized as follows: In Section3.2, we study the stability of the solutions of the first model, while Section3.3 is devoted to the analysis of the second model.
3.2
The third order model
In this section, we consider the Taylor series expansion of (8) up to the first-order terms in τθ, τqand τν. That is
q+ τqqt = − [k (θx+ τθθxt)+ k∗(νx+ τννxt)]
= − τ∗
νθx+ kτθθxt+ k∗νx, (3.1)
where we have used the relation νt = θ with τ∗ν = k + k∗τν.
Equation (3.1) together with the heat conservation law:
ρcυθt(x, t)+ qx(x, t)= 0, (3.2)
leads to the following equation
τqρcυθttt + ρcυθtt− k∗θxx−τ∗νθtxx− kτθθttxx = 0, θ (x, 0) = θ0(x), θt(x, 0) = θ1(x), θtt(x, 0) = θ2(x), (3.3)
where x ∈ R and t ≥ 0. All the coefficients are positive constants. Our goal now is to show the optimal decay rate for the solution of (3.3).
3.2.1
The energy method in the Fourier space
This subection is devoted to the proof of the pointwise estimates of the Fourier image of the solution of (3.3). For simplicity, we write our system (3.3) as a first-order (in time) system. Indeed, we introduce the following variables:
y= θt+ τqθtt, w= θx+ τqθtx, z = θtx, (3.4)
then, the resulting system takes the form
wt− yx = 0, ρcυyt+ kτθ τq + k∗τ q−τ∗ν ! zx− k∗wx − kτθ τq yxx = 0, τqzt− yx+ z = 0, x ∈ R, t > 0, (3.5)
with the initial data
(w, y, z) (x, 0) = (y0, w0, z0) (x). (3.6)
Taking the Fourier transform of system (3.5), we obtain ˆ wt− iξˆy = 0, ρcυˆyt+ iξ kτθ τq + k∗τ q−τ∗ν ! ˆz − iξk∗wˆ + ξ2kτθ τq ˆy= 0, τqˆzt− iξˆy + ˆz = 0, ξ ∈ R, t > 0. (3.7)
The initial data (3.6) takes the form
( ˆw, ˆy, ˆz) (ξ, 0) = ( ˆw0, ˆy0, ˆz0) (ξ) . (3.8)
Let us define the energy functional of system (3.7)-(3.8) as follows: ˆ E(ξ, t) = k∗| ˆw|2+ ρcυ|ˆy|2+τq(τ ∗ ν− k∗τq) − kτθ |ˆz|2. (3.9) Lemma 3.1. Let ˆU(ξ, t) = ( ˆw, ˆy, ˆz)T(ξ, t) be the solution of (3.7)-(3.8), then for any t ≥0, the identity d dtEˆ(ξ, t) = kτθ τq + k∗τ q−τ ∗ ν |ˆz|2−ξ2kτθ τq |ˆy|2, (3.10) holds.
Proof. Multiplying the first equation in (3.7) by k∗¯ˆw, the second equation by ¯ˆy, the third equation by (kττν
q + k
∗τ
q −τ∗ν)¯ˆz, adding the resulting equalities and taking the real part,
then (3.10) follows. This finishes the proof of Lemma3.1. To ensure that the energy is positive and non-increasing function, it is necessary to impose the assumption
τ∗ ν > kτθ τq + k∗τ q. (3.11)
Lemma 3.2. Assume that (3.11) is satisfied. Let ˆU(ξ, t) = ( ˆw, ˆy, ˆz) be the solution of system (3.7)-(3.8). Then for any t ≥ 0 and ξ ∈ R, we have the following pointwise estimate Uˆ (ξ, t) 2 ≤ Ce−cρ(ξ)t Uˆ (ξ, 0) 2 , (3.12)
where
ρ (ξ) = ξ2
1+ ξ2. (3.13)
Here C and c are two positive constants.
Proof. Multiplying the first equation in (3.7) by −iρcυξ¯ˆy and the second equation by
iξ ¯ˆw, adding the resulting equalities and taking the real part, we have d dtRe(iρcυξ ¯ˆwˆy) + ξ 2(k∗ | ˆw|2−ρcυ|ˆy|2)= Re ξ2kτθ τq + k∗τ q−τ∗ν ˆz ¯ˆw ! − Re iξ3kτθ τq ˆy ¯ˆw !
Applying Young’s inequality, we obtain, for any ε > 0 d dtF(ξ, t)+ (k ∗ −ε)ξ2| ˆw|2 ≤ C(ε)(ξ2+ ξ4) |ˆy|2+ C(ε)ξ2|ˆz|2, (3.14) where F(ξ, t)= Re(iρcυξ ¯ˆwˆy). (3.15)
We define the Lyapunov functional L(ξ, t) as:
L(ξ, t)= 1
1+ ξ2F(ξ, t)+ N ˆE(ξ, t), (3.16)
where N is a large positive constant that will be chosen later. Taking the derivative of L(ξ, t) with respect to t, we obtain
d dtL(ξ, t)+ ξ2 1+ ξ2(k ∗−ε) | ˆw|2+ ( N −kτθ τq − k∗τq+ τ∗ν ! − C(ε) ) |ˆz|2 +ξ2 Nkτθ τq − C(ε) ! |ˆy|2 ≤ 0, ∀t ≥ 0. (3.17)
Keeping in mind (3.11), choosing ε small enough such that ε < k∗and N large enough such that N > max C() (τ∗ ν − kττqθ − k ∗τ q) ,C() τq kτθ . With these choices, (3.17) takes the form
d dtL(ξ, t)+ ηΦ(ξ, t) ≤ 0, ∀t ≥ 0, (3.18) where Φ(ξ, t) = ξ2 1+ ξ2 | ˆw| 2+ ξ2|ˆy|2+ |ˆz|2 (3.19)
and η is a positive constant . On the other hand, it is not hard to see that for N large enough there exist two positive constants δ1and δ2such that for all t ≥ 0, we have
δ1E(ξ, t) ≤ L(ξ, t) ≤ δˆ 2E(ξ, t).ˆ (3.20)
Also, from (3.19) and (3.9), we deduce that Φ(ξ, t) ≥ c ξ2
1+ ξ2E(ξ, t),ˆ ∀t ≥ 0. (3.21)
Now, combining (3.18), (3.20) and (3.21), we have dL(ξ, t) dt ≤ − ηc δ2 ξ2 (1+ ξ2)L(ξ, t), ∀t ≥ 0. (3.22)
Integrating (3.22) with respect to t and exploiting once again (3.20), we get (3.12). This
completes the proof of Lemma3.2.
3.2.2
Decay estimates
In this subsection, we prove the decay estimates of the L2-norm for the solution of
(3.5)-(3.6). Thus, we have
Theorem 3.3. (L1-initial data) Let s be a nonnegative integer and assume that U
0 =
(w, y, z)T
∈ Hs(R) ∩ L1(R) . Assume that (3.11) holds. Then the solution U = (w, y, z)T of system (3.5)-(3.6) satisfies the following decay estimates:
∂ k xU(t) L2 ≤ C(1+ t) −1/4−k/2kU 0kL1 + Ce−ct ∂ k xU0 L2, (3.23)
Proof. The proof is essentially based on the pointwise estimate in Lemma3.2. Indeed, applying the Plancherel theorem, we may write
∂ k xU(t) 2 L2 = ˆ R |ξ|2k |U(ξ, t)|2dξ ≤ C ˆ R |ξ|2ke−cρ(ξ)t U(ξ, 0)ˆ 2 dξ ≤ C ˆ |ξ|≤1 |ξ|2k e−cρ(ξ)t U(ξ, 0)ˆ 2 dξ + C ˆ |ξ|≥1 |ξ|2k e−cρ(ξ)t U(ξ, 0)ˆ 2 dξ = L1+ L2. (3.24)
The integral here is divided into two parts: the low-frequency part (|ξ| ≤ 1) and the high-frequency part (|ξ| ≥ 1). As ρ(ξ) ≥ 12ξ2 for |ξ| ≤ 1, then we have for the low-frequency
part that L1 ≤ C Uˆ0 2 L∞ ˆ |ξ|≤1 |ξ|2k e−cξ2tdξ. (3.25)
Using the inequality
1
ˆ
0
|ξ|σ
e−cξ2tdξ ≤ C (1 + t)−(σ+1)2 , (3.26)
we deduce from (3.25) that
L1 ≤ C(1+ t) −1
2−kkU
0k2L1. (3.27)
For the high-frequency part, L2we have ρ(ξ) ≥ 12, and therefore
L2 ≤ Ce−ct ˆ |ξ|≥1 |ξ|2k U(ξ, 0)ˆ 2 dξ ≤ Ce−ct ∂ k xU0 2 2. (3.28)
Inserting (3.27) and (3.28) into (3.24), we obtain the estimate (3.23). In the next theorem, we show that the decay rate given in Theorem3.3can be improved for initial data in some weighted spaces of L1and with zero total mass.
Theorem 3.4. (L1,γ-initial data) Let γ ∈ [0, 1]. Let s be a nonnegative integer and assume U0 = (w, y, z) T ∈ Hs (R) ∩ L1,γ(R) such that´ R U0(x)dx= 0 . Assume that (3.11)
holds. Then the solution U = (w, y, z)Tof system (3.5)-(3.6) satisfies the following decay estimates: ∂ k xU(t) L2 ≤ C(1+ t) −1/4−(k+γ)/2kU 0kL1,γ + Ce−ct ∂ k xU0 L2 (3.29)
for k ≤ s and C, c are two positive constants.
The proof of Theorem3.4can be done following the same ideas as in [45] and [49]. We omit the details.
3.3
The Fourth order model
Retaining terms of the order of τ2qin the Taylor expansion of (8), we obtain
q+ τqqt + τ2qqtt = − τ∗νθx+ kτθθxt+ k∗νx. (3.30)
Taking the divergence of both sides in (3.30) and combining the result with (3.2), we get τqρcυθttt+ ρcυθtt+ τ2 q 2ρcυθtttt− k ∗θ xx−τ∗νθtxx− kτθθttxx= 0, θ (x, 0) = θ0(x), θt(x, 0) = θ1(x), θtt(x, 0) = θ2(x), θttt(x, 0)= θ3(x), (3.31) where x ∈ R and t ≥ 0. All the coefficients are positive constants. We introduce the following variables: y= θt+τqθtt+ τ2 q 2θttt, w= θx+τqθxt+ τ2 q 2θxtt, z= θxt+ τq 2θxtt, h= θxt, ϕ = θxtt.
Then, the resulting system takes the form wt− yx = 0, τqzt− yx+ h = 0, ρcυyt− k∗wx− 2kτθ τq zx+ 2kτθ τq + k∗τ q−τ∗ν ! hx+ k∗τ2 q 2 ϕx = 0, x ∈ R, t > 0, ht−ϕ = 0, τ2 q 2ϕt+ τqϕ + h − yx = 0, (3.32) with the initial data
(w, z, y, h, ϕ) (x, 0) = (w0, z0, y0, h0, ϕ0) (x). (3.33)
Taking the Fourier transform of system (3.32), we obtain ˆ wt− iξˆy = 0, τqˆzt− iξˆy + ˆh = 0,
ρcυˆyt− iξk∗w − iξˆ
2kτθ τq ˆz+ iξ 2kτθ τq + k∗τ q−τ∗ν ! ˆh+ iξk ∗τ2 q 2 ϕ = 0, ξ ∈ R, t > 0ˆ ˆht− ˆϕ = 0, τ2 q 2ϕˆt+ τqϕ + ˆh − iξˆy = 0,ˆ (3.34) The initial data (3.33) takes form
ˆ w, ˆz, ˆy, ˆh, ˆϕ (ξ, 0) =wˆ0, ˆz0, ˆy0, ˆh0, ˆϕ0 (ξ). (3.35)
Let us define the energy functional of system (3.34)-(3.35) as follows:
ˆ E (ξ, t) = 1 2 k∗| ˆw|2+ τqτ∗ν − k∗τq |ˆz|2+ ρcυ|ˆy|2+ kτθ− k ∗τ 2 q 2 ˆh 2 + τ 2 q 2 kτθ− τqτ∗ν 2 ! | ˆϕ|2 . (3.36)
Lemma 3.5. Let ˆV(ξ, t) = w, ˆz, ˆy, ˆh, ˆϕˆ T(ξ, t) be the solution of (3.34)-(3.35), then for any t ≥0, the identity
d dt ˆ E (ξ, t) = −τq kτθ− τqτ∗ν 2 ! | ˆϕ|2−τ∗ν− k∗τq ˆh 2 , (3.37) holds.
Proof. Multiplying the first equation in (3.34) by k∗¯ˆw, the second equation byτ∗ν− k∗τ q ¯ˆz,
the third equation by ¯ˆy, the fourth equation by
kτθ− k∗ τ
2 q
2 ¯ˆh and the last equation by
kτθ− τqτ
∗ ν
2 ¯ˆϕ adding the results and taking the real part to get
d dt ˆ E (ξ, t) = Re( 2kττ θ q −τ∗ν+ k∗τq ! iξ¯ˆyˆz − 2kτθ τq + k∗τ q−τ∗ν ! iξ¯ˆyˆh ) −Reτ∗ν− k∗τq ¯ˆzˆh +Re τqτ∗ν 2 − k ∗τ 2 q 2 ϕ¯ˆhˆ −τq kτθ− τqτ∗ν 2 ! | ˆϕ|2 (3.38) +Re ( kτθ+ k∗τ q 2 − τqτ∗ν 2 ! iξˆy ¯ˆϕ ) . Now, we have Reiξ¯ˆyˆz = Re iξ ¯ˆut+ τq¯ˆutt+ τ2 q 2 ¯ˆuttt iξ ˆut+ τq 2 ˆutt = −ξ2
| ˆut|2+ τqRe ¯ˆuttˆut +
τ2 q 2Re ¯ˆutttˆut + τq 2Re ¯ˆutˆutt + τ2 q 2 | ˆutt|+ τ3 q 4Re ¯ˆutttˆutt , and Reiξ¯ˆyˆh = −ξ2Re ¯ˆut+ τq¯ˆutt+ τ2 q 2 ¯ˆuttt ˆut = −ξ2
| ˆut|2+ τqRe ¯ˆuttˆut +
τ2 q 2Re ¯ˆutttˆut . Also, Re ¯ˆzˆh = ξ2Re ¯ˆut+ τq 2 ¯ˆutt ˆut = ξ2| ˆu t|2+ τq 2Re ¯ˆuttˆut .
Furthermore, Reϕ¯ˆhˆ = ξ2Re ¯ˆutˆutt . Finally, Reiξˆy ¯ˆϕ = ξ2Re ˆut+ τqˆutt+ τ2 q 2 ˆuttt ¯ˆutt = ξ2
Reˆut¯ˆutt + τq| ˆutt|2+
τ2 q 2Re ˆuttt¯ˆutt .
Inserting the above identities into (3.38), then (3.37) is fulfilled. To ensure that the energy is positive and non-increasing fuction, it is necessary to impose the assumption
τ∗
ν > k∗τq and kτθ >
τqτ∗ν
2 . (3.39)
Lemma 3.6. Let ˆV(ξ, t) = w, ˆz, ˆy, ˆh, ˆϕˆ T(ξ, t) be the solution of system (3.34)-(3.35). Assume that (3.39) holds. Then for any t ≥0 and ξ ∈ R, we have the following pointwise estimate Vˆ (ξ, t) 2 ≤ Ce−˜cρ2(ξ)t Vˆ (ξ, 0) 2 , (3.40) where ρ2(ξ) = ξ2 1+ ξ2. (3.41)
Here C and ˜c are two positive constants.
Proof. Multiplying the third equation in (3.34) by iξ ¯ˆwand the first equation by −iξρcυ¯ˆy,
adding the resulting equalities and taking the real part, we have d dtRe iξρcυˆy ¯ˆw −ρcυξ2|ˆy|2+ ξ2k∗| ˆw|2 = −Re 2kττ θ q ξ2 ˆz ¯ˆw ! + Re 2kττ θ q ξ2ˆh ¯ˆw ! + Re k∗τqξ2ˆh ¯ˆw (3.42) −Reτ∗νξ2ˆh ¯ˆw + Re k∗τ2q 2 ξ 2 ˆ ϕ ¯ˆw .
