Spectra of Gurtin-Pipkin type of integro-differential equations and applications to waves in graded viscoelastic structures

In this paper, we study spectral properties and spectral enclosures for the Gurtin-Pipkin type of integro-differential equations in several dimensions. The analysis is based on an operator function and we consider the relation between the studied operator function and other formulations of the spectral problem. The theory is applied to wave equations with Boltzmann damping.


Introduction
Integro-differential equations with unbounded operator coefficients are used to describe viscoelasticity, heat transfer with finite propagation speed, systems with thermal memory, and acoustic waves in composite media [13,4,9].
In this paper, we consider spectral properties and spectral enclosures of the second-order Gurtin-Pipkin type of integro-differential equations (1.1) u tt ptq`T a uptq`ż where T a and T b are unbounded self-adjoint operators with compact resolvents in a separable Hilbert space H. Assume that the constants in the exponential kernel K satisfy a j ą 0 and 0 ă b 1 ă b 2 ă¨¨¨ă b N . After applying the Laplace transform f pλq " ş 8 0 f ptqe´λ t dt to (1.1) with homogeneous initial conditions, we formally obtain the symbol of (1.1), which for λ P D :" Czt´b 1 ,´b 2 , . . . ,´b N u takes the form (1.2) T pλq " λ 2`T a´TbK pλq,Kpλq " In the spectral analysis of (1.2) we need to deal with the following three difficulties: (i) the essential spectrum is non-trivial, (ii) the domain of T depend on λ, and (iii) the spectrum is not real. Spectral properties of other classes of rational operator functions have been studied in e.g. [1,15,12,18,5,8].
The point spectrum of (1.2) and solvability of (1.1) when T b is a positive multiple of T a were studied extensively; see [9,19] and the references therein. Those papers derive results for general kernels K, using methods from analytic function theory.
An alternative approach is to base the analysis on the system operator [3]. Spectral properties of the system operator that corresponds to (1.1) were in [11] considered in the space L 2 p0, 1q with T a " a d dx , T b " b d dx for positive constants a and b. A closely related one-dimensional wave equation was studied in [20]. They derived explicit expressions on the essential spectrum, studied the asymptotics of the eigenvalues, and the Riesz basis property. In all these studies is T b " bT a for some positive constant b, which implies that (1.2) only depends on one unbounded operator. However, there are many applications with two unbounded operator coefficients in (1.2), e.g. contact problems where one material is elastic with no dissipation and one material is viscoelastic [16]. Moreover, interest for functionally graded materials in elastic structures is growing. Those materials are designed by varying the microstructure from one material to another material with a specific gradient. The gradient is then, on a macroscopic scale, modeled by a smooth function. Under some geometric conditions, it has been shown that the solution of the wave equation with a graded Boltzmann damping decays exponentially to zero [17]. However, there are to my knowledge no spectral analysis results for this class of problems.
In this paper, we derive spectral results for the operator function (1.2). The results for graded materials include an explicit formula for the essential spectrum and spectral enclosures for the eigenvalues. We consider also the case when T b " bT a for some positive constant b and apply the results to wave equations in several space variables with Boltzmann damping.
Throughout this article LpH 1 , H 2 q and BpH 1 , H 2 q denote the collection of linear and bounded operators between the Hilbert spaces H 1 and H 2 , respectively. For convenience we use the notation LpH 1 q :" LpH 1 , H 1 q. Assume that the operator function T : D Ñ LpHq with the (in general) λ-dependent domain dom T is closed. Then, the spectrum of an operator function T is defined as σpT q " tλ P D : 0 P σpT pλqqu.
The essential spectrum σ ess pAq of a closed operator A is defined as the set of all λ P C such that A´λ is not a Fredholm operator. Hence, the essential spectrum of the operator function T is the set σ ess pT q " tλ P D : 0 P σ ess pT pλqqu.
Let D T , DT Ă C and consider the operator functions with domains dom T and domT , respectively. Then T andT are called equivalent on D Ă D T X DT [10,7] if there exists invertible operator functions Furthermore, T andT are called equivalent after operator extension [7] if for some bounded operator functionsÊ andF with bounded inverse. Equivalent operators share many properties. In particular, T pλq is closed (closable) if and only ifT pλq is closed (closable). Let T andT denote the closure of two operator functions that are equivalent on D. Then, if we restrict the functions to D, the spectrum is preserved: σ p pT q " σ p pT q, σ c pT q " σ c pT q, σ r pT q " σ r pT q, where σ p , σ c , and σ r denote the point, continuous, and residual spectrum, respectively.
Assume that the operator functionT : D Ñ LpH 'Ĥq can be represented as The domain ofT is, unless otherwise stated, the natural domain of the operator function, domT pλq " dom Apλq X dom Cpλq ' dom Bpλq X dom Dpλq.

The spectrum and numerical range of T
In this section, we prove theorems that in the application part of the paper are used to study wave equations with graded material properties and Boltzmann damping. For that reason, we introduce in the theorem below a continuous function b on a bounded and sufficiently smooth domain Ω Ă R n . This function corresponds in the application to a graded material. Moreover, 1´bpxqKp0q ą 0 for any x P Ω in the studied physical problem and we will therefore restrict the analysis to functions that satisfy the inequality.
The operator function T : D Ñ LpHq in (1.2) is always defined on the maximal domain D :" Czt´b 1 ,´b 2 , . . . ,´b N u.
Theorem 2.1. Assume that L : H Ñ H is a T a -compact operator, where T a is a positive self-adjoint operator and T´1 a is compact. Furthermore, (1.2) is a closed operator function in H, where the self-adjoint operator T b can be written in the form T b " bT a`L for some b P CpΩq. Assume 0 ď b min ď bpxq ď b max , for x P Ω and 1 ą b max ř N j"1 a j and let f pλ, xq :" 1´bpxqKpλq. Then (i): σ ess pT q Ă p´b N , 0q and σ ess pT q " tλ P C : f pλ, xq " 0 for some x PΩu, whereT pλq is a Fredholm operator if f pλ, xq is non-zero for x P Ω. Note that f pλ, x 0 q ś N j"1 pλ`b j q for a fixed x 0 P Ω is an N -th degree polynomial in λ. Hence, f pλ, x 0 q " 0 has at most N solutions. By assumption is f p0, x 0 q ą 0, f pλ, x 0 q Ñ 8, λ Ñ´bj , and f pλ, x 0 q Ñ´8, λ Ñ´bj . Hence, f pλ, x 0 q " 0 has N real solutions. Take positive constants , δ and x 0 PΩ such that f pλ, x 0 q " 0 and |f pλ, x 0 q| ă for all x P Ω δ with Ω δ :" tx P Ω : |x´x 0 | ă δu. Let u δ denote a smooth function with support in Ω δ and }u δ } " 1. Then Hence, λ P σ ess pT q Ă p´b N , 0q.
(ii) Take λ P σ r pT q and set V :" ran T pλq. Since, T is self-adjoint it follows from the projection theorem that Hence,λ is an eigenvalue of T , but (i) implies that λ P σ r pT q is real. The claim follows since λ P σ r pT q is not an eigenvalue.
The concept of Jordan chains can be formulated for general bounded analytic operator functions [14]. We study an unbounded operator function but the spectral problem can also as in Theorem 2.1 be formulated in terms of a bounded operator function. LetT pλ 0 qu 0 " 0. The vectors u 1 , u 2 , . . . u m´1 are associated with the The sequence tu j u m´1 j"0 is called a Jordan chain of length m and the maximal length of a chain of an eigenvector and associated vectors is called the multiplicity of the eigenvector [14].
Then, the length of the corresponding Jordan chain is one.
Note that pbu 0 , u 0 q " b when b is constant, which simplifies the condition (2.1). A frequently studied case is when T b " bT a for some positive constant b. The operator function can then be written in the form Let dom T pλq " H if 1´bKpλq " 0 and dom T pλq " dom T a otherwise. Assume 1´bKpλq ‰ 0 and set (2.2) Gpλq :"´λ 2 1´bKpλq .
Then λ P σpT q if and only if Gpλq P σpT a q for some λ P D.
2.1. The enclosure of the numerical range. The numerical range of T : D Ñ LpHq is the set W pT q " tλ P D : Du P dom T zt0u, }u} " 1, so that pT pλqu, uq " 0u, where p¨,¨q and }¨} denote the inner product and norm in H, respectively. We assume that T satisfies the assumptions in Theorem 2.1 with T b " bT a . Hence 0 R W pT p0qq, which implies σpT q Ă W pT q [14, Theorem 26.6].
Properties of the numerical range, including the number of components, are of fundamental importance, but it is difficult to study this set directly. Therefore, we considered in [6] a systematic approach to derive a computable enclosure of W for a particular class of rational operator functions. In this paper, we use a similar idea to derive an enclosure of W pT q, but we incorporate the special structure with the two (in general) unbounded operator coefficients T a and T b . Take u P dom T , with }u} " 1 and set Note that α only depend on W pT a q, which in many cases is easy to find. Lemma 2.3. Assume that T satisfies the assumptions in Theorem 2.1 with T b " bT a and set Then σ ess pT q Ă W α,β pT q X R Ă rc 0 , c 1 s Ă p´b N , 0s. Moreover, W α,β pT q is symmetric with respect to R.
Proof. We have λ P W α,β pT q if and only if gpλq P W pT a q for someb. Note that gpλq Ñ´8, λ Ñ˘8, which implies that W α,β pT q X R is contained in a bounded interval. Set From the assumption follows rp0q ą 0 and we have r 1 pλq ą 0 for λ ě 0. Moreover, rpλq ą 0 for λ ă´b N . Define the pN`2q-degree polynomial g α,β pλq " rpλq ś N j"1 pλ`b j q. The polynomial g α,β has real coefficients, which implies the symmetry with respect to R.
In the following lemma, C´denote the set of all complex numbers with nonpositive real part. Lemma 2.4. Assume that T satisfies the assumptions in Theorem 2.1 with T b " bT a and set λ :" x`iy, x, y P R. Then W α,β pT q Ă C´and λ " x`iy P W α,β pT qzR if and only if for some β P rb min α, b max αs, with α P W pT a q.
Proof. The system (2.5) follows directly by taking the real and imaginary parts of (2.4) and W α,β pT q Ă C´is a consequence of (2.5).
Lemma 2.5. Set I α " tiy P iR : y "˘?α , for some α P W pT a qu. Then Proof. The claims follow directly from (2.5).
Note that we for many configurations do not expect spectrum on iR even if b min " 0. The reason is that it has been shown that the solution of the corresponding time dependent problem decays exponentially to zero under some additional "geometric" conditions on b [17].
In the next subsection we provide more detailed spectral results for the case with one rational term.

2.2.
A one-pole case. In this subsection, we consider the rational operator function with one pole. Assume that T satisfies the assumptions in Theorem 2.1 with T b " bT a and define for λ P C the operator polynomial p α,β pλq " λ 3`b 1 λ 2`α λ`b 1 α´a 1 b 1 β and define the enclosure W α,β pP q of W pP q as W α,β pP q :" tλ P C : p α,β pλq " 0, α P W pT a q, β P rb min α, b max αsu.
Then, the numerical range W pP q is composed of at most three components and the following lemma provides, for the one-pole case, more explicit constants than Lemma 2.3.
Lemma 2.6. The enclosure W α,β pP q has the following properties (i): W α,β pP q X R Ă rc 0 , c 1 s, where rc 0 , c 1 s Ă r´b 1 , 0s are given by gpλq ě W min pT a q for someb ) , c 1 "´b 1`bmax a 1 b 1 .
(ii): r´b 1`bmin a 1 b 1 ,´b 1`bmax a 1 b 1 s Ă W α,β pP q if and only if W pT a q is unbounded.
Then the numerical range is contained in the union of S 0 " rc 0 , c 1 s, S`" ts P C : Re s P rd 0 , d 1 s, Im s ědu, S´" ts P C : Re s P rd 0 , d 1 s, Im s ď´du.
Proof. We prove the first two claims by studying the function g defined in (2.3).
(i): The claim follows from the assumptions 1´b max a 1 ą 0, W min pT a q ą 0 and the definition of g. (ii): The function b is continuous and g is singular when λ "´b 1`b a 1 b 1 for someb P rb min , b max s. (iii): Set ppλq " p α,β pλq and write p in the form where c " b 1 p1`ba 1 qα,b P rb min , b max s. This polynomial can be written in the form ppλq " pλ´pu`ivqqpλ´pu´ivqqpλ´wq for some u, v, w P R. Identification of the coefficients give b 1 "´2u´w, α " u 2`v2`2 uw, αb 1 p1`ba 1 q "´wpu 2`v2 q.

On equivalence and linearization of T
In this section, we consider the relation between the operator function T in (1.2) and the linearization of T in a general setting. Let H be a Hilbert space and assume that T a with domain dom T a is self-adjoint in H and bounded by η from below. Furthermore, T b with domain dom T b is assumed to be non-negative and self-adjoint in H.
SetĤ "Ĥ 1 'Ĥ 1 '¨¨¨'Ĥ N and let B " rB 1 B 2 . . . B N s :Ĥ Ñ H denote a densely defined operator. Then T : D Ñ LpHq can be written in the form and D " diag pb 1 , b 2 , . . . , b N q. LetH " H 'Ĥ and define the operator function P : C Ñ LpHq as with the natural domain. This operator function is formally related to T by the relation " However, the relation is not an equivalence when B is unbounded since the outer operators in the product are not bounded.
Define the operator functions Those functions are equivalent on C sincẽ Moreover, by applying the theory developed in [7], we obtain the equivalencẽ P W pλq "ẼpλqT pλqF pλq, where The function P W can be written in the form and we have therefore shown that P after operator function extension with W is equivalent toT pλq. Hence, under the assumption that the operator functions are closable we have the following properties σ p pT qzt0u " σ p pP qzt0u, σ c pT qzt0u " σ c pP qzt0u, σ r pT qzt0u " σ r pP qzt0u. Proof. Write P in the form Hence, P pλq is a bounded perturbation of a symmetric and upper-dominant block operator matrix, which implies that P is closable [2]. Let A denote the closure of A. Then, the self-adjoint operator A is given by Hence, P pλq " A`Bpλq with domain dom A is closed. Since P is closable it follows from the equivalence discussed above thatT is closable.
Assume that T b is bounded by zero from below, T a ą µ ą 0, and T b is T abounded. Then, we can as an alternative to (3.1) consider the bounded operator function (3.6)T pλq " I`λ 2 T´1 a´B pD`λq´1B˚, whereB " rB 1B2 . . .B N s,B j " a a j b j pT´1 a T b q 1{2 and D " diag pb 1 , b 2 , . . . , b N q. The rational operator functionT is then, after extension, equivalent to a block operator matrix on the form (3.2) with bounded entries.
3.1. One pole. LetH " H 'Ĥ 1 and define the closed operator B :Ĥ 1 Ñ H such that BB˚v 1 " T b v 1 . Assume that the block operator function P : C Ñ LpHq, Proof. Assume´b 1 P σ p pP q. Then it exists a non-zero v " rv 1 , v 2 s T PH such that Take v 1 " 0. Then Bv 2 " 0, which since B is injective implies v 2 " 0. Assume v 1 ‰ 0. Taking the inner product with v 1 in (3.8) we find that 0 " ppT a`b 2 1 qv 1 , v 1 q`pv 1 , B˚v 1 q but (3.9) implies that B˚v 1 " 0 and T a`b  " v 1 pb 1`λ q´1B˚v 1  is an eigenvector of P at λ.
(ii) The claim follows immediately from the definition of P pλq. Proof. The claims follow from the definitions of P andT .
We can write the rational operator function (3.1) in the alternative form T pλq " T a`λ 2´B pb 1`λ q´1Ĉ, T a "BĈ and use the approach in [7] to associate an operator pencil in H 'H. By switching order of the first two spaces in H 'H " H ' H 'Ĥ 1 , we obtaiñ T pλq " shows thatT is equivalent toS´λ, whereS is an abstract version of the system operator considered in [11]. The generalization to arbitrary rational terms is straightforward. The system operator in [11] was derived by introducing new variables and we showed above how the system operator is related to the rational operator function T .

The elastic wave equation with Boltzmann damping
Viscoelastic materials such as synthetic polymers, wood, tissue, and metals at high temperature are common in nature and engineering. They are used to absorb chock, to isolate from vibrations, and to dampen noise. In this section we study the wave equation in Ω Ă R n with Boltzmann damping [4]. The wave equation with Boltzmann damping can for u " uptq : r0, 8q Ñ L 2 pΩq be written in the form with upx, tq " 0, x P BΩ and upx, 0q " u 0 pxq, u t px, 0q " u 1 pxq [4,16]. Let b be a L 8 -function on a bounded Ω Ă R n with a Lipschitz continuous boundary BΩ and let a denote a positive constant. Assume that 0 ă b min ă bpxq ă b max and define the form t b : L 2 pΩqˆL 2 pΩq Ñ C, t b ru, vs " a ż Ω b∇u¨∇vdx with domain dom t b " tu P L 2 pΩq : t b ru, us ă 8u. The domain of t b is a Hilbert space with respect to t b r¨,¨s, dom t b is dense in L 2 pΩq, and t b is symmetric. From Kato, p 331 follows that where, the self-adjoint operator T b : L 2 pΩq Ñ L 2 pΩq with domain dom T b " tu P dom t b : T 1{2 b u P dom T b u is taken as the realization of T b "´adivb∇. Similarly, T a denotes the self-adjoint realization of´a∆.
Space constant Boltzmann damping. Spectral analysis of (4.1) was in [11] presented for the Ω Ă R and b constant. In the following, we consider the generalization to Ω Ă R n , n " 1, 2, 3 with one rational term and derive enclosures of the numerical range. Set H :" L 2 pΩq ' L 2 pΩq ' L 2 pΩq n .
Let d 0 denote the constant in Lemma 2.6. By taking the imaginary part of (2.2) with N " 1, we conclude that a sequence of eigenvalues tλ n u 8 n"1 with Im λ n Ñ˘8 only is possible if Re λ n Ñ d 0 when n Ñ 8. In the case Ω " p0, 1q, it was proved in [11] that it exists a branch of eigenvalues λ n with λ n " d 0˘i ?
The condition (2.1) is satisfied, which implies that eigenvectors corresponding to real eigenvalues have no associated vectors. The eigenvectors of T are related to P (Proposition 3.3),T (Proposition 3.4), andS (3.13).