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On the time decay in phase–lag thermoelasticity with two temperatures

  • Received: 01 June 2019 Revised: 01 November 2019
  • Primary: 74F05, 74H40; Secondary: 74H20, 34B35, 35P20

  • The aim of this paper is to study the time decay of the solutions for two models of the one-dimensional phase-lag thermoelasticity with two temperatures. The first one is obtained when the heat flux vector and the inductive temperature are approximated by a second-order and first-order Taylor polynomial, respectively. In this case, the solutions decay in a slow way. The second model that we consider is obtained taking first-order Taylor approximations for the inductive thermal displacement, the inductive temperature and the heat flux. The decay is, therefore, of exponential type.

    Citation: Antonio Magaña, Alain Miranville, Ramón Quintanilla. On the time decay in phase–lag thermoelasticity with two temperatures[J]. Electronic Research Archive, 2019, 27: 7-19. doi: 10.3934/era.2019007

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  • The aim of this paper is to study the time decay of the solutions for two models of the one-dimensional phase-lag thermoelasticity with two temperatures. The first one is obtained when the heat flux vector and the inductive temperature are approximated by a second-order and first-order Taylor polynomial, respectively. In this case, the solutions decay in a slow way. The second model that we consider is obtained taking first-order Taylor approximations for the inductive thermal displacement, the inductive temperature and the heat flux. The decay is, therefore, of exponential type.



    The Fourier formulation to describe heat conduction is widely used by mathematicians, physicists and engineers. For this model, the heat flux is proportional to the gradient of the temperature. Unfortunately, this formulation jointly with the usual energy equation

    c˙θ+ div q=0,(c>0) (1)

    leads to the instantaneous propagation of heat, a drawback of the model because this fact is incompatible with real observations. In the above equation q=(qi) is the heat flux vector and θ is the temperature. In order to overcome this drawback, alternative proposals have been stated.

    In 1995, Tzou proposed a theory in which the heat flux and the gradient of the temperature have a delay in the constitutive equations [32]. When this consideration is taken into account, it is usual to speak of phase-lag theories. In that case, the constitutive equations are given by:

    qi(x,t+τ1)=kθ,i(x,t+τ2),k>0, (2)

    where τ1 and τ2 are the delay parameters which are assumed to be positive. As usual, the notation θ,i means the derivative of θ with respect to the variable xi, and repeated subscripts means summation. The derivative with respect to the time is denoted using a dot over the function.

    This equation suggests that the temperature gradient established across a material volume at position x and time t+τ2 results in a heat flux to flow at a different time t+τ1. These delays can be understood in terms of the microstructure of the material. This theory is usually known as dual-phase-lag.

    In 2007, Choudhuri [7] suggested an extension of Tzou's theory in which the heat flux is described using the following constitutive equations:

    qi(x,t+τ1)=k1α,i(x,t+τ3)k2θ,i(x,t+τ2), (3)

    where ˙α=θ. The variable α is called the thermal displacement, and was introduced by Green and Naghdi [10,11]. The parameter τ3 is another delay parameter. Choudhuri's version is commonly known as three-dual-phase-lag.

    It is worth noting that both proposals, those of Tzou and Choudhuri, lead to ill-posed problems in the sense of Hadamard. To be more precise, it has been shown that combining equation (2) (or (3)) with the energy equation (1) leads to the existence of a sequence of elements in the point spectrum such that its real part tends to infinity [8].

    These two aforementioned theories have several derivations when the heat flux and the gradients of the temperature and the thermal displacement are replaced by Taylor approximations. In fact, one can think that Choudhuri's proposal aims to recover Green and Naghdi theories when different Taylor approximations are considered. This new approach gives rise to different equations (depending on the selected Taylor approximation) to describe heat conduction that have been analyzed by many authors (see, for example, [1,3,12,19,23,26,27,28,29,30,31,34]).

    In order to obtain a heat conduction theory with delays but without such an explosive behavior, Quintanilla [24,25] combined the delay parameters of Tzou and Choudhuri with the two-temperatures theory proposed by Chen and Gurtin [4,5,6,33]. The basic constitutive equations read

    qi(x,t+τ1)=k1β,i(x,t+τ3)k2T,i(x,t+τ2), (4)

    where α=βmΔβ, θ=TmΔT and m is a positive constant.

    This theory has also been extended to the thermoelasticity context [24,25]. To do so, one must assume the equation of motion

    tji,j=ρ¨ui, (5)

    the energy equation

    ˙η=qi,i (6)

    and the constitutive equations

    tji=2μeij+λerrδij+aθδijη=aeii+cθ (7)

    where tji represents the stress tensor, η is the entropy, (ui) is the displacement vector, eij is the strain tensor, λ and μ are the Lamé constants and a is related with the thermal expansion constant and ρ and c are the mass density and the thermal capacity, respectively.

    It is worth noting that these new thermomechanical theories have attracted a lot of attention recently [2,9,14,20,21,34].

    In this work we restrict our attention to the homogeneous one-dimensional case. Therefore, the system of equations that we want to study is given by

    tx=ρ¨u˙η=qx (8)

    with the following constitutive equations:

    q(t+τ1)=k1βx(t+τ3)k2Tx(t+τ2)t=μux+aθη=aux+cθ (9)

    In this paper we assume that the delay parameters τ1, τ2 and τ3 are nonnegative and, in each section, we will impose several conditions on them to guarantee the stability or instability of the solutions. A similar assumption is made on k1 and k2.

    In a recent paper [15], it was proved that the Lord-Shulman thermoelasticity combined with the two-temperatures theory leads to the slow decay of the solutions, that is: the decay cannot be controlled in a uniform way by means of a negative exponential. Nevertheless, the Green-Lindsay thermoelastic theory with two-temperatures leads to the exponential decay. The aim of this paper is to continue this line of research. To be more precise, in this work we consider two third-order in time heat conduction models with two temperatures. The first one comes from the dual-phase-lag theory (see approach [24]), and the other one from the three-dual-phase-lag theory (see [25]). We prove the slow decay of solutions for the first model and the exponential decay for the second.

    The plan of the paper is the following. In the next section we consider the dual-phase-lag thermoelasticity taking a second-order Taylor approximation for the heat flux and a first-order approximation for the inductive temperature. We first prove the well-posedness of the problem using the semigroup arguments in a convenient Hilbert space. Then we show the slow decay of the solutions by proving that elements of the point spectrum can be found as close as desired to the imaginary axis. In Section 3 we study the three-dual-phase-lag thermoelasticity introducing first-order Taylor approximations for the heat flux, for the inductive thermal displacement and for the inductive temperature. By using semigroup arguments, we prove again the existence and uniqueness of solutions. Next we show the exponential decay of the solutions by means of the semigroup of linear operators theory.

    In this first case we assume that k1=0 and k2>0 in the basic constitutive equations. Notice that taking k1=0 is equivalent to consider the dual-phase-lag case. We take a second-order Taylor approximation for the heat flux qi and a first-order Taylor approximation for the inductive temperature T.

    q(x,t+τ1)q(x)+τ1˙q(x)+τ212¨q(x),T(x,t+τ2)T(x)+τ2˙T(x). (10)

    Replacing the above expressions into the constitutive equations, we obtain the following system of equations for our model:

    {ρ¨u=μuxx+aθxc(˙θ+τ1¨θ+τ212θ)=k2Txx+k2τ2˙Txx+a(˙ux+τ1¨ux+τ212ux)θ=TmTxx. (11)

    Here u is the displacement, θ is the temperature and T is the inductive temperature. As usual, ρ represents the mass density, μ is the elasticity, c is the thermal capacity, k2 plays a similar role to the thermal conductivity, a is the coupling coefficient between the displacement and the temperature, τ1 and τ2 are two delay parameters and m is a positive constant related with the two temperatures theory. We also assume that 2τ2>τ1. This assumption comes from a previous work and it is related with the exponential stability of the heat equation (see [17]). To be precise, the exponential stability of solutions was proved assuming this condition. It would be interesting to know if this property is conserved when the elasticity is also considered.

    We study the system in [0,π]×[0,).

    To have a well-posed problem we need to impose initial and boundary conditions. We assume null Dirichlet boundary conditions, that is,

    u(0,t)=u(π,t)=T(0,t)=T(π,t)=0 for t[0,). (12)

    As far as the initial conditions are concerned, we assume that

    u(x,0)=u0(x),˙u(x,0)=v0(x),θ(x,0)=θ0(x),˙θ(x,0)=ϕ0(x),¨θ(x,0)=ψ0(x) for x(0,π). (13)

    We study the problem determined by system (11), the boundary conditions (12) and the initial conditions (13).

    We will transform the given problem into an abstract problem involving a convenient Hilbert space.

    First, we note that Idmxx:TTmxxT=θ is an isomorphism on W2,2W1,20 and takes values in L2, where W2,2,W1,20 and L2 are the usual Hilbert spaces. We shall denote by Φ(θ)=T the inverse operator.

    From the definition of θ and in view of the boundary conditions we see that

    θ2=T2+2mTx2+m2Txx2. (14)

    Therefore, the L2 norm of θ is equivalent to the W2,2 norm of T.

    Let us denote v=˙u, ϕ=˙θ and ψ=˙ϕ. We introduce also the following notation:

    ˆf=f+τ1˙f+τ212¨f.

    Therefore, system (11) may be written as

    {˙ˆu=ˆv˙ˆv=1ρ(μˆuxx+a(θx+τ1ϕx+τ212ψx))˙θ=ϕ˙ϕ=ψ˙ψ=2cτ21(k2Φ(θ)xx+k2τ2Φ(ϕ)xx)+2acτ21ˆvxaτ21ϕ2τ1ψ (15)

    If the above system is solved, then we will find u from ˆu solving a second-order ordinary differential equation.

    To ease the notation, we remove the hat from variables u and v and rewrite the system as follows:

    {˙u=v˙v=1ρ(μuxx+a(θx+τ1ϕx+τ212ψx))˙θ=ϕ˙ϕ=ψ˙ψ=2cτ21(k2Φ(θ)xx+k2τ2Φ(ϕ)xx)+2acτ21vxaτ21ϕ2τ1ψ (16)

    To prove the existence and uniqueness of solutions we consider the Hilbert space

    H={U=(u,v,θ,ϕ,ψ):uW1,20,v,θ,ϕ,ψL2}

    with the inner product defined by

    U,U=12π0(ρv¯v+μux¯ux+c(θ+τ1ϕ+τ212ψ)(¯θ+τ1ϕ+τ212ψ)+k2(τ1+τ2)(Φ(θ)x¯Φ(θ)x+mΦ(θ)xx¯Φ(θ)xx)+k2τ21τ22(Φ(ϕ)x¯Φ(ϕ)x+mΦ(ϕ)xx¯Φ(ϕ)xx)+k2τ212(Φ(θ)x¯Φ(ϕ)x+Φ(ϕ)x¯Φ(θ)x+mΦ(θ)xx¯Φ(ϕ)xx+mΦ(ϕ)xx¯Φ(θ)xx))dx. (17)

    Here, and from now on, the bar means the conjugate complex. Notice that the norm induced by this inner product is equivalent to the usual one in H.

    To propose a synthetic expression to the above problem, we define the matrix operator:

    A=(0I000μρD20aρIaτ1ρDaτ212ρD000I00000I02acτ21D2k2cτ21D2Φ2k2τ2cτ21D2Φ2τ21I2τ1I). (18)

    Here I is the identity operator and D denotes the derivative with respect to x.

    Therefore, our problem can be written as

    dUdt=AU,U(0)=(u0,v0,θ0,ϕ0,ψ0). (19)

    Notice that the domain of A is the set D={UH such that AUH}, which is a dense subspace of H.

    Lemma 2.1. The operator A is dissipative. That is:

    AU,U0

    for every UD.

    Proof. If we take into account the evolution equations and the boundary conditions we see that

    AU,U=k2π0(|Φ(θ)x|2+m|Φ(θ)xx|2)dxk2τ12(2τ2τ1)π0(|Φ(ϕ)x|2+m|Φ(ϕ)xx|2)dx.

    As we assume that 2τ2>τ1, the lemma is proved.

    Lemma 2.2. 0 belongs to the resolvent of A.

    Proof. We have to prove that for any F=(f1,f2,f3,f4,f5)H the equation AU=F has a solution. If we write this equation term by term we get:

    v=f1μuxx+a(θx+τ1ϕx+τ212ψx)=ρf2ϕ=f3ψ=f4k2Φ(θ)xx+k2τ2Φ(ϕ)xx+avxcϕcτ1ψ=cτ212f5} (20)

    We obtain v, ϕ and ψ straight away. Therefore, we have to solve the system given by

    μuxx+aθx=ρf2aτ1f3,xaτ212f4,xk2Φ(θ)xx=cτ212f5af1,x+cf3+cτ1f4k2τ2Φ(f3)xx} (21)

    If we assume homogeneous boundary conditions on Φ(θ), we can solve the second equation of the above system. Once we have θ, substituting it in the first equation we obtain u.

    It is also clear that the inequality UKF holds for a positive constant K independent of U.

    As a consequence of the above lemmas and the Lumer–Phillips corollary to the Hille–Yosida Theorem (see [18], page 136) we obtain the well–posedness.

    Theorem 2.3. The operator A generates a contractive semigroup in H, and for each U(0)D there exists a unique solution U(t)C1([0,),H)C0([0,),D) to the problem determined by the system (11) with boundary conditions (12) and initial conditions (13).

    Remark 1. The continuous dependence of solutions on initial data and supply terms (in case they were assumed) can also be obtained.

    These facts prove that the problem is well-posed in the sense of Hadamard.

    Remark 2. We could have also considered the problem determined by system (11) with the following boundary conditions:

    u(0,t)=u(π,t)=Tx(0,t)=Tx(π,t)=0 for t[0,). (22)

    In this case, the operator Φ acts on

    L2={θL2:π0θdx=0},

    and takes values in W2,2L2{T:Tx(0)=Tx(π)=0}. Nevertheless, Φ is still an isomorphism and the equality (14) also holds.

    We will prove that a uniform rate of decay of exponential type cannot be obtained for the solutions of system (11) with the initial conditions (13) and the boundary conditions (22).

    Theorem 2.4. Let (u,T) be a solution of the problem determined by (11), (13) and (22). Then (u,T) decays in a slow way.

    Proof. We will prove that there exists a solution of system (11) of the form

    u=K1eωtsin(nx),T=K2eωtcos(nx),

    such that (ω)>ϵ for all positive ϵ. Hence, a solution ω as close as desired to the imaginary axis can be found. Imposing that u and T are as above and replacing them in (11) the following homogeneous system in the unknowns K1 and K2 is obtained:

    (A1A2A3A4)(K1K2)=(00)

    where

    A1=an(mn2+1)A2=μn2+ω2ρA3=cω(mn2+1)(τ21ω2+2τ1ω+2)+2k2n2(τ2ω+1)A4=anω(τ21ω2+2τ1ω+2)

    This system will have nontrivial solutions if and only if the determinant of the coefficients matrix is equal to zero. We denote by p(x) the determinant once ω is replaced by x. Straightforward calculations (made using Mathematica) show that p(x) is a fifth degree polynomial:

    p(x)=a0x5+a1x4+a2x3+a3x2+a4x+a5,

    where

    a0=cτ21ρ(mn2+1)a1=2cτ1ρ(mn2+1)a2=n2τ21(mn2+1)(a2+cμ)+2ϱ(cmn2+c+k2n2τ2)a3=2n2(τ1(mn2+1)(a2+cμ)+k2ϱ)a4=2n2((mn2+1)(a2+cμ)+k2μn2τ2)a5=2k2μn4

    To prove that p(x) has roots as close as we want to the complex axis, we will show that for any ϵ>0 there are roots of p(x) located on the right-hand side of the vertical line (z)=ϵ. This fact will be shown if the polynomial p(xϵ) has a root with positive real part. To prove that, we use the Routh–Hurwitz theorem. It assesses that, if b0>0, then all the roots of polynomial

    b0x5+b1x4+b2x3+b3x2+b4x+b5

    have negative real part if, and only if, all the leading minors of the matrix

    (b1b0000b3b2b1b00b5b4b3b2b100b5b4b30000b5)

    are positive. We denote by Li, for i=1,2,3,4,5, the leading minors of this matrix.

    Direct computations show that the second leading minor, L2, is a sixth degree polynomial on n:

    L2=2cm2τ41ϵρ(a2+cμ)n6+R(n),

    where R(n) is a polynomial on n of degree 4. Thus, for n large enough, L2 is negative and p(xϵ) has a root with positive real part. (We have used Mathematica to compute L2.)

    This argument shows that the solutions of system (11) decay in a slowly way, or, in other words, that a uniform rate of decay of exponential type for all the solutions can not be obtained.

    We want to point out that this result differs considerably from the one known for the usual dual-phase-lag thermoelasticity: if only one temperature is considered, the solutions decay in an exponential way [26]. Let us also highlight the following fact. If in the second equation of system (11) variable T is replaced by θ, then the resulting equation would be hyperbolic and, therefore, in some sense we can say that the second equation of system (11) is a combination of an hyperbolic equation with the two temperatures theory. We have seen that this combination, when it is coupled in the usual way with the elasticity, drives the solutions of the system to the slow decay. This result was already observed in [15] for another combination of an hyperbolic equation with the two temperatures theory.

    We consider now another system. We assume that k1>0 and k2>0. We take a first-order Taylor approximation for qi, for β and for T. Therefore, we assume that

    q(x,t+τ1)q(x)+τ1˙q(x),β(x,t+τ3)β(x)+τ3˙β(x),T(x,t+τ2)T(x)+τ2˙T(x). (23)

    Substituting these expressions into the constitutive equations, we obtain the following system:

    {ρ¨u=μuxx+aθxc(¨θ+τ1θ)=k1Txx+τ4˙Txx+k2τ2¨Txx+a(¨ux+τ1ux)θ=TmTxx, (24)

    where τ4=τ3k1+k2.

    As in the previous case, ρ, μ and m are positive. The delay parameters are also positive and τ4k1τ1>0. This last assumption comes again from the exponential stability of the heat equation (see [17]). The sign of a is not relevant, but it must be different from zero. The parameters k1 and k2 are also assumed positive.

    We assume the same boundary and initial conditions as in the previous section.

    As before, we introduce suitable notation. In this case we write ˆf=f+τ1˙f. Moreover, we use w=˙v. Therefore, we consider the system given by

    {˙ˆv=ˆw˙ˆw=1ρ(μˆvxx+a(ϕx+τ1ψx))˙θ=ϕ˙ϕ=ψ˙ψ=1cτ1(k1Φ(θ)xx+τ4Φ(ϕ)xx+k2τ2Φ(ψ)xx))+acτ1ˆwx1cψ (25)

    If this system is solved, then we can also find v by solving an ordinary differential equation.

    To ease the notation we remove the hat from the variables and we concentrate in the following system of equations:

    {˙v=w˙w=1ρ(μvxx+a(ϕx+τ1ψx))˙θ=ϕ˙ϕ=ψ˙ψ=1cτ1(k1Φ(θ)xx+τ4Φ(ϕ)xx+k2τ2Φ(ψ)xx))+acτ1wx1cψ (26)

    To prove the existence and uniqueness of solutions we consider the Hilbert space

    H={U=(v,w,θ,ϕ,ψ):vW1,20,w,θ,ϕ,ψL2}

    with the inner product defined by

    U,U=12π0(ρw¯w+μvx¯vx+c(ϕ+τ1ψ)(¯ϕ+τ1ψ)+k1(Φ(θ)x+τ1Φ(ϕ)x)(¯Φ(θ)x+τ1Φ(ϕ)x)+mk1(Φ(θ)xx+τ1Φ(ϕ)xx)(¯Φ(θ)xx+τ1Φ(ϕ)xx)+(τ1(τ4k1τ1)+k2τ2)(Φ(ϕ)x¯Φ(ϕ)x+mΦ(ϕ)xx¯Φ(ϕ)xx))dx. (27)

    Again, this inner product defines a norm which is equivalent to the usual norm in H.

    We abuse the notation a little bit and denote the following matrix operator again by A:

    A=(0I000μρD200aρDaτ1ρD000I00000I0acτ1Dk1cτ1D2Φτ4cτ1D2Φk2τ2cτ1D2Φ1cI). (28)

    Therefore, our problem can be written as

    dUdt=AU,U(0)=(v0,w0,θ0,ϕ0,ψ0). (29)

    Notice that the domain of A is the set D={UH such that AUH}, which is a dense subspace of H.

    Lemma 3.1. The operator A is dissipative. That is:

    AU,U0

    for every UD.

    Proof. In view of the evolution equations and the boundary conditions we see that

    AU,U=(τ4k1τ1)π0(|Φ(ϕ)x|2+m|Φ(ϕ)xx|2)dxk2τ1τ2π0(|Φ(ψ)x|2+m|Φ(ψ)xx|2)dx.

    As we assume that τ4>k1τ1, k2, τ1 and τ2 positive, the lemma is proved.

    Lemma 3.2. 0 belongs to the resolvent of A.

    Proof. We have to prove that for any F=(f1,f2,f3,f4,f5)H the equation AU=F has a solution. If we write this equation term by term we get:

    w=f1μvxx+a(ϕx+τ1ψx)=ρf2ϕ=f3ψ=f4k1Φ(θ)xx+τ4Φ(ϕ)xx+k2τ2Φ(ψ)xx+awxτ1ψ=cτ1f5} (30)

    As in the previous section, we obtain w, ϕ and ψ straight away. Therefore, we have to solve the system given by

    μvxx=ρf2a(f3,x+τ1f4,x)k1Φ(θ)xx=cτ1f5af1,x+τ1f4τ4Φ(f3)xxk2τ2Φ(f4)xx} (31)

    If we assume homogeneous Dirichlet boundary conditions, we can solve this system as in the previous case. Hence, we can get θ by means of the isomorfism. Again, an inequality of the type UKF can be obtained.

    As a consequence of the above lemmas and the Lumer–Phillips corollary to the Hille–Yosida Theorem we obtain the well–posedness in the sense of Hadamard.

    Theorem 3.3. The operator A generates a contractive semigroup, S(t)={etA}, in H, and for each U(0)D there exists a unique solution U(t)C1([0,),H)C0([0,),D) to the problem determined by the system (24) with boundary conditions (12) and initial conditions (13).

    We have now the basic tools to prove the main result of this section. Before doing this, we recall the caractherization stated in the book of Liu and Zheng that ensures the exponential decay (see [13], [16] or [22]).

    Theorem 3.4. Let S(t)={eAt}t0 be a C0-semigroup of contractions on a Hilbert space. Then S(t) is exponentially stable if and only if the following two conditions are satisfied:

    (i) iRρ(A), (here ρ(A) means the resolvent of A).

    (ii) ¯lim|λ|(iλIA)1L(H)<.

    Theorem 3.5. The operator A defined at (28) generates a semigroup which is exponentially stable.

    Proof. Following the arguments given by Liu and Zheng ([16], page 25), the proof consists of the following steps:

    (ⅰ) Since 0 is in the resolvent of A, using the contraction mapping theorem, we have that for any real λ such that |λ|<||A1||1, the operator iλIA=A(iλA1I) is invertible. Moreover, ||(iλIA)1|| is a continuous function of λ in the interval (||A1||1,||A1||1).

    (ⅱ) If sup{||(iλIA)1||,|λ|<||A1||1}=M<, then by the contraction theorem, the operator

    iλIA=(iλ0IA)(I+i(λλ0)(iλ0IA)1),

    is invertible for |λλ0|<M1. It turns out that, by choosing λ0 as close to ||A1||1 as we can, the set {λ,|λ|<||A1||1+M1} is contained in the resolvent of A and ||(iλIA)1|| is a continuous function of λ in the interval (||A1||1M1,||A1||1+M1).

    (ⅲ) Let us assume that the intersection of the imaginary axis and the spectrum is not empty. Then there exists a real number ϖ with ||A1||1|ϖ|< such that the set {iλ,|λ|<|ϖ|} is in the resolvent of A and sup{||(iλIA)1||,|λ|<|ϖ|}=. Therefore, there exist a sequence of real numbers λn with λnϖ,|λn|<|ϖ| and a sequence of vectors Un=(vn,wn,θn,ϕn,ψn) in the domain of the operator A and with unit norm such that

    (iλnIA)Un0. (32)

    If we write (32) in components, we obtain the following conditions:

    iλnvnwn0, in W1,2 (33)
    iλnwn1ρ(μD2vn+a(Dϕn+τ1Dψn))0, in L2 (34)
    iλnθnϕn0, in L2 (35)
    iλnϕnψn0, in L2 (36)
    iλnψn1cτ1(k1D2Φ(θn)+τ4D2Φ(ϕn)+k2τ2D2Φ(ψn)+aDwn)+1cψn0, in L2 (37)

    In view of the dissipative term of the operator (see the proof of Lemma 3.1), we see that

    DΦ(ϕn),D2Φ(ϕn),DΦ(ψn),D2Φ(ψn)0 in L2.

    Therefore, from (14) we conclude that ϕn0 in W2,2 and, consequently, θn,ψn0 in L2. Now, from (37) we obtain

    iλnψnacτ1Dwn0, in L2,

    which, after simplifying, implies that

    iψnacτ1Dvn0, in L2.

    Then, as a is different from 0, Dvn0 in L2. Finally, from (33) and (34) we conclude that wn0 in W1,2. These behaviors contradict the hypothesis that Un has unit norm.

    We can prove the second condition of the Theorem 3.4 following a similar argument.

    Remark 3. The analysis proposed in this section can be easily adapted to the boundary conditions

    u(0,t)=u(π,t)=Tx(0,t)=Tx(π,t)=0.

    That means that we can obtain the existence, the uniqueness and the exponential decay of the solutions for the problem determined by system (24) with the initial conditions given by (13) and the above boundary conditions.

    We also point out that this behavior differs from the one obtained in the previous section. This is because now the heat conduction is described by the combination of a parabolic equation with the two temperatures theory.

    In this paper we have analised two systems of equations for the phase-lag thermoelasticicty with two temperatures, theory which is currently being studied from different points of view. We have focused on the decay of solutions. To be precise, we have analysed the following situations:

    ● Dual-phase-lag. We have introduced a second order Taylor approximation for the heat flux and a first-order approximation for the inductive temperature and we have obtained system (11). The second equation of this system can be seen as a combination of a hyperbolic equation with the two temperatures theory. We have proved that the solutions of this thermoelastic system of equations decay in a slow way.

    ● Three-dual-phase-lag. In this case we have considered first order Taylor approximations for the heat flux, for the inductive temperature and for the inductive thermal displacement, and we have obtained system (24). In this case, the second equation can be seen as the combination of a parabolic equation with the two temperatures theory. We have proved that the solutions of this thermoelastic system of equations decay exponentially.

    These results seem to suggest a different behavior for the decay of the solutions depending on the type of equation (hyperbolic or parabolic) that is combined with the two temperatures theory.

    We would like to thank an anonymous referee for his useful comments.



    [1] Abdallah I. A. (2009) Dual phase lag heat conduction and thermoelastic properties of a semi-infinite medium induced by ultrashort pulsed laser. Progress in Physics 3: 60-63.
    [2] Banik S., Kanoria M. (2012) Effects of three-phase-lag on two temperatures generalized thermoelasticity for an infinite medium with a spherical cavity. Applied Mathematics and Mechanics 33: 483-498.
    [3] Borgmeyer K., Quintanilla R., Racke R. (2014) Phase-lag heat conduction: Decay rates for limit problems and well-posedness. J. Evolution Equations 14: 863-884.
    [4] Chen P. J., Gurtin M. E. (1968) On a theory of heat involving two temperatures. J. Applied Mathematics and Physics (ZAMP) 19: 614-627.
    [5] Chen P. J., Gurtin M. E., Williams W. O. (1968) A note on non-simple heat conduction. J. Applied Mathematics and Physics (ZAMP) 19: 969-970.
    [6] Chen P. J., Gurtin M. E., Williams W. O. (1969) On the thermodynamics of non-simple materials with two temperatures. J. Applied Mathematics and Physics (ZAMP) 20: 107-112.
    [7] Choudhuri S. K. R. (2007) On a thermoelastic three-phase-lag model. J. Thermal Stresses 30: 231-238.
    [8] Dreher M., Quintanilla R., Racke R. (2009) Ill-posed problems in thermomechanics. Applied Mathematics Letters 22: 1374-1379.
    [9] Ezzat M. A., El-Karamany A. S., Ezzat S. M. (2012) Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer. Nuclear Engineering and Design 252: 267-277.
    [10] A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253–264.

    10.1080/01495739208946136

    MR1175235

    [11] A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189–208.

    10.1007/BF00044969

    MR1236373

    [12] Hader M. A., Al-Nimr M. A., Abu Nabah B. A. (2002) The Dual-Phase-Lag heat conduction model in thin slabs under a fluctuating volumetric thermal disturbance. Int. J. Thermophysics 23: 1669-1680.
    [13] Huang F. L. (1993) Strong asymptotic stability of linear dynamical systems in Banach spaces. J. Differential Equations 104: 307-324.
    [14] Quintanilla R., Jordan P. M. (2009) A note on the two-temperature theory with dual-phase-lag decay: Some exact solutions. Mechanics Research Communications 36: 796-803.
    [15] Leseduarte M. C., Quintanilla R., Racke R. (2017) On (non-)exponential decay in generalized thermoelasticity with two temperatures. Applied Mathematics Letters 70: 18-25.
    [16] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999.

    MR1681343

    [17] Magaña A., Miranville A., Quintanilla R. (2018) On the stability in phase-lag heat conduction with two temperatures. J. of Evolution Equations 18: 1697-1712.
    [18] J. E. Marsden and T. J. R. Hughes, Topics in the mathematical foundations of elasticity, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. II, 30-285, Res. Notes in Math., 27, Pitman, Boston, Mass.-London, 1978.

    MR576233

    [19] Miranville A., Quintanilla R. (2011) A phase-field model based on a three-phase-lag heat conduction. Applied Mathematics and Optimization 63: 133-150.
    [20] Mukhopadhyay S., Prasad R, Kumar R. (2011) On the theory of Two-Temperature Thermoelaticity with Two Phase-Lags. J. Thermal Stresses 34: 352-365.
    [21] Othman M. A., Hasona W. M., Abd-Elaziz E. M. (2014) Effect of rotation on micropolar generalized thermoelasticity with two temperatures using a dual-phase-lag model. Canadian J. Physics 92: 149-158.
    [22] Prüss J. (1984) On the spectrum of C0-semigroups. Trans. Amer. Math. Soc. 284: 847-857.
    [23] Quintanilla R. (2002) Exponential stability in the dual-phase-lag heat conduction theory. J. Non-Equilibrium Thermodynamics 27: 217-227.
    [24] Quintanilla R. (2008) A well-posed problem for the Dual-Phase-Lag heat conduction. J. Thermal Stresses 31: 260-269.
    [25] Quintanilla R. (2009) A well-posed problem for the three-dual-phase-lag heat conduction. J. Thermal Stresses 32: 1270-1278.
    [26] Quintanilla R., Racke R. (2006) Qualitative aspects in dual-phase-lag thermoelasticity. SIAM J. Appl. Math. 66: 977-1001.
    [27] Quintanilla R., Racke R. (2006) A note on stability of dual-phase-lag heat conduction. Int. J. Heat Mass Transfer 49: 1209-1213.
    [28] Quintanilla R., Racke R. (2007) Qualitative aspects in dual-phase-lag heat conduction. Proc. Royal Society London A 463: 659-674.
    [29] Quintanilla R., Racke R. (2008) A note on stability in three-phase-lag heat conduction. Int. J. Heat Mass Transfer 51: 24-29.
    [30] Quintanilla R., Racke R. (2015) Spatial behavior in phase-lag heat conduction. Differential and Integral Equations 28: 291-308.
    [31] Rukolaine S. A. (2014) Unphysical effects of the dual-phase-lag model of heat conduction. Int. J. Heat and Mass Transfer 78: 58-63.
    [32] Tzou D. Y. (1995) A unified approach for heat conduction from macro to micro-scales. ASME J. Heat Transfer 117: 8-16.
    [33] Warren W. E., Chen P. J. (1973) Wave propagation in two temperatures theory of thermoelaticity. Acta Mechanica 16: 83-117.
    [34] Zhang Y. (2009) Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues. Int. J. of Heat and Mass Transfer 52: 4829-4834.
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