TIME DECAY IN DUAL-PHASE-LAG THERMOELASTICITY: CRITICAL CASE

. This note is devoted to the study of the time decay of the one-dimensional dual-phase-lag thermoelasticity. In this theory two delay para- meters τ q and τ θ are proposed. It is known that the system is exponentially stable if τ q < 2 τ θ [22]. We here make two new contributions to this problem. First, we prove the polynomial stability in the case that τ q = 2 τ θ as well the optimality of this decay rate. Second, we prove that the exponential stability remains true even if the inequality only holds in a proper sub-interval of the spatial domain, when τ θ is spatially dependent.


1.
Introduction. The Fourier constitutive law for the heat flux vector proposes the well-known linear parabolic equation for the heat conduction. This equation has received criticism from a physical point of view because it implies that the thermal disturbances at some point will be felt instantly anywhere for every distance. Different theories describing the heat conduction have been established to save this drawback. The most well known one is the Maxwell-Cattaneo law that proposes a hyperbolic damped equation for the heat conduction. Two hyperbolic theories of thermoelasticity have been proposed for this heat conduction. They are the theory of Lord and Shulman [16] and the theory of Green and Lindsay [6], which are under intensive study in the recent years. In the 1990's Green and Naghdi also proposed three different thermoelastic theories [7,8,9] based on the axioms of the continuum mechanics. The main difference in these theories was the kind of heat conduction they proposed. More details of the above theories can be found in the books and review articles [10,11,13,23].
In 1995 Tzou [24] suggested an alternative to modify of the classical Fourier law. He introduced delay parameters in the constitutive equations. To be precise, the q(x, t + τ q ) = −k∇θ(x, t + τ θ ), k > 0. (1) Here θ is the temperature, q is the heat flux vector and τ θ and τ q are two delay parameters. This equation assumes that the temperature gradient across a material volume at position x and at time t + τ θ results in the heat flux to flow at a different instant t + τ q . However the proposition of this theory only builds on an intuitive point of view, and there is no axiomatic mathematical foundation for it. In fact, when we adjoin this equations with the classical energy equatioṅ it has been proved the existence of a sequence of solutions T n (x, t) = exp(ω n t)Ψ n (x), n ≥ 1, such that the real part of ω n becomes positively unbounded [4]. This says that the mathematical problem is ill posed in the sense of Hadamard. Therefore, we see an explosive behavior of the solutions which is not a suitable property for a heat conduction theory. Nevertheless this theory has caught researchers' attention when equation (1) is replaced by its Taylor approximations to the delay parameters. For instance, by taking the first order approximation of the flux vector and zeroth order approximation of the temperature gradient we recover the Maxwell-Cattaneo proposition; and by taking first order approximation to both, we recover the theory proposed by Morro, Payne and Straughan [18] which is considered in the book by Flavin and Rionero [5]. Several other order of approximations have proposed some new and stimulating heat conduction equations to be studied from a mathematical viewpoint. In this paper we will adopt the following approximation proposed by Tzou,θ To clarify the stability of this equation is the aim of many research works [1,19,21,15] (among others). It is known that the solutions of this equation decay exponentially when τ q < 2τ θ and polynomially when τ q = 2τ θ . Moreover, they are unstable when τ q > 2τ θ . Thus, τ q = 2τ θ is called the critical case of stability when the phase-lag τ q and τ θ are constants.
In this paper we concentrate our attention to the thermoelastic theory based on the heat conduction equation (3). The system was proposed by Chandrasekharaiah [3] and studied in [22]. There Quintanilla and Racke considered the one-dimensional problem and obtained by means of the energy arguments the exponential decay of solutions in case that τ q < 2τ θ . Instability of solutions was also proved when τ q > 2τ θ [20] by means of the spectral analysis. We would like to continue the study of this problem and concentrate our attention to two cases. First, for constant phase-lag, we want to clarify whether τ q = 2τ θ is still the critical case. The second problem corresponds to the case that the phase-lag τ θ depends on the material point. A new case arises between τ q < 2τ θ (x) and τ q = 2τ θ (x) on the entire spatial domain, i.e., τ q < 2τ θ (x) only on a proper sub-domain.
To prove our main theorems we will use the semigroup arguments. For the first case we apply a theorem obtained by Borichev and Tomilov [2] which characterize the polynomial decay of solutions by the resolvent operator behavior on the imaginary axis. This kind of treatment has been applied to a large class of problems in the literature. For the second case we adopt the semigroup arguments similar to the one in [12]. It is worth noting that special multipliers are introduced to handle 2 τ 2 u(0, t) = θ(0, t) = u(L, t) = θ(L, t) = 0, where we have deleted the tilde to simplify the notation. This system will be the target of our investigation in this paper. Define We say that a(x) is locally positive if for some x 1 , x 2 ∈ [0, L]. For the simplicity of notation, we use prime to denote the derivative about x hereafter. In order to obtain a properly defined energy for this system, we multiply equation (8) and (9) by θ 0 u t and Note that and Then the sum of (13) and (14) yields where Let v = u t , φ = θ t , ψ = θ tt , and z = (u, v, θ, φ, ψ) T . Define the state space , equipped with the inner product which induces the energy norm in (18). Then, we can rewrite (8)-(9) as a first order evolution equation in H where It is obvious that D(A) is dense in H. From (17), we also have the dissipativeness of A. Moreover, 0 ∈ ρ(A) which will be proved in next section where we show the imaginary axis is in ρ(A). By Theorem 1.2.4 in [17], we have Theorem 2.1. The operator A defined above is the infinitesimal generator of a C 0 -semigroup e At of contractions in the Hilbert space H.
Our main tools are the following two theorems.
Here, polynomial stability of order 1 k means that there exists a positive constant C > 0 such that for all z 0 ∈ D(A).
Throughout this paper, we use ·, · and · for the standard inner product and norm in L 2 .
3. Polynomial stability for a(x) ≡ 0. In this section, we consider the case i.e., τ θ and τ q are positive constants, and τ θ = τq 2 . Theorem 3.1. The semigroup e tA is polynomially stable of order γ = 1 2 when 2τ θ = τ q , i.e., for all z 0 ∈ D(A), there is a constant C > 0 such that the solution z of (19) satisfies Proof. Assume that (25) is false. Then by the uniform boundedness theorem, there exist a sequence β n → ∞ and a unit norm vector sequence which implies that For the simplicity of notation, we omit the subscript n hereafter. Let γ = 2. From dissipation, we have Combining (35) and (32), we also have which further implies that, due to (33), Take the inner product of ψ β 2 and φ β 2 with (33) and (34) in L 2 (Ω), respectively. We have Integrating by parts shows that last three terms on the left-hand side of (39) converges to zero due to (35) and (36). It is clear that the second and third term on the left-hand side of (39) also converges to zero. Therefore, (38),(39) now lead to Dividing (34) by β 3 and using (30), it can be simplified into The inner product of (41) with u in L 2 (Ω) leads to It follows from (31) and (35)-(37) that 1 β u is bounded. Hence, the second term on the left-hand side of (42) converges to zero. On the other hand, the boundary terms can be estimated by the Gagliardo-Nirenberg inequality as follows.
To verify condition (24), we assume iω ∈ σ(A). Then, there exist a sequence β → ω and a unit norm vector sequence z = (u, v, θ, φ, ψ) T ∈ D(A) such that iβψ It follows from the dissipation of A that since β is bounded in (50) and (51). By the Poincaré inequality, we also have Replace v in (52) by iβu . Repeating the same argument between (41) and (47), we obtain u , v = o(1).
Theorem 3.2. The polynomial decay rate found in Theorem 3.1 is optimal.
Proof. Notice that when m = 0 in system (8)-(9), the system decoupled into a dissipative phase-lag heat equation and a conservative wave equation. It has been proved in [15] that the energy of that heat equation, which coincides the heat energy defined in this paper, decays at a rate of 1 √ t , same as the one we derived in Theorem 3.1. If we can show 1 √ t is the optimal energy decay rate for the heat equation, then the energy decay rate in Theorem 3.1 can not be faster.
Denote by A 1 the restriction of A to the last three components where m = 0, and by H 1 the last three spaces in H. Consider the sequence F n = (0, 0, sin p n x) T where p n = nπ L . This sequence is bounded in H 1 . Indeed, for all n.
We will find solution in the form z n = (θ n , φ n , ψ n ) T to the equation Eliminating φ n , ψ n in (57), we obtain where τ is used to denote τ q and 2τ θ . Due to the boundary conditions, the functions solves the equation (58) if and only if A n satisfies We choose β n such that Then, This gives A n = − τ 2 . Now, the vector function z n = (A n sin p n x, iA n β n sin p n x, −β 2 n A n sin p n x) T is the solution to the resolvent equation (57). Therefore, which implies that for any > 0, we can find β n → ∞ such that 1 This shows that 1 √ t is the optimal decay rate. Remark 1. It is worthwhile to point out that by coupling the dual-phase-lag heat equation with a conservative wave equation, the optimal polynomial decay rate remains the same. The heat dissipation is passed to the conservative equation most efficiently.
4. Exponential stability for locally positive a(x). In this section, we consider the case of locally positive a(x). Assume that there is a positive constants C such that |a (x)| ≤ Ca x 2 ). But if a(x) = 0 at the interface, it imposes a restriction on the slope of a(x) near the interface. In this case, condition (64) can be satisfied, for example, if a (x i ), i = 1, 2 are finite. We believe that condition (64) is only needed because of the multiplier method we will use in the proof of next theorem. Proof. Since (22) has been proved in last section, we only need to verify condition (23). Assume that it is false. Then by the uniform boundedness theorem, there exists a sequence β → ∞ and a unit norm vector sequence z = (u, v, θ, φ, ψ) T ∈ D(A) such that iβψ From the dissipation, We are going to derive z H = o(1) from (66)-(71), which is a contradiction. The rest of the proof is divided into two parts. First, we show z H is of o(1) locally. More precisely, a 1 2 (x)z H = o(1). Then, we extend the local result to global. This strategy is adopted from [14].

ZHUANGYI LIU AND RAMON QUINTANILLA
The inner product of (74) with a(x)u in L 2 (Ω) leads to The first term on the left-hand side of (75) is of o(1) because u is bounded and (73). The third term can be estimated as follows.
β u is bounded which can be seen from (67). Here, we also used (71). The boundary terms in (75) vanish if a(0) = a(L) = 0. Otherwise, by the Gagliardo-Nirenberg inequality, for ξ = 0, L. Therefore, we obtain Finally, we take the inner product of (67) with 1 β a(x)v in L 2 (Ω), i.e., It follows from the local results (71),(78), and the equations (66),(69), that the above two inner product terms converge to zero. We arrive at Step 2 In this step, we will establish the following identity First, we substitute (66) into (67) to get For any nonnegative function q 1 (x) ∈ C 1 (Ω) with q 1 (0) = 0, we take the real part of the inner product of (82) with 2q 1 (x)u in L 2 (Ω). Then, since we have θ = o(1) in (71), f 1 = o(1), and Next, Substitute (69) into (70) and use the fact that φ = o(1) to get For any function q(x) ∈ C 1 [0, L] with q(0) = 0, taking the real part of the inner product of (85) with 2q( since Note that two terms in (86) can be simplified and removed as follows.
due to the local dissipation of φ in (71). Furthermore, the last term on the left-hand side of (83) and (86) are related. If we chose Here, we remark that the inner product of ψ with u , and v with φ are not small. However, by using related multipliers q 1 (x) and q(x), the sum of these two terms is small as shown in (89). On the other hand, we also have We now add (83) and (86). Taking into account of (88)-(90), this gives us (81).
Step 3 Pick q(x) = due to the condition (64). Therefore, by the local dissipation derived in Step 1, which further implies that v , ψ = o(1) (96) due to (66) and (69). We have reached the promised contradiction. The proof above can be repeated with very minor changes, such as the definition of the state space, the boundary terms from integration by parts. In fact, we don't need to use the Gagliardo-Nirenberg inequality to estimate boundary terms since they vanish in the above two cases.
Remark 4. The most general case corresponds to that both delay parameters depend on the material point. However, the analysis seems to be very cumbersome. We are unable to find a proper dissipative energy even just for the dual phase-lag heat equation (3). It is our plan to address this problem in future studies.