Asymptotic analysis of a nonsimple thermoelastic rod

The asymptotic analysis of a one-dimensional nonsimple thermoelastic 
problem is considered in this paper. By a detailed spectral 
analysis, the asymptotic expressions for eigenvalues and 
eigenfunctions of the considered system are developed. It is shown 
that the eigenfunctions form a Riesz basis on the Hilbert space and 
the eigenvalues asymptotically 
 fall on two branches. One branch is along the negative horizontal axis in the 
 complex plane and the other branch is asymptotic to a vertical line that is parallel 
to the imaginary axis. 
 This gives the spectrum-determined growth condition for the $C_0-$semigroup associated to the system, 
 and consequently, the asymptotic and the 
exponential stability of the solutions are deduced. The approach 
developed in this paper 
 confirms the already-existing results; furthermore, 
it can be extended to a larger field of applications such as 
coupled system of rod or beam with diffusion equation. The method 
will be illustrated by an example of thermoelastic beam equations 
with Dirichlet boundary conditions.


1.
Introduction. In general, the one-dimensional linear thermoelasticity problem for a homogeneous rod of uniform cross section with Dirichlet-Dirichlet boundary condition can be formulated as (see [12,18]) where u = u(x, t) represents the displacement of the solid elastic material and θ = θ(x, t) represents the absolute temperature. The coupling constant γ is a measure of the mechanical-thermal coupling present in the system. The constant c can be viewed as the small amplitude wave speed about a uniform temperature. Many important qualitative properties of system (1) have been published in recent years. By using a spectral approach, it is shown in [12] that the semigroup associated to system (1) is uniformly exponentially stable. Later the same result has been obtained by Liu and Zheng [18] by frequency domain multiplier method. By different spectral approaches (see for example [12] and [10]) it was shown that 1476 MONCEF AOUADI AND TAOUFIK MOULAHI there are two branches of eigenvalues for the system (1), which have the following asymptotic expansions where k is a large positive integer. It is seen from (2) that the first branch of eigenvalues is produced by the heat equation while the second one is associated with the elastic vibrations. In [19], numerical simulations show that the eigenfrequency of system (1) approaches asymptotically to a vertical line that is parallel to the imaginary axis and contained in the open left half of the complex plane. Guo and Yung [10] proved this numerical conjecture theoretically. A significant result of Guo and Chen [9] showing that there is a real eigenvalue for the system (1) that is greater than the dominant eigenvalue of "pure" heat equation. A more generalized result was proved in [8] that there is a set of generalized eigenfunctions of the system (1), which forms a Riesz basis for the state space. By Riesz basis property, the dynamic behavior of the system (1) can be expressed in terms of its eigenfrequencies. Moreover, the Riesz basis property concludes the spectrumdetermined growth condition which implies automatically the exponential stability. This is one of the hard and important problems in the stability analysis of infinitedimensional systems. From the well-known fact that ω(A) = max{s(A), ω ess (A)} and (2), where ω(A), s(A) and ω ess (A) denote the growth order of the semigroup e At , the spectral bound and essential bound of A, respectively, we see that the spectrum-determined-growth condition is always true for system (1). In [14], it is proved that the asymptote of the complex eigenvalues given in (2) is also the spectrum-determined-growth condition of A (the linear operator of the system), i.e., On the other hand, it was shown in [12] that for the system (1) with Dirichlet-Neumann or Neumann-Dirichlet boundary conditions, there is a sequence of generalized eigenfunctions which forms a Riesz basis for the Hilbert state space. The success in obtaining this result lies in the simplicity of the corresponding characteristic equation as well as the explicit structure of the eigenfunctions. However, for the Dirichlet-Dirichlet boundary condition, the characteristic equations become a complicated transcend equation, and the eigenfunctions satisfy a fourth-order ordinary differential equation. For this end, Guo in [10] and [8] proposed another way to analyze the asymptotic behavior. This paper concerns the appropriate spectral eigenvalues estimates, showing that the eigenfunctions forms a Riesz basis, proving that the spectrum-determined growth condition holds and deducing the exponential stability for a nonsimple thermoelastic rod (see system (8) below). Our main tool in proving this, is a result due to Hansen [12] (see Proposition 3.1 below). More precisely, we treat the case where classic thermoelasticity involves higher order gradients of displacement. This makes the spectral analysis of the corresponding system more difficult and the proofs of basic theorems, as presented in [12], need further elaboration. Hence, the elaborated method in this paper allows the study of spectral properties of linear operators containing higher order gradients of displacement. The high order derivatives clarify the possible configurations of the materials more and more finely by the values of the successive higher gradients [3,7,15]. Moreover, in recent years, the spectral analysis becomes an important tool in the study of control problems (see for example [17]) and L p − L q decay estimations (see for example [24]) for classic thermoelastic materials. In other words, our study can be useful in the study of control and L p − L q decay estimations problems of nonsimple materials which is a topic that has not yet been addressed in the literature despite its various applications in applied sciences. This work continues the task developed in several recent papers where it has been tried to clarify the time decay of solutions for nonsimple thermoelastic problems [1,2,4,5,16,21,23].
This paper is organized as follows. In Section 2, we state the basic equations and semigroup setting of the nonsimple thermoelastic problem. In Section 3, it is shown that the corresponding eigenfunctions forms a Riesz basis for the Hilbert state space. In Section 4, we shall prove the existence of eigenvalues and show that there are two branches of eigenvalues for our system. One branch is along the negative horizontal axis in the complex plane and the other branch is asymptotic to a vertical line that is parallel to the imaginary axis. In section 5, we deduce the spectrum-determined growth condition, the asymptotic and the exponential stability. In the last section we conclude by confirming the already-existing results and by illustrating the method on a example of thermoelastic beam equations with Dirichlet boundary conditions. 2. Basic equations and semigroup setting. We shall employ the usual summation and differentiation conventions: Latin subscripts are understood to range over the integers (1, 2, 3), summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding cartesian coordinate. When supply terms are not present, the linear evolution equations of the general three-dimensional theory of nonsimple thermoelastic solids are given by [3,7,15] ρ and the constitutive equations for isotropic and centrosymmetric solids are (u i,rr δ jk + 2u r,rj δ ij + u j,rr δ ij ) + 2 (u r,ri δ jk + u r,rj δ ik ) + 2 3 u k,rr δ ij + 2 4 u k,ji + 5 (u i,jk + u j,ik ), where β = (3λ + 2µ)ς, ς is the coefficient of linear thermal expansion, λ and µ are Lamé's constants. T is the absolute temperature of the medium, T 0 is the reference uniform temperature of the body chosen such that |(T − T 0 )/T 0 | 1. q i is the heat conduction vector, k is the coefficient of thermal conductivity, c E is the specific heat at constant strain. σ ij , T ijk are the stress and the hyper-stress tensors, u i is the displacement vector, e ij are the components of the strain tensor, ρ is the mass density and i (i = 1, · · · , 5) are constitutive coefficients.
In the one-dimensional case, the evolution equations (4) become and the constitutive equations (5) take the form If we substitute the above constitutive equations into the evolution equations (6), we obtain the system of field equations in (0, By the change of variables the equations (7) become in (0, 1) × R + (dropping the asterisks for convenience) where The coupling constants γ and c are generally small in comparison to 1. The constant γ is a measure of the mechanical-thermal coupling present in the system, while constant c can be viewed as the small amplitude wave speed about a uniform temperature. The positive constant α is the coefficient of the fourth order spatial derivative characterizing the nonsimple elastic solid. If α = 0, the system (8) reduces to the classic thermoelastic system (1). In terms of the new variables y = (u x , u t , θ) = (y 1 , y 2 , y 3 ), the system (8) becomesẏ where We are going to study the system (9) with the following initial conditions and boundary conditions equipped with the inner product where y = (y 1 , y 2 , y 3 ) and z = (z 1 , z 2 , z 3 ). In view of (13), the corresponding norm in H is given by The initial-boundary value problem (9)-(11) is equivalent to the problem where y 0 = (y 0 1 , y 0 2 , y 0 3 ) and A : D(A) ⊂ H → H is the linear operator defined by The adjoint operator of A is easily calculated and given by for z ∈ D(A * ) = D(A).
Proposition 2.1. Let A and H be defined as before. Then A generates a C 0 − semigroup of contractions S(t) on H. Hence, the system (15) is well-posed, i.e., for any y 0 = (y 0 1 , y 0 2 , y 0 3 ) ∈ H, system (15) has a unique weak solution is the classical solution to (15).
3. Riesz basis property. In this section we will show that the eigenfunctions of A form a Riesz basis. Let us first recall that a set of vectors {f k } are said to be a Riesz basis for the Hilbert space H if there exists a bounded and invertible operator L from H to H such that f k = Le k , where {e k } is an orthogonal basis for H. We refer the reader to [25] for details. Our main tool in proving this is the following result [12].
Then {Φ k j } also is a Riesz basis for H. We can now state the main result of this section. Proposition 3.2. Let A (resp. A * ) be the operator defined as in (16) (resp. 17). Then the eigenfunctions of A (resp. A * ) form a Riesz basis for H. and By duality it will be enough to show that the eigenfunctions of A * form a Riesz basis. One can easily verify the following With respect to the orthogonal basis {E k j } 1≤j≤3 , A * has been decoupled into a chain of 3 × 3 blocks along the diagonal. If we set Σ k = (E k 1 , E k 2 , E k 3 ) and ξ = (ξ 1 , ξ 2 , ξ 3 ) T we can rewrite (19) as where Thus if (ξ, λ) is an eigenpair of R k , then (Σ k ξ, µ k λ) is an eigenpair of A * . The characteristic equation of R k is The coupling constants γ and c are generally taken small in comparison to 1, so δ = (γc) 2 = O(1). In particular we have δ = O(k −2 ) as k → ∞ which implies that (λ + µ k )(λ 2 + s 2 k ) → 0. Asymptotically, the roots of (22) consist of a real root λ k and a nonreal complex conjugate pair (σ k , σ k ) : The asymptotic expansions of λ k and σ k will be given in the following sections for all k ∈ N * .
Since the eigenfunctions of R k are simple then R k is diagonalisable, that is The columns of M k are the eigenvectors of R k , corresponding to simple eigenvalues (σ k , σ k , λ k ). Hence M k , is nonsingular for all k. In fact, from (23) we have that s 2 k +σ 2 k γc 2 → 0 and If we denote by Φ σ k , Φ σ k and Φ λ k (k ∈ N * ) the eigenvectors of R k corresponding to σ k , σ k and λ k , respectively, then Σ k Φ σ k , Σ k Φ σ k and Σ k Φ λ k are the corresponding eigenvectors of A * . Thus we have Furthermore, from (20) we get Since {E k j } is a Riesz basis (being an orthogonal basis) for H (defined in (12)), we may apply Proposition 3.1, from which it follows that Σ k Φ σ k ∪ Σ k Φ σ k ∪ Σ k Φ λ k is a Riesz basis for H. Thus the eigenfunctions of A * (resp. A) form a Riesz basis for H.
Remark 3.1. Likewise, for k ∈ N * it can be shown that Σ k Φ −σ k , Σ k Φ −σ k , and Σ k Φ −λ k (i.e., replace σ k by −σ k into Φ −σ k and so forth) are eigenvectors of A corresponding to the eigenvalues µ k σ k , µ k σ k and µ k λ k , respectively. The union of these eigenvectors is a Riesz basis for H which is dual to the Riesz basis defined in (25), i.e., if v k = n j where 0 < c 0 < |c n k | < C 0 (see Young [25]). 4. The asymptotic expansion of eigenvalues. In this section we obtain uniform and asymptotic estimates on the location of the spectrum of A. By virtue of (22) we need only to approximate the roots of the characteristic equation for R k In the following we give some properties of the function f k on the interval I = (−µ k − r, −µ k + r) where r is a positive real given by Is it clear that the definition of r always implies the condition In the following we construct a numerical sequence converging to the real solution of (26).
Proposition 4.1. Let f k denotes the characteristic polynomial of (26). Then f k is strictly increasing on I = (−µ k − r, −µ k + r) such that Furthermore there exists a positive constant M = 6r such that f k is M -Lipschitz on I.
Proof. As f k ∈ C ∞ (I) and f k (λ) = 6λ + 2µ k vanishes only at λ = − µ k 3 ∈ I (by force of (28)), we have that f k (λ) < 0 on I, then f k is strictly decreasing on I and consequently On the other hand, for all k ∈ N * , we have that This inequality together with (30) gives (29), which implies that f k is strictly increasing on I. For all λ, ς ∈ I, we have that which implies the existence of a positive constant M = 6r such that f k is M -Lipschitiz on I. Proof. Since f k ∈ C ∞ (I) and is strictly increasing on I (by Proposition 4.1), then f k is one to one on I. To prove the proposition it remains just to check that since γ is taken small in comparison to 1.
Since f k is M -Lipschitz on I, it is not hard to show that Applying the finite-increment theorem to ϕ on [0, 1] and using (31), yields Lemma 4.1. Let f k denotes the characteristic polynomial given by (26), then the sequence (λ n k ) n≥0 defined as follows: has the following properties for all n ≥ 1, There exists a positive constant K such that Proof. To prove (34), we first obtain a first order approximation λ 1 k to the real root λ k of f k (λ) by using one step of Newton's method with the initial guess of from which we get By this last identity, the expression of f k (λ 1 k ) becomes , and |λ 1 which implies To prove (35), we use Newton's sequence (33) which can be written as Applying (32) to (39) for ς = λ n k and λ = λ n−1 k , then (35) follows. From (29), (33) and (35) we get Hence (36) follows for Lemma 4.2. The sequence (λ n k ) n≥0 given by (33) satisfies and converges to λ k the unique real root of (26) on I.
We now use (41) to prove that the sequence (33) converges to λ k . For all m ∈ N * and p ∈ N * we have, .
From (38) 1 and (40), we can easily see that This implies that (λ n k ) n∈N is a Cauchy sequence on I and consequently converges to λ k the real root of (26).
We now give the asymptotic expansion of λ k . Theorem 4.3. If λ k denotes the real root of (26) then Proof. By steps of Newton's method with the initial guess of λ 0 k = −µ k , we get and Since the sequence (λ n k ) n≥0 defined by (33) converges to λ k , the sequence Λ n k converges to Λ k = λ 1 k − λ k . From the finite-increment theorem, there exists .
We summarize this in the following proposition.  (16) and (45) holds. The eigenvalues of A consist of a sequence of conjugate pairs {σ k ,σ k } ∞ k=1 and a real sequence . The corresponding set of eigenfunctions {Φσ k , Φσ k , Φλ k } is given by Proof. The eigenvalues of A (or A * ) given by (59) and (60) are obtained by multiplying σ k , σ k and λ k by µ k . Since Σ k Φ σ k , Σ k Φ σ k and Σ k Φ λ k are the corresponding of A (or A * ), we infer from (24) and (25) that According to Remark 3.1, the eigenfunctions Φσ k , Φσ k and Φλ k corresponding to the eigenvalues (59) and (60) are Transposing this last expression we get the desired result.
Remark 4.2. Asymptotically, the eigenvalues of A are given bŷ 5. Stability of the system. In this section, we shall discuss the stability of the system (9)-(11) based on the distribution of the spectrum of A.
Theorem 5.1. The solution to system (9)-(11) decays exponentially to zero, that is to say, there exist constants M > 1 and such that e At ≤ M e −ωt , or equivalently E(t) ≤ M E(0)e −ωt where E(t) is the energy function of system (9)-(11) defined by (14).