Correct solvability of hyperbolic Volterra equations with kernels depending on the parameter
aa r X i v : . [ m a t h - ph ] D ec Correct solvability of hyperbolic Volterra equations withkernels depending on the parameter
Romeo Perez Ortiz ∗ , Victor V. Vlasov † Faculty of Mechanics and MathematicsMoscow Lomonosov State UniversityVorobievi Gori, Moscow, 119991, Russia [email protected]@mech.math.msu.su
Abstract
We study the correct solvability of an abstract functional differentialequations in Hilbert space, which includes integro-differential equationsdescribing evolution of thermal phenomena, heat transfer in materialswith memory or sound propagation in viscoelastic media.
Mathematics Subject Classification(2014): 34D05, 34C23.Keywords: Functional differential equations, Integrodifferential equations, Sobolev space, Gurtin-Pipkinheat equation. ∗ Supported by the Mexican Center for Economic and Social Studies (CEMEES, by its Spanish acronym). † Supported by the Russian Foundation for Basic Research, project N14-01-00349a and N13-01-00384a . Introduction
We study functional differential and integro-differential equations with unbounded operatorcoefficients in a Hilbert space. The main part of the equation under consideration is an abstracthyperbolic-type equation, disturbed by terms involving Volterra operators. These equationscan be regarded as an abstract form of the Gurtin-Pipkin equation (see [4, 8] for more details),which describes evolution of thermal phenomena, heat transfer in materials with memory orsound propagation in viscoelastic media. It also arises in homogenization problems in porousmedia (Darcy law). Countless examples about Gurtin-Pipkin type equation are studied in [1].It is shown that the initial boundary value problems for these equations are well-solvable inSobolev spaces on the positive half-axis (see, for instance, [9, 10, 11, 12, 13, 18]).For a self-adjoint positive operator A considered, we can take, in particular, the operator Au = − µ ∆ u − ( λ + µ ) ▽ (div u ) where µ , λ are the Lame coefficients or A = − ∆ with differentboundary conditions (for more details, see [2, 3]). Actually, there is an extensive literatureon abstract integro-differential equations (see [4, 5, 8, 9, 11, 12, 13, 17, 18] and the referencestherein).Vlasov and Rautian in [9] established well-defined solvability of initial boundary value prob-lems in weighted Sobolev space on the positive semi-axis for the case ξ = 1. In the presentpaper we establish also the well-defined solvability, but for the case ξ ∈ (0 , ξ = 1 , ξ = 0 are included. Thepresent work is a natural extension of the results [9, 11, 12].It is important to mention here that in the paper [13] we provided a theorem about thewell-defined solvability for the case ξ ∈ (0 ,
1) and there we mentionated that the result was thesame as in [9, Theorem 1]. In reality the result is more general and different. In the section 3we provide the details.This paper consists of four sections. Section 1 is a brief introduction to the subject anddescription of some applications of Gurtin-Pipkin type equation. Section 2 contains definitionsof Sobolev space and the formulation of correct solvability theorem. In the section 3 is providedthe proof of correct solvability theorem for the case ξ ∈ (0 , a . b means a ≤ Cb , C > a ≈ b means a . b . a .
2. Correct Solvability
Let H be a separable Hilbert space and A a self-adjoint positive operator in H with acompact inverse. We associate the domain dom( A β ) of the operator A β , β >
0, with a Hilbertspace H β by introducing on dom( A β ) the norm k·k = k A β ·k , equivalent to the graph norm of theoperator A β . Denote by { e j } ∞ j =1 an othonormal basis formed by eigenvectors of A correspondingto its eigenvalues a j such that Ae j = a j e j , j ∈ N . The eigenvalues a j are enumerated inincreasing order with their multiplicity, that is, they satisfy: 0 < a ≤ a ≤ · · · ≤ a n · · · ; where1 n → ∞ as n → + ∞ .By W n ,γ ( R + , A n ) we denote the Sobolev space that consists of vector-functions on the semi-axis R + = (0 , ∞ ) with values in H and norm k u k W n ,γ ( R + ,A n ) ≡ (cid:18)Z ∞ exp( − γt ) (cid:0) k u ( n ) ( t ) k H + k A n u ( t ) k H (cid:1) dt (cid:19) / , γ ≥ . A complete description of the space W n ,γ ( R + , A n ) and its properties are given in the monograph[17, Chap. I]. Now, on the semi-axis R + = (0 , ∞ ) consider the problem d udt + A u − Z t K ( t − s ) A ξ u ( s ) ds = f ( t ) , t ∈ R + , (2.1) u (+0) = ϕ , u (1) (+0) = ϕ , < ξ < . (2.2)It is assumed that the vector-valued function A − ξ f ( t ) belongs to L ,γ ( R + , H ) for some γ ≥ K ( t ) admits the representation K ( t ) = ∞ X j =1 c j exp( − γ j t ) , (2.3)where c j > γ j +1 > γ j > j ∈ N , γ j → + ∞ ( j → + ∞ ) and it is assumed that ∞ X j =1 c j γ j < . (2.4)Note that if the condition (2.4) is satisfied, then K ∈ L ( R + ) and k K k L <
1. Now, if,moreover, we take into consideration the condition ∞ X j =1 c j < + ∞ , (2.5)then the kernel K belongs to the space W ( R + ). Definition 2.1.
A vector-valued function u is called a strong solution of problem (2.1) and(2.2) if for some γ ≥ u ∈ W ,γ ( R + , A ) satisfies the equation (2.1) almost everywhere on thesemi-axis R + , as well as the initial condition (2.2).In the paper [9, Theorem 1] was shown the existence of strong solution u and the well-defined solvability of system (2.1) − (2.2) for ξ = 1. The result provided below is more generalthan the result obtained in [9]. But it is important to note that for ξ = 1 we obtain the sameresult. Theorem 2.1.
Suppose for some γ ≥ , A − ξ f ( t ) ∈ L ,γ ( R + , H ) for all ξ ∈ [0 , . Supposealso that the condition (2.4) is satisfied. Then ) If condition (2.5) holds and ϕ ∈ H , ϕ ∈ H , ξ ∈ [0 , then for any γ > γ the problems(2.1) and (2.2) have a unique solution in space W ,γ ( R + , A ) and this solution satisfies theestimate k u k W ,γ ( R + ,A ) ≤ d (cid:16) k A − ξ f k L ,γ ( R + ,H ) + k A ϕ k H + k Aϕ k H (cid:17) . (2.6) with a constant d that does not depend on the vector-valued function f and the vectors ϕ , ϕ .2) If condition (2.5) does not hold (that is, K ( t ) / ∈ W ( R + ) ) and ϕ ∈ H ξ , ϕ ∈ H ξ , ξ ∈ (0 , then for any γ > γ the problems (2.1) and (2.2) have a unique solution in space W ,γ ( R + , A ) and this solution satisfies the estimate k u k W ,γ ( R + ,A ) ≤ d (cid:16) k A − ξ f k L ,γ ( R + ,H ) + k A ξ ϕ k H + k A ξ ϕ k H (cid:17) . (2.7) with a constant d that does not depend on the vector-valued function f and the vectors ϕ , ϕ .Remark . We note, moreover, that the solution u ( t ) ∈ W ,γ ( R + , A ξ ) if A ξ ∈ L ,γ ( R + , H )for all ξ ∈ (0 ,
3. Proof of Theorem 2.1
Proof in the case of homogeneous initial conditions ϕ = ϕ = 0. For this purpose, we needto establish well-defined solvability of Cauchy problem for hyperbolic equations on the basisof the Laplace transformation. Before proceeding, is appropriate to mention some well-knownfacts that will be used later. Definition 3.1.
The
Hardy space H (Re ζ > γ, H ) is defined as the class of functions ˆ f ( ζ )taking values in H , holomorphic (analytic) on the half-plane { ζ ∈ C : Re ζ > γ ≥ } endowedwith the norm k ˆ f k H (Re ζ>γ,H ) = (cid:18) sup Re ζ>γ Z + ∞−∞ k ˆ f ( x + iy ) k H dy (cid:19) / < + ∞ , ( ζ = x + iy ) . (3.1)Let us formulate the well-known Paley-Wiener theorem about the Hardy space H (Re ζ > γ, H ). Theorem 3.1. ( Paley-Wiener )1. The space H (Re ζ > γ, H ) coincides with the set of vector-valued functions (Laplace trans-forms), which admit the representation ˆ f ( ζ ) = 1 √ π Z + ∞ exp( − ζ t ) f ( t ) dt, (3.2) where f ( t ) ∈ L ,γ ( R + , H ) , ζ ∈ C , Re ζ > γ ≥ . For any ˆ f ( ζ ) ∈ H (Re ζ > γ, H ) there is one and only one representation of the form (3.2),where f ( t ) ∈ L ,γ ( R + , H ) . Moreover, the following inversion formula holds: f ( t ) = 1 √ π Z + ∞−∞ ˆ f ( γ + iy ) exp(( γ + iy ) t ) dt, t ∈ R + , γ ≥
3. For ˆ f ( ζ ) ∈ H (Re ζ > γ, H ) and f ( t ) ∈ L ,γ ( R + , H ) connected by the representation (3.2),the following relation holds: k ˆ f ( ζ ) k H (Re ζ>γ,H ) ≡ sup Re ζ>γ Z + ∞−∞ k ˆ f ( x + iy ) k H dy = Z + ∞ e − γt k f ( t ) k H dt ≡ k f ( t ) k L ,γ ( R + ,H ) . (3.4) This Theorem is well-known for scalar functions, but can be easily extended to the case offunctions with values in a separable Hilbert space (see, for instance, [18]).We begin proving the Theorem 2.1 in the homogeneous initial conditions case ϕ = ϕ = 0.We note that the Laplace transform ˆ u ( ζ ) of any strong solution of equation (2.1) with the initialcondition (2.2) has the form ˆ u ( ζ ) = L − ( ζ ) ˆ f ( ζ ) , (3.5)where the operator-valued function L ( ζ ) is the symbol of equation (2.1) and can be representedas L ( ζ ) = ζ I + A − ∞ X k =1 c k ζ + γ k ! A ξ , < ξ < . (3.6)It is assumed here that there is γ ∗ ≥ u ( t ) ∈ W ,γ ∗ ( R + , A ). This condition isnecessary to apply the Laplace transform to the equation (2.1).If we can prove that the vector-valued function of equation (3.5) is such that A ˆ u ( ζ ) and ζ ˆ u ( ζ ) belong to the Hardy space H (Re ζ > γ, H ) for some γ > γ ≥
0, then by the Paley-Wiener theorem, we be able to prove that A u ( t ) and d u ( t ) /dt belong to L ,γ ( R + , H ) and,therefore we will have shown that u ( t ) ∈ W ,γ ( R + , A ). That is, the solvability of system(2.1) − (2.2) in the space W ,γ ( R + , A ) will be established.With this in mind, let us consider the projection ˆ u n ( ζ ) of the vector-valued function ˆ u ( ζ )to the one-dimensional subspace spanned by the vector e n :ˆ u n ( ζ ) = ℓ − n ( ζ ) ˆ f n ( ζ ) , (3.7)where ˆ f n ( ζ ) = ( ˆ f ( ζ ) , e n ) and ℓ n ( ζ ) := ( L ( ζ ) e n , e n ) = ζ + a n − ∞ X k =1 c k ζ + γ k ! a ξn A ˆ u ( ζ ) to the one-dimensional space spanned by e n has the form (cid:0) A ˆ u ( ζ ) e n , e n (cid:1) = a ξn ˆ g n ( ζ ) ℓ n ( ζ ) , ≤ ξ ≤ g n ( ζ ) is the n-th coordinate of the vector-valued function ˆ g ( ζ ) = A − ξ ˆ f ( ζ ). Accordingto the conditions of the Theorem 2.1, the vector-valued function g ( t ) = A − ξ f ( t ) belongs tothe space L ,γ ( R + , H ), and therefore, its Laplace transform ˆ g ( ζ ) = A − ξ ˆ f ( ζ ) belongs to Hardyspace H (Re ζ > γ , H ).In order to prove that A ˆ u ( ζ ) belongs to H (Re ζ > γ, H ), it is suffices to establish theestimate sup Re ζ>γn ∈ N (cid:12)(cid:12)(cid:12)(cid:12) a ξn ℓ n ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const, for all ξ ∈ [0 ,
1] (3.9)which is uniform with respect to ζ (Re ζ > γ ) and n ∈ N .For this purpose, consider the function m n ( ζ ) = ℓ n ( ζ ) a n . Let us estimate m n ( ζ ) from below bymeans of its real and imaginary parts:Re m n ( ζ ) = x − y a n + 1 − a − ξ ) n ∞ X k =1 c k ( x + γ k )( x + γ k ) + y ! , ζ = x + iy Im m n ( ζ ) = 2 xya n + ya − ξ ) n ∞ X k =1 c k ( x + γ k ) + y ! . First, we find a lower bound for | Im m n ( ζ ) | for | y | > x , where x > γ > γ ≥ | Im m n ( ζ ) | > x | y | a n + 1 | y | a − ξ ) n ∞ X k =1 c k (1 + γ k | y | ) + 1 ! > γy + k ( γ ) a ξn | y | a n , where k ( γ ) = c (1+ γ γ ) +1 . Hence, for | y | > x with x > γ > γ ≥ | ℓ n ( ζ ) | = 1 a n | Im m n ( ζ ) | < | y | γy + k ( γ ) a ξn < a ξn p γ · k ( γ ) . (3.10)Second, we estimate | Re m n ( ζ ) | from below for | y | < x , where x > γ > γ ≥
0. Note that1 a − ξ ) n ∞ X k =1 c k ( x + γ k )( x + γ k ) + y ! < a − ξ ) n ∞ X k =1 c k x + γ k ! < a − ξ ) n ∞ X k =1 c k γ k ! < .
5t follows that | Re m n ( ζ ) | > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − a − ξ ) n ∞ X k =1 c k γ k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > . Therefore, for | y | < x , with x > γ > γ ≥
0, we have1 | ℓ n ( ζ ) | < a n | Re m n ( ζ ) | < a n (cid:12)(cid:12)(cid:12) − a − ξ ) n P ∞ k =1 c k γ k (cid:12)(cid:12)(cid:12) . (3.11)From the estimates (3.10) and (3.11) we obtain (cid:12)(cid:12)(cid:12)(cid:12) a ξn ℓ n ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) < p γ · k ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) a ξn ℓ n ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) < a − ξn (cid:12)(cid:12)(cid:12) − a − ξ ) n P ∞ k =1 c k γ k (cid:12)(cid:12)(cid:12) < a − ξ (cid:12)(cid:12)(cid:12)(cid:12) − a − ξ )1 P ∞ k =1 c k γ k (cid:12)(cid:12)(cid:12)(cid:12) . Therefore,sup Re ζ>γn ∈ N (cid:12)(cid:12)(cid:12)(cid:12) a ξn ℓ n ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) < (cid:18)p γ · k ( γ ) , a − ξ (cid:12)(cid:12)(cid:12)(cid:12) − a − ξ )1 P ∞ k =1 c k γ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , for all ξ ∈ [0 , . (3.12) Remark . The estimate (3.12) implies thatsup Re ζ>γ (cid:13)(cid:13) A ξ L − ( ζ ) (cid:13)(cid:13) ≤ const. (3.13)The Hardy space H (Re ζ > γ, H ) is invariant with respect to multiplication of functions ofthe form a ξn ℓ n ( ζ ) , since they are analytic and bounded in view of (3.12). Therefore, the inclusionˆ g ( ζ ) = A − ξ ˆ f ( ζ ) ∈ H (Re ζ > γ, H ) implies that A ˆ u ( ζ ) belongs to H (Re ζ > γ, H ).Let us establish the estimate for the norm of vector-valued function A u ( t ) ∈ L ,γ ( R + , H ).From (3.5), it follows that A ˆ u ( ζ ) = A L − ( ζ ) ˆ f ( ζ ) = A ξ L − ( ζ ) A − ξ ˆ f ( ζ ) . (3.14)This function can be represented in the form A ˆ u ( ζ ) = ∞ X k =1 a ξk ℓ k ( ζ ) · a − ξk ˆ f k ( ζ ) e k . According to the hypothesis of Theorem 2.1, the vector-valued function A − ξ f ( t ) ∈ L ,γ ( R + , H ).6herefore, by the Paley-Wiener theorem, A − ξ ˆ f ( ζ ) ∈ H (Re ζ > γ , H ) and k A − ξ f k L ,γ ( R + ,H ) = k A − ξ ˆ f k H (Re ζ>γ ,H ) . By (3.12) and Paley-Wiener theorem, the following relations hold: k A u ( t ) k L ,γ ( R + ,H ) = k A ˆ u ( ζ ) k H (Re ζ>γ,H ) = k A ξ L − ( ζ ) A − ξ ˆ f ( ζ ) k H (Re ζ>γ,H ) But k A ξ L − ( ζ ) A − ξ ˆ f ( ζ ) k H (Re ζ>γ,H ) = sup Re ζ>γ Z + ∞−∞ ∞ X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ξk ℓ k ( ζ ) · a − ξk ˆ f k ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy ≤ sup Re ζ>γk ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ξk ℓ k ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · sup Re ζ>γ Z + ∞−∞ ∞ X k =1 (cid:12)(cid:12)(cid:12) a − ξk ˆ f k ( ζ ) (cid:12)(cid:12)(cid:12) ! dy (3.15) ≤ sup Re ζ>γk ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ξk ℓ k ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k A − ξ ˆ f ( ζ ) k H (Re ζ>γ,H ) ≤ d k A − ξ f k L ,γ ( R + ,H ) . where d = sup Re ζ>γk ∈ N (cid:12)(cid:12)(cid:12) a ξk ℓ k ( ζ ) (cid:12)(cid:12)(cid:12) . Hence, we obtain the inclusion A u ( t ) ∈ L ,γ ( R + , H ) and theestimate k A u k L ,γ ( R + ,H ) ≤ d (cid:0) k A − ξ f k L ,γ ( R + ,H ) (cid:1) . (3.16)is valid.Now let us prove that ζ ˆ u also belongs to H (Re ζ > γ, H ) . SetΨ( ζ ) := ∞ X k =1 c k ζ + γ k . Note that for Re ζ > γ we can write I = ζ L − ( ζ ) + (cid:0) − Ψ( ζ ) A − − ξ ) (cid:1) A L − ( ζ ) . Hence, for Re ζ > γ , we obtainˆ f ( ζ ) = ζ L − ( ζ ) ˆ f ( ζ ) + (cid:0) − Ψ( ζ ) A − − ξ ) (cid:1) A L − ( ζ ) ˆ f ( ζ )= ζ ˆ u ( ζ ) + (cid:0) − Ψ( ζ ) A − − ξ ) (cid:1) A ξ L − ( ζ ) A − ξ ˆ f ( ζ ) . (3.17)It is our understanding that A − ξ ˆ f ( ζ ) ∈ H (Re ζ > γ, H ) and sup Re ζ>γ (cid:13)(cid:13) A ξ L − ( ζ ) (cid:13)(cid:13) ≤ const. From (3.17), it follows that ζ ˆ u ( ζ ) = ˆ f ( ζ ) − (cid:0) − Ψ( ζ ) A − − ξ ) (cid:1) A ξ L − ( ζ ) A − ξ ˆ f ( ζ ) . (3.18)7herefore, the function ζ ˆ u n ( ζ ) can be represented in the form ζ ˆ u n ( ζ ) = ˆ f n ( ζ ) − (cid:18) − Ψ( ζ ) a − ξ ) n (cid:19) a ξn ℓ n ( ζ ) ˆ g n ( ζ ) (3.19)Under the assumptions imposed on the sequences { c k } ∞ k =1 and { γ k } ∞ k =1 , the function Ψ( ζ ) isanalytic and bounded on the half-plane { ζ : Re ζ > γ } . Indeed, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Ψ( ζ ) a − ξ ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ( ζ ) a − ξ ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < a − ξ ) n ∞ X k =1 c k γ k (cid:12)(cid:12)(cid:12) ζγ k + 1 (cid:12)(cid:12)(cid:12) < a − ξ ) n ∞ X k =1 c k γ k (cid:16) Re ζγ k + 1 (cid:17) < a − ξ ) n ∞ X k =1 c k γ k (cid:16) γγ k + 1 (cid:17) < a − ξ ) n ∞ X k =1 c k γ k < . (3.20) Since the vector-function A − ξ f ( t ) ∈ L ,γ ( R + , H ) then its Laplace transform A − ξ ˆ f ( ζ ) belongsto Hardy space H (Re ζ > γ, H ), that is, k A − ξ ˆ f k H (Re ζ>γ,H ) = sup Re ζ>γ Z + ∞−∞ k A − ξ ˆ f ( x + iy ) k H dy < + ∞ , ζ = x + iy Hence we obtain the following relation k A − ξ ˆ f k H (Re ζ>γ,H ) = sup Re ζ>γn ∈ N Z + ∞−∞ ∞ X n =1 | a − ξn ˆ f n ( x + iy ) | ! dy> sup Re ζ>γn ∈ N a − ξ )1 Z + ∞−∞ ∞ X n =1 | ˆ f n ( x + iy ) | ! dy = a − ξ )1 sup Re ζ>γ Z + ∞−∞ k ˆ f ( x + iy ) k H dy = a − ξ )1 k ˆ f ( ζ ) k H (Re ζ>γ,H ) . (3.21)Therefore, ˆ f ( ζ ) ∈ H (Re ζ > γ, H ) and f ( t ) ∈ L ,γ ( R + , H ). Now, taking into account (3.12)and (3.21) we obtain the following estimate: sup Re ζ>γn ∈ N Z + ∞−∞ | ζ ˆ u n ( ζ ) | dy < sup Re ζ>γn ∈ N a − ξ )1 + 4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ξn ℓ n ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · sup Re ζ>γn ∈ N Z + ∞−∞ | ˆ g n ( ζ ) | dy < + ∞ . Therefore ζ ˆ u n ( ζ ) ∈ H (Re ζ > γ, C ) and d dt u n ( t ) ∈ L ,γ ( R + , C ).8y (3.12) and Paley-Wiener theorem we have: (cid:13)(cid:13)(cid:13)(cid:13) d dt u ( t ) (cid:13)(cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) = k ζ ˆ u ( ζ ) k H (Re ζ>γ,H ) = k ˆ f ( ζ ) − (cid:0) − Ψ( ζ ) A − − ξ ) (cid:1) A ξ L − ( ζ ) A − ξ ˆ f ( ζ ) k H (Re ζ>γ,H ) < a − ξ )1 k A − ξ ˆ f ( ζ ) k H (Re ζ>γ,H ) + 4 k A ξ L − ( ζ ) A − ξ ˆ f ( ζ ) k H (Re ζ>γ,H ) ≤ a − ξ )1 k A − ξ ˆ f ( ζ ) k H (Re ζ>γ,H ) + 4 sup Re ζ>γk ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ξk ℓ k ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k A − ξ ˆ f ( ζ ) k H (Re ζ>γ,H ) < a − ξ + 2 sup Re ζ>γk ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ξk ℓ k ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k A − ξ ˆ f ( ζ ) k H (Re ζ>γ,H ) ≤ d k A − ξ f k L ,γ ( R + ,H ) , (3.22)where d = (cid:16) a − ξ + 2 d (cid:17) . Hence, we obtain the inclusion d dt u ( t ) ∈ L ,γ ( R + , H ) and theestimate (cid:13)(cid:13)(cid:13)(cid:13) d dt u ( t ) (cid:13)(cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) ≤ d (cid:0) k A − ξ f k L ,γ ( R + ,H ) (cid:1) (3.23)is valid. Accordingly, combining the estimates (3.16) and (3.23), we come to the desired in-equality k u ( t ) k W ,γ ( R + ,A ) ≤ d (cid:0) k A − ξ f k L ,γ ( R + ,H ) (cid:1) . (3.24)with a constant d independent of f . Implying that the equation (2.1) has a solution u ( t ), whichbelongs to space W ,γ ( R + , A ) and the estimate (3.24) holds.Now, let us prove that the solution u ( t ) the initial conditions u (+0) = 0 and u (1) (+0) = 0. Remark . If ϕ ( ζ ) ∈ H (Re ζ > γ, C ), then for any η > γ there is a sequence { η k } ∞ k =1 suchthat lim k →∞ η k = + ∞ and lim k →∞ Z ηγ | ϕ ( x ± iη k ) | dx = 0 . Indeed, for any η > γ and η k >
0, we have Z η k − η k (cid:18)Z ηγ | ϕ ( x ± iη k ) | dx (cid:19) dy ≤ Z ηγ (cid:18)Z + ∞−∞ | ϕ ( x ± iη k ) | dy (cid:19) dx < + ∞ . η > { η k } ∞ k =1 such that lim k →∞ η k = + ∞ andlim k →∞ Z ηγ | ϕ ( x ± iη k ) | dx = 0 . Now it remains to use the Cauchy inequality.The above reasoning shows that u ( t ) ∈ L ,γ ( R + , H ), and therefore, ˆ u ( ζ ) ∈ H (Re ζ > γ, H )and ˆ u n ( ζ ) ∈ H (Re ζ > γ, C ). Moreover, let us prove that ζ ˆ u n ( ζ ) ∈ H (Re ζ > γ, C ). Indeed, sup Re ζ>γn ∈ N Z + ∞−∞ | (Re ζ + iy )ˆ u n (Re ζ + iy ) | dy = sup Re ζ>γn ∈ N Z + ∞−∞ | (Re ζ + iy ) ˆ u n (Re ζ + iy ) | (Re ζ ) + y dy< γ sup Re ζ>γn ∈ N Z + ∞−∞ | ζ ˆ u n ( ζ ) | dy < + ∞ , since ζ ˆ u n ( ζ ) ∈ H (Re ζ > γ, C ). By the Paley-Wiener theorem, we obtain ˆ u n (+0) = 1 √ π lim η k →∞ Z η k − η k ˆ u n ( x + iy ) dy = 1 √ πi lim η k →∞ Z γ + iη k γ − iη k ˆ u n ( ζ ) dζ ˆ u (1) n (+0) = 1 √ π lim η k →∞ Z η k − η k ( x + iy )ˆ u n ( x + iy ) dy = 1 √ πi lim η k →∞ Z γ + iη k γ − iη k ζ ˆ u n ( ζ ) dζ. The functions ˆ u n (+0) and ˆ u (1) n (+0) are analytics on the right half-plane Re ζ > γ ≥
0, andtherefore, by the Cauchy theorem, for any η > γ , we have Z γ + iη k γ − iη k ˆ u n ( ζ ) dζ = (cid:18)Z η − iη k γ − iη k − Z η + iη k γ + iη k + Z η + iη k η − iη k (cid:19) ˆ u n ( ζ ) dζ = Z ηγ ˆ u n ( x − iη k ) dx − Z ηγ ˆ u n ( x + iη k ) dx + i Z η k − η k ˆ u n ( η + iy ) dy. and Z γ + iη k γ − iη k ζ ˆ u n ( ζ ) dζ = (cid:18)Z η − iη k γ − iη k − Z η + iη k γ + iη k + Z η + iη k η − iη k (cid:19) ζ ˆ u n ( ζ ) dζ = Z ηγ ( x − iη k )ˆ u n ( x − iη k ) dx − Z ηγ ( x + iη k )ˆ u n ( x + iη k ) dx ++ i Z η k − η k ( η + iy )ˆ u n ( η + iy ) dy. According to Remark 3.2, we havelim k →∞ Z ηγ | ˆ u n ( x ± iη k ) | dx = 0lim k →∞ Z ηγ | ( x ± iη k )ˆ u n ( x ± iη k ) | dx = 0 , ζ ˆ u n ( ζ ) ∈ H (Re ζ > γ, C ). Therefore, for η > γ , we find that | ˆ u n (+0) | ≤ √ π lim η k →∞ Z η k − η k | ˆ u n ( η + iy ) | dy = 1 √ π Z + ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) ( η + iy ) ˆ u n ( η + iy )( η + iy ) (cid:12)(cid:12)(cid:12)(cid:12) dy ≤ √ π (cid:18)Z + ∞−∞ | ( η + iy ) ˆ u n ( η + iy ) | dy (cid:19) / (cid:18)Z + ∞−∞ dy ( η + y ) (cid:19) / . η / | ˆ u (1) n (+0) | ≤ √ π lim η k →∞ Z η k − η k | ( η + iy )ˆ u n ( η + iy ) | dy = 1 √ π Z + ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) ( η + iy ) ˆ u n ( η + iy ) η + iy (cid:12)(cid:12)(cid:12)(cid:12) dy ≤ √ π (cid:18)Z + ∞−∞ | ( η + iy ) ˆ u n ( η + iy ) | dy (cid:19) / (cid:18)Z + ∞−∞ dyη + y (cid:19) / . η / It follows that for η → + ∞ , u (+0) = 0 and u (1) (+0) = 0.Finally, let us prove that the solution u ( t ) satisfies the equation (2.1). By Paley-Wienertheorem, we have u ( t ) = 1 √ π lim η → + ∞ Z η − η L − ( γ + iy ) ˆ f ( γ + iy )e ( γ + iy ) t dy = 1 √ πi lim η → + ∞ Z γ + iηγ − iη L − ( ζ ) ˆ f ( ζ )e ζt dζ. Hence, we get d dt u ( t ) = 1 √ πi lim η → + ∞ Z γ + iηγ − iη ζ L − ( ζ ) ˆ f ( ζ )e ζt dζ (3.25) A u ( t ) = 1 √ πi lim η → + ∞ Z γ + iηγ − iη A L − ( ζ ) ˆ f ( ζ )e ζt dζ (3.26) Z t K ( t − s ) A ξ u ( s ) ds = 1 √ πi lim η → + ∞ Z t ∞ X k =1 c k e − γ k ( t − s ) (cid:18)Z γ + iηγ − iη A ξ L − ( ζ ) ˆ f ( ζ )e ζt dζ (cid:19) ds = 1 √ πi lim η → + ∞ Z γ + iηγ − iη A ξ L − ( ζ ) ˆ f ( ζ ) ∞ X k =1 c k Z t e − γ k ( t − s ) e ζt ds ! dζ = 1 √ πi lim η → + ∞ Z γ + iηγ − iη ∞ X k =1 c k ζ + γ k ! A ξ L − ( ζ ) ˆ f ( ζ )e ζt dζ. (3.27) From (3.25)-(3.27), it follows that u ( t ) satisfies the equation (2.1).Let us turn to the proof of Theorem 2.1 in the case of nonhomogeneous initial conditions.11n the system (2.1)-(2.2), set u ( t ) = cos( At ) ϕ + A − sin( At ) ϕ + ω ( t ) . Then for the function ω ( t ) we obtain the system d dt ω ( t )+ A ω ( t ) − Z t K ( t − s ) A ξ ds = f ( t ) , t ∈ R + ω (+0) = ω (1) (+0) = 0 , where f ( t ) = f ( t ) − h ( t ) and h ( t ) = Z t K ( t − s ) A ξ (cid:0) cos( At ) ϕ + A − sin( At ) ϕ (cid:1) ds (3.28) A − ξ h ( t ) = Z t K ( t − s ) (cid:0) A ξ cos( At ) ϕ + A ξ sin( At ) ϕ (cid:1) ds (3.29)Let us prove that the vector-valued function f ( t ) satisfies the conditions of Theorem 2.1 withhomogeneous initial conditions. Indeed, k A − ξ f ( t ) k L ,γ ( R + ,H ) ≤ k A − ξ f ( t ) k L ,γ ( R + ,H ) + k A − ξ h ( t ) k L ,γ ( R + ,H ) . Let us estimate the norm k A − ξ h ( t ) k L ,γ ( R + ,H ) .1. Suppose that the condition (2.5) holds. The direct integration shows that Z t e − γ k ( t − s ) cos( As ) ds = (cid:0) A + γ k (cid:1) − { γ k { cos( At ) − exp( − γ k t ) } + A sin( At ) } (3.30) Z t e − γ k ( t − s ) sin( As ) ds = (cid:0) A + γ k (cid:1) − { A { exp( − γ k t ) − cos( At ) } + γ k sin( At ) } (3.31)In what follows, we need the following argument. Remark . The following inequality holds k (cid:0) A + γ k (cid:1) − k H . γ − k k A − k H . (3.32)Indeed, note that for any vector v ∈ H , k v k H = 1, the following relations are valid: k (cid:0) A + γ k (cid:1) − k H = ∞ X n =1 (cid:0) a n + γ n (cid:1) − | v n | . ∞ X n =1 ( a n γ n ) − | v n | = γ − k k A − v k H , where v n = ( v, e n ). This implies (3.32), since operator A is self-adjoint and γ k is positive.Now, using (3.28), (3.30), (3.31), the inequalities k cos( At ) k ≤ k sin( At ) k ≤ k A − ξ h ( t ) k L ,γ ( R + ,H ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 c k Z t e − γ k ( t − s ) n A ξ cos( At ) ϕ + A ξ sin( At ) ϕ o ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) ≤ (cid:13)(cid:13)(cid:13) R v A ξ { γ k { cos( At ) − exp( − γ k t ) } + A sin( At ) } ϕ (cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) ++ (cid:13)(cid:13)(cid:13) R v A ξ { A { exp( − γ k t ) − cos( At ) } + γ k sin( At ) } ϕ (cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) = (cid:13)(cid:13)(cid:13) cos( At ) R v γ k A ξ ϕ (cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) + (cid:13)(cid:13)(cid:13) e − γ k t R v γ k A ξ ϕ (cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) ++ (cid:13)(cid:13)(cid:13) sin( At ) R v A ξ ϕ (cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) + (cid:13)(cid:13)(cid:13) e − γ k t R v A ξ ϕ (cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) ++ (cid:13)(cid:13)(cid:13) cos( At ) R v A ξ ϕ (cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) + (cid:13)(cid:13)(cid:13) sin( At ) R v γ k A ξ ϕ (cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 c k γ k γ − k A − A ξ ϕ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 c k (cid:0) A + γ k (cid:1) − A ξ ϕ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 c k (cid:0) A + γ k (cid:1) − A ξ ϕ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 c k γ k γ − k A − A ξ ϕ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H . ∞ X k =1 c k (cid:16) k A ξ ϕ k H + k A ξ ϕ k H (cid:17) where R v = P ∞ k =1 c k ( A + γ k ) − . Thus, from (3.24) and the last estimate, we get k ω k W ,γ ( R + ,H ) ≤ d k A − ξ f k L ,γ ( R + ,H ) ≤ d (cid:16) k A − ξ f ( t ) k L ,γ ( R + ,H ) + k A ξ ϕ k H + k A ξ ϕ k H (cid:17) . (3.33) The estimation of k v ( t ) k W n ,γ ( R + ,A ) := k cos( At ) ϕ k W n ,γ ( R + ,A ) + k A − sin( At ) ϕ k W n ,γ ( R + ,A ) isobtained inmediatly. Indeed, k cos( At ) ϕ k W n ,γ ( R + ,A ) = (cid:18)Z ∞ e − γt (cid:0) k A cos( At ) ϕ k H + k A cos( At ) ϕ k H (cid:1) dt (cid:19) / < (cid:18) Z ∞ e − γt (cid:0) k A ϕ k H (cid:1) dt (cid:19) / < γ / k A ϕ k H k A − sin( At ) ϕ k W n ,γ ( R + ,A ) = (cid:18)Z ∞ e − γt (cid:0) k A sin( At ) ϕ k H + k A sin( At ) ϕ k H (cid:1) dt (cid:19) / < (cid:18) Z ∞ e − γt (cid:0) k Aϕ k H (cid:1) dt (cid:19) / < γ / k Aϕ k H This mean that ϕ ∈ dom( A ) = H and ϕ ∈ dom( A ) = H . Hence, from (3.33) and obtainedabove it follows that k u ( t ) k W ,γ ( R + ,H ) ≤ d (cid:0) k A − ξ f ( t ) k L ,γ ( R + ,H ) + k A ϕ k H + k Aϕ k H (cid:1) for all ξ ∈ [0 , . with a constant d that does not depend on the vector-valued function f and the vectors ϕ , ϕ .13herefore, we have obtained the first estimate of Theorem 2.1.2. Suppose, now, that the condition (2.5) does not hold. Then (2.4) and (3.28) imply that k A − ξ h ( t ) k L ,γ ( R + ,H ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t ∞ X k =1 c k e − γ k ( t − s ) (cid:8) A ξ cos( At ) ϕ + A ξ sin( At ) ϕ (cid:9) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ,γ ( R + ,H ) ≤ √ γ ∞ X k =1 c k γ k (cid:0) k A ξ ϕ k H + k A ξ ϕ k H (cid:1) . Therefore, for all ξ ∈ [0 ,
1] we have k ω k W ,γ ( R + ,H ) ≤ d k A − ξ f k L ,γ ( R + ,H ) ≤ d (cid:0) k A − ξ f ( t ) k L ,γ ( R + ,H ) + k A ξ ϕ k H + k A ξ ϕ k H (cid:1) . Consequently, the second estimate of Theorem 2.1 is calculated: k u ( t ) k W ,γ ( R + ,H ) ≤ d (cid:0) k A − ξ f ( t ) k L ,γ ( R + ,H ) + k A ξ ϕ k H + k A ξ ϕ k H (cid:1) for all ξ ∈ (0 , , with a constant d that does not depend on the vector-valued function f and the vectors ϕ , ϕ .
4. Comments and observations
Let us mention some results closely related to those obtained in the present paper. In [10, 11]well-defined solvability is studied for problems of the form (2.6) and (2.7) under the hypothesisthat the kernel K ( t ) is of class W ( R + ). In [9] and in the presente paper was included the casewhen K ( t ) / ∈ W ( R + ) (see the second estimate of Theorem 2.1).The proof in [10, 11] about the existence theorem essentially utilizes the decay to theLaplace transform b K ( ζ ), but does not involve the assumption that the function K ( t ) can berepresented as an exponential series of the form (2.3) (case that was considered in [9] and inthe present work). It is noteworthy here that the correct solvability of hyperbolic Volterraequations considered here is a general and natural extension of what was presented in [9].Hereinafter, we provide some corrections and accuracies of paper [13]. Lemma 4.1.
Consider the function ℓ n,N ( ζ ) = ζ + a n − a − ξ ) n N X k =1 c k ζ + γ k ! := ζ + a n f n,N ( ζ ) , k = 1 , ...N. (4.1)14 hen the zeroes of the function ℓ n,N form a set of real zeroes { µ n,k } Nk =1 , such that − γ k < µ n,k < x n,k < − γ k − < µ n,k − < x n,k − < · · · < − γ < µ n, < x n, < , (4.2) µ n,k − x n,k = O a − ξ ) n ! , ≤ ξ < / , k = 1 , , · · · N, a n → ∞ ,µ n,k − x n,k = O (cid:18) a n (cid:19) , ξ = 1 / , k = 1 , , · · · N, a n → ∞ ,µ n,k − x n,k = O (cid:18) a ξn (cid:19) , / < ξ ≤ , k = 1 , , · · · N, a n → ∞ , where x n,k are real zeroes of the function f n,N ( ζ ) together with a pair of complex-conjugatezeroes µ ± n , µ + n = µ − n , which admit the asymptotic representation for all ξ ∈ (0 , and a n → ∞ µ ± n ( ξ ) = −
12 1 a − ξ ) n N X j =1 c j + O a − ξ ) n ! ± i (cid:18) a n + O (cid:18) a − ξn (cid:19)(cid:19) , ≤ ξ < , (4.3) µ ± n ( ξ ) = −
12 1 a n N X j =1 c j + O (cid:18) a n (cid:19) ± i ( a n + O (1)) , ξ = , (4.4) µ ± n ( ξ ) = −
12 1 a − ξ ) n N X j =1 c j + O (cid:18) a ξn (cid:19) ± i (cid:18) a n + O (cid:18) a ξ − n (cid:19)(cid:19) , < ξ ≤ In [13] was assumed the following condition holds:sup k { γ k ( γ k +1 − γ k ) } = + ∞ (4.6) Theorem 4.1.
Suppose that conditions (2.5) and (4.6) are satisfied. Then the zeros of themeromorphic function ℓ n ( ζ ) form a countable set of real zeroes { µ n,k } ∞ k =1 , which satisfy − γ k < µ n,k < − γ k − , lim n →∞ µ n,k = − γ k , (4.7) together with a pair of complex-conjugate zeroes µ ± n , µ + n = µ − n which admit the following asymp-totic representation µ ± n ( ξ ) = −
12 1 a − ξ ) n ∞ X j =1 c j + O a − ξ ) n ! ± i (cid:18) a n + O (cid:18) a − ξn (cid:19)(cid:19) , ≤ ξ < (4.8) µ ± n ( ξ ) = −
12 1 a n ∞ X j =1 c j + O (cid:18) a n (cid:19) ± i ( a n + O (1)) , ξ = , (4.9) µ ± n ( ξ ) = −
12 1 a − ξ ) n ∞ X j =1 c j + O (cid:18) a ξn (cid:19) ± i (cid:18) a n + O (cid:18) a ξ − n (cid:19)(cid:19) , < ξ ≤ Under the conditions of Theorem 4.1, the spectrum σ ( L ) of the operator-valued function15 ( ζ ) coincides with the set of zeroes { µ n,k } ∞ , ∞ n,k =1 and { µ ± n } ∞ n =1 of function ℓ n ( ζ ), that is, σ ( L ) = (cid:0) ∪ ∞ n,k =1 µ n,k (cid:1) ∪ ( ∪ ∞ n µ ± n ) Condition 1.
In the case when the condition (2.5) is not satisfied, the sequences { c k } ∞ k =1 , { γ k } ∞ k =1 have the following asymptotic representation c k = Ak α + O (cid:0) k α +1 (cid:1) , γ k = Bk β + O ( k β − ),( k → ∞ ) where c k > γ k +1 > γ k >
0, constants
A > , B >
0, 0 < α ≤ α + β > P ∞ k =1 c k γ k < γ k +1 − γ k ≈ k β − p is satisfied for some p ∈ N , p ≥ k → ∞ then thecondition (4.6) of Ivanov is satisfied when p < β . Theorem 4.2.
Assume that the Condition 1 is satisfied. Then the zeroes of the function ℓ n ( ζ ) form a countable set of real zeroes { λ n,k | k ∈ N } such that · · · − γ k < − λ n,k < · · · < − γ < , lim n →∞ λ n,k = − γ k (4.11) together with a pair of complex-conjugate zeroes λ ± n , λ + n = λ − n which admit the asymptoticrepresentations when a n → ∞ : λ ± n ( ξ ) = AD B r − βa r +1 − ξn + O (cid:18) a δ n (cid:19) ± i (cid:18) a n + AD B r − βa r +1 − ξn + O (cid:18) a δ n (cid:19)(cid:19) , r ∈ (cid:0) , (cid:1) , (4.12) λ ± n ( ξ ) = AD B r − βa r +1 − ξn + O a − ξ ) n ! ± i (cid:18) a n + AD B r − βa r +1 − ξn + O (cid:18) a δ n (cid:19)(cid:19) , r ∈ (cid:2) , (cid:1) , (4.13) λ ± n ( ξ ) = − Aβ ln a n a − ξ ) n + O a − ξ ) n ! ± ia n , r = 1 , (4.14) where δ = min { − ξ ) , r + 3 − ξ } , δ = 2 r + 3 − ξ k = 1 , , r := α + β − β , the constant D = D + iD depends of r and is defined as follows: D := i Z ∞ dtt r ( i + t ) = 12 (cid:18)Z ∞ dtt r (1 + t ) + i Z ∞ dtt r − (1 + t ) (cid:19) = π (cid:0) i π (1 − r ) (cid:1) sin( πr ) . Under the conditions of Theorem 4.2 , the spectrum σ ( L ) of the operator-valued function L ( ζ ) coincides with the set of zeroes { λ n,k } ∞ , ∞ n,k =1 and { λ ± n } ∞ n =1 of function ℓ n ( ζ ), that is, σ ( L ) = (cid:0) ∪ ∞ n,k =1 λ n,k (cid:1) ∪ ( ∪ ∞ n λ ± n ) References [1] Giovambattista Amendola, Mauro Fabrizio and John Murrough Golden, Thermodynam-ics of Materials with Memory, Theory and Applications, Springer New York DordrechtHeidelberg London, 2012. 162] Jaime E. Mu˜noz Rivera, Maria Grazia Naso and Elena Vuk, Asymptotic behavior of theenergy for electromagnetic systems with memory.
Mathematical Methods in the AppliedSciences , 27, 819-841, 2004.[3] Jaime E. Mu˜noz Rivera, Maria Grazia Naso, On the Decay of the Energy for Systems withMemory and Indefinite Dissipation.
Asymptotic Analysis , 49, 189-204, 2006.[4] L. Pandolfi and S. Ivanov, Heat equations with memory: lack of controllability to the rest.
Journal of Mathematical Analysis and Applications , 355:1-11, 2009.[5] L. Pandolfi, The controllability of the Gurtin-Pipkin equations: a cosine operator approach.
Applied Mathematics and Optimization , 52:143-165, 2005.[6] Mauro Fabrizio and Barbara Lazzari, On the existence and the asymptotic stability ofsolutions for linearly viscoelastic solids.
Archive for Rational Mechanics and Analysis .Springer-Verlag, 116, 139-152, 1991.[7] M. A. Biot, Generalized theory of acoustic propagation in porous dissipative media,
Journalacoustical society of america , 34, 1254-1264, 1962.[8] A. C. Pipkin and M. E. Gurtin, A General theory of heat conduction with finite wavespeeds.
Archive for Rational Mechanics and Analysis , 31:13, 1968.[9] V. V. Vlasov and N. A. Rautian, Well-defined solvability and spectral analysis of abstracthyperbolic integrodifferential equations.
Journal of Mathematical Sciences , 179:25, 2011.[10] V. V. Vlasov, J. Wu, Spectral analysis and solvability of abstract hyperbolic equationsaftereffect, Diff. Uravn, 45, No. 4, 524-533 (2009).[11] G. R. Kabiroba, V. V. Vlasov and J. Wu. Well-defined and spectral properties of abstracthyperbolic integrodifferential equations with aftereffect.
Journal of Mathematical Sciences ,170:15, 2010.[12] V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, Spectral analysis and correct solvabilityof abstract integrodifferential equations arising in thermophysics and acoustics.
Journal ofMathematical Sciences , Vol. 190, No. 1, 2013.[13] Romeo Perez Ortiz, Victor V. Vlasov, Spectra of the Gurtin-Pipkin type equations withthe kernel, depending on the parameter. http://arxiv.org/abs/1403.4382 [14] S. A. Ivanov, “Wave Type” Spectrum of the Gurtin-Pipkin Equation of the Second Order,http://arxiv.org/abs/1002.2831[15] S. A. Ivanov, T. L. Sheronova, Spectrum of the heat equation with memory,http//arxiv.org/abs/0912.1818v1[16] Enrique Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory,