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
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
This equation suggests that the temperature gradient established across a material volume at position
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
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
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
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
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
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+τ212⃛θ)=k2Txx+k2τ2˙Txx+a(˙ux+τ1¨ux+τ212⃛ux)θ=T−mTxx. | (11) |
Here
We study the system in
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
From the definition of
‖θ‖2=‖T‖2+2m‖Tx‖2+m2‖Txx‖2. | (14) |
Therefore, the
Let us denote
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)+2acτ21ˆvx−aτ21ϕ−2τ1ψ | (15) |
If the above system is solved, then we will find
To ease the notation, we remove the hat from variables
{˙u=v˙v=1ρ(μuxx+a(θx+τ1ϕx+τ212ψx))˙θ=ϕ˙ϕ=ψ˙ψ=2cτ21(k2Φ(θ)xx+k2τ2Φ(ϕ)xx)+2acτ21vx−aτ21ϕ−2τ1ψ | (16) |
To prove the existence and uniqueness of solutions we consider the Hilbert space
H={U=(u,v,θ,ϕ,ψ):u∈W1,20,v,θ,ϕ,ψ∈L2} |
with the inner product defined by
⟨U,U∗⟩=12∫π0(ρv¯v∗+μux¯u∗x+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
To propose a synthetic expression to the above problem, we define the matrix operator:
A=(0I000μρD20aρIaτ1ρDaτ212ρD000I00000I02acτ21D2k2cτ21D2Φ2k2τ2cτ21D2Φ−2τ21I−2τ1I). | (18) |
Here
Therefore, our problem can be written as
dUdt=AU,U(0)=(u0,v0,θ0,ϕ0,ψ0). | (19) |
Notice that the domain of
Lemma 2.1. The operator
ℜ⟨AU,U⟩≤0 |
for every
Proof. If we take into account the evolution equations and the boundary conditions we see that
ℜ⟨AU,U⟩=−k2∫π0(|Φ(θ)x|2+m|Φ(θ)xx|2)dx−k2τ12(2τ2−τ1)∫π0(|Φ(ϕ)x|2+m|Φ(ϕ)xx|2)dx. |
As we assume that
Lemma 2.2.
Proof. We have to prove that for any
v=f1μuxx+a(θx+τ1ϕx+τ212ψx)=ρf2ϕ=f3ψ=f4k2Φ(θ)xx+k2τ2Φ(ϕ)xx+avx−cϕ−cτ1ψ=cτ212f5} | (20) |
We obtain
μuxx+aθx=ρf2−aτ1f3,x−aτ212f4,xk2Φ(θ)xx=cτ212f5−af1,x+cf3+cτ1f4−k2τ2Φ(f3)xx} | (21) |
If we assume homogeneous boundary conditions on
It is also clear that the inequality
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
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
L2⋆={θ∈L2:∫π0θdx=0}, |
and takes values in
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
Proof. We will prove that there exists a solution of system (11) of the form
u=K1eωtsin(nx),T=K2eωtcos(nx), |
such that
(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)=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
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
Direct computations show that the second leading minor,
L2=−2cm2τ41ϵρ(a2+cμ)n6+R(n), |
where
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
We consider now another system. 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+τ1⃛θ)=k1Txx+τ4˙Txx+k2τ2¨Txx+a(¨ux+τ1⃛ux)θ=T−mTxx, | (24) |
where
As in the previous case,
We assume the same boundary and initial conditions as in the previous section.
As before, we introduce suitable notation. In this case we write
{˙ˆv=ˆw˙ˆw=1ρ(μˆvxx+a(ϕx+τ1ψx))˙θ=ϕ˙ϕ=ψ˙ψ=1cτ1(k1Φ(θ)xx+τ4Φ(ϕ)xx+k2τ2Φ(ψ)xx))+acτ1ˆwx−1cψ | (25) |
If this system is solved, then we can also find
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))+acτ1wx−1cψ | (26) |
To prove the existence and uniqueness of solutions we consider the Hilbert space
H={U=(v,w,θ,ϕ,ψ):v∈W1,20,w,θ,ϕ,ψ∈L2} |
with the inner product defined by
⟨U,U∗⟩=12∫π0(ρw¯w∗+μvx¯v∗x+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(τ4−k1τ1)+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
We abuse the notation a little bit and denote the following matrix operator again by
A=(0I000μρD200aρDaτ1ρD000I00000I0acτ1Dk1cτ1D2Φτ4cτ1D2Φk2τ2cτ1D2Φ−1cI). | (28) |
Therefore, our problem can be written as
dUdt=AU,U(0)=(v0,w0,θ0,ϕ0,ψ0). | (29) |
Notice that the domain of
Lemma 3.1. The operator
ℜ⟨AU,U⟩≤0 |
for every
Proof. In view of the evolution equations and the boundary conditions we see that
ℜ⟨AU,U⟩=−(τ4−k1τ1)∫π0(|Φ(ϕ)x|2+m|Φ(ϕ)xx|2)dx−k2τ1τ2∫π0(|Φ(ψ)x|2+m|Φ(ψ)xx|2)dx. |
As we assume that
Lemma 3.2. 0 belongs to the resolvent of
Proof. We have to prove that for any
w=f1μvxx+a(ϕx+τ1ψx)=ρf2ϕ=f3ψ=f4k1Φ(θ)xx+τ4Φ(ϕ)xx+k2τ2Φ(ψ)xx+awx−τ1ψ=cτ1f5} | (30) |
As in the previous section, we obtain
μvxx=ρf2−a(f3,x+τ1f4,x)k1Φ(θ)xx=cτ1f5−af1,x+τ1f4−τ4Φ(f3)xx−k2τ2Φ(f4)xx} | (31) |
If we assume homogeneous Dirichlet boundary conditions, we can solve this system as in the previous case. Hence, we can get
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
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
Theorem 3.5. The operator
Proof. Following the arguments given by Liu and Zheng ([16], page 25), the proof consists of the following steps:
(ⅰ) Since
(ⅱ) If
iλI−A=(iλ0I−A)(I+i(λ−λ0)(iλ0I−A)−1), |
is invertible for
(ⅲ) Let us assume that the intersection of the imaginary axis and the spectrum is not empty. Then there exists a real number
‖(iλnI−A)Un‖→0. | (32) |
If we write (32) in components, we obtain the following conditions:
iλnvn−wn→0, in W1,2 | (33) |
iλnwn−1ρ(μD2vn+a(Dϕn+τ1Dψn))→0, in L2 | (34) |
iλnθn−ϕn→0, in L2 | (35) |
iλnϕn−ψn→0, in L2 | (36) |
iλnψn−1cτ1(k1D2Φ(θn)+τ4D2Φ(ϕn)+k2τ2D2Φ(ψn)+aDwn)+1cψn→0, 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
iλnψn−acτ1Dwn→0, in L2, |
which, after simplifying, implies that
iψn−acτ1Dvn→0, in L2. |
Then, as
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.
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