Spectral analysis and representation of solutions of

Volterra integro-dierential equations with

fractional exponential kernels

N. A. Rautian and V. V. Vlasov

Lomonosov Moscow State University

OTKR-2019, December 19-22, 2019

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Introduction

We study integro-dierential equations with unbounded operator coecientsin Hilbert space. The main part of these equations is an abstract hyperbolicequations, disturbed by the terms containing abstract integral Volterraoperators. The equations mentioned above are the abstract form of theintegro-dierential equation of Gurtin-Pipkin describing the process of heatpropagation in media with memory, process of wave propagation in the visco-elastic media, and also arising in the problems of porous media (Darci law).1) We obtain correct solvability of the initial value problems for the describedequations in the weighted Sobolev spaces on the positive semiaxis.2) We study the asymptotic behavior of solutions of integro-dierentialequations on the basis of spectral analysis of their symbols.3) To this end, we obtain representations of strong solutions of theseequations in the form of a sum of terms corresponding to the real and nonrealparts of the spectrum of the operator functions that are the symbols of theseequations. These representations are new for the considered class of integro-dierential equations.

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Let us H be a separable Hilbert space and A be a self-adjoint positiveoperator A∗ = A > κ0 (κ0 > 0) acting in the spaceH and having a compactinverse operator. Let us B be a symmetric operator (Bx, y) = (x,By),acting in the space H having the domain Dom (B) (Dom (A) ⊆ Dom (B)).Moreover B be a nonnegative operator that is (Bx, x) > 0 for any x, y ∈Dom (B) and satisfying to inequality ‖Bx‖ 6 κ ‖Ax‖, 0 < κ < 1 for anyx ∈ Dom (A) and I be the identity operator acting in the space H.We consider the following problem for a second-order integro-dierentialequation on the semiaxis R+ = (0,∞):

d2u(t)

dt2+Au(t)+Bu(t)−

∫ t

0K(t− s)Au(s)ds−

∫ t

0Q(t− s)Bu(s)ds =

= f(t), t ∈ R+, (1)

u(+0) = ϕ0, u(1)(+0) = ϕ1. (2)

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Assume that the scalar functions K(t) and Q(t) that are the kernels ofintegral operators admits the following representations:

K(t) =

∫ ∞0

e−tτdµ(τ), Q(t) =

∫ ∞0

e−tτdη(τ), (3)

where dµ and dη are the positive measures corresponding to an increasingright-continuous distribution functions µ and η respectively. The integral isunderstood in the Stieltjes sense.

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We assume that the following conditions are satised:∫ ∞0

dµ(τ)

τ< 1,

∫ ∞0

dη(τ)

τ< 1, (4)

Here the supports µ and η belong to the interval (d0,+∞), d0 > 0. Theconditions (4) means that K(t), Q(t) ∈ L1(R+), ‖K‖L1

< 1, ‖Q‖L1< 1.

If conditions (4) are supplemented with the conditions

K(0) =

∫ ∞0

dµ(τ) ≡ Varµ|∞0 < +∞,

Q(0) =

∫ ∞0

dη(τ) ≡ Var η|∞0 < +∞. (5)

then the kernels K(t) and Q(t) belong to the space W 11 (R+).

Further we will assume that the following condition is satised

inf||x||=1,

x∈Dom(A)

((A+B)x, x) > 1. (6)

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The equation (1) can be regarded as an abstract form of dynamicalviscoelastic integro-dierential equation where operators A and B aregenerated by the following dierential expressions

A = −ρ−1µ (∆u+ (1/3)grad(divu)) , B = −ρ−1λ(1/3)grad(divu),

here u = ~u(x, t) ∈ R3 is displacement vector of viscoelastic hereditaryisotropic media that ll the bounded domain Ω ⊂ R3 with smooth boundary,∂Ω, ρ is a constant density, ρ > 0, Lame parameters λ, µ are the positiveconstants, K(t), Q(t) are the relaxation functions characterizing hereditaryproperties of media. On the domain boundary ∂Ω the Dirichlet condition

u|∂Ω = 0. (7)

is satised. The Hilbert space H can be realized as the space of threedimensional vector-functions L2(Ω). The domain Dom(A) belongs to theSobolev space W 2

2 (Ω) of vector functions satisfying the condition (7). SeeA.A. Ilyushin, B.E. Pobedrya Bases of the mathematical theory of

thermoviscoelasticity. - M. Nauka, 1970,R.M. Christensen Theory of viscoelasticity. An introduction. - Academic PressNew York and London, 1971

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In case operator B = 0, positive and self-adjoint operator A can be realizedas operator Ay = −y′′(x), where x ∈ (0, π), y(0) = y(π) = 0, or theoperator Ay = −∆y with Dirichlet conditions on the bounded domain Q ⊂Rn with smooth boundary (H = L2(Q)) or more general elliptic self-adjointoperators in the space L2(Q). The equation (1) can be regarded as anabstract form of the Gurtin-Pipkin equation that describes heat transfer inmaterials with memory with nite speed. See

Gurtin M. E., Pipkin A. C. General theory of heat conduction with nite wavespeed. // Archive for Rational Mechanics and Analysis. 1968. 31. P. 113126.

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Applying the Laplace transform to the equation (1) with zero initialconditions we obtain the following operator-valued function

L(λ) = λ2I +A+B − K(λ)A− Q(λ)B, (8)

which are the symbol (analogue of the characteristic quasi-polynomial) ofthe equation (1). Here K(λ) and Q(λ) are the Laplace transforms of kernelsK(t) and Q(t) respectively, having the following representations

K(λ) =

∫ ∞0

dµ(τ)

λ+ τ, Q(λ) =

∫ ∞0

dν(τ)

λ+ τ, (9)

Denition

The set of values λ ∈ C is called the resolvent set R(L) of operator-valuedfunction L(λ) if there exists L−1(λ) is bounded for any λ ∈ R(L). The setσ(L) = λ ∈ C\R(L) |L(λ) exists is called the spectra of operator-valuedfucntion L(λ).

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Denote byA0 := A+B. It is follows from the properties of operatorsA andBthat the operator A0 is positive and self-adjoint. Moreover A0 is reversible,operators AA−1

0 , BA−10 are bounded and operator A−1

0 is compact (seemonograph T. Kato Perturbation Theory for Linear Operators// Springer-Verlag Berlin Heidelberg New York, 1980).Let us denote by Wn

2,γ (R+, A0) the Sobolev space of the vector-valuedfunctions on the positive semiaxis R+ = (0,∞) with the values in the spaceH equiped by the norm

‖u‖Wn2,γ(R+,A0) ≡

(∫ ∞0

e−2γt

(∥∥∥u(n)(t)∥∥∥2

H+ ‖A0u(t)‖2H

)dt

)1/2

,

γ ≥ 0.

For more detail description of the space Wn2,γ (R+, A0) see the monograph

J. L. Lions and E. Magenes Nonhomogeneous Boundary-Value Problems andApplications // Springer-Verlag, Berlin-Heidelberg-New York. 1972, chapter1. For n = 0 we haveW 0

2,γ (R+, A0) ≡ L2,γ (R+, H), and for γ = 0 we shallwrite Wn

2,0 = Wn2 .

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Correct solvability

We establish well-dened solvability of initial boundary value problem (1),(2) in weighted Sobolev spaces on the positive semi-axis and examine thespectra localization of operator-valued functions L(λ) representing symbolof the equation (1).

Denition

Vector-valued function u is called the strong solution of the problem (1),(2), if it belongs to the space W 2

2,γ(R+, A0) for some γ > 0, satises theequation (1) almost everywhere on the semiaxis R+, and also initialconditions (2).

Let us convert the domain Dom(Aβ0 ) of the operator Aβ0 , (β > 0) into the

Hilbert space Hβ , by introducing the norm ‖ · ‖β = ‖Aβ0 · ‖ on the space

Dom(Aβ0 ) which is equivalent the graph norm of the operator Aβ0 .The following theorem present the result on the correct solubility of theproblem (1), (2).

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Here we use the approach similar to the approach of M.S. Agranovich andM.I. Vishik in their famous article Elliptic problems with a parameter and

parabolic problems of general type.// (Russian) Uspehi Mat. Nauk 19, 1964. 3 (117), 53â161.

Theorem (1)

Suppose that f (1)(t) ∈ L2,γ0 (R+, H) for some γ0 > 0 and the conditions

(4), (5) are satised, moreover ϕ0 ∈ H1, ϕ1 ∈ H1/2. Then there exists

such γ1 ≥ γ0 that the problem (1), (2) has the unique solution in the space

W 22,γ (R+, A0) for arbitrary γ > γ1. Moreover the following estimate is valid

‖u‖W 22,γ(R+,A0) ≤ d

(∥∥∥f (1)(t)∥∥∥L2,γ(R+,H)

+ ‖A0ϕ0‖H +∥∥∥A1/2

0 ϕ1

∥∥∥H

)(10)

with a constant d that does not depend on vector-function f and vectors

ϕ0 and ϕ1.

The results about correct solvability of integro-dierential equations of thesecond oder with operator coecients were obtained by L. Pandol, R. Miller,N.D. Kopachevsky, D. Zakora, E. Syomkina.

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Denition

Vector-valued function u is called the weak (generalized) solution of the

problem (1), (2) if it belongs to the space W 12,γ(R+, A

1/20 ) for some γ > 0,

satises the initial condition u(+0) = ϕ0 and also satises the followingidentity

−⟨u′(t), v′(t)

⟩L2,γ

+⟨A

1/20 u(t), A

1/20 v(t)

⟩L2,γ

+ 2γ⟨u′(t), v(t)

⟩L2,γ−

−⟨∫ t

0K(t− s)A−1/2

0 Au(s)ds,A1/20 v(t)

⟩L2,γ

−

−⟨∫ t

0Q(t− s)A−1/2

0 Bu(s)ds,A−1/20 v(t)

⟩L2,γ

−

− 〈f(t), v(t)〉L2,γ− (ϕ1, v(0)) = 0 (11)

for every v(t) ∈W 12,γ(R+, A

1/20 ), satisfying the condition

limt→+∞

v(t)e−γt = 0.

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Theorem (2)

Suppose that f(t) ∈ L2,γ0 (R+, H) for some γ0 > 0 and the conditions

(4), (5) are satised, moreover ϕ0 ∈ H1/2, ϕ1 ∈ H. Then there exists such

γ1 ≥ γ0 that the problem (1), (2) has the unique solution in the space

W 12,γ

(R+, A

1/20

)for arbitrary γ > γ1. Moreover the following estimate is

valid

‖u‖W 1

2,γ

(R+,A

1/20

) ≤ d(‖f(t)‖L2,γ(R+,H) +∥∥∥A1/2

0 ϕ0

∥∥∥H

+ ‖ϕ1‖H)

(12)

with a constant d that does not depend on vector-function f and vectors

ϕ0 and ϕ1.

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Spectral Analysis

We formulate the results about the spectrum localization of operator-function L(λ) when the measures dµ(τ), dη(τ) have compact supports.

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Theorem (3, The main spectral theorem)

Suppose that conditions (4), (5) holds and the supports of measures

dµ(τ), dη(τ) belong to the segment [d1, d2], 0 < d1 < d2 < +∞. Then for

any arbitrary small number θ0 > 0 there exists such number R0 > 0 that

spectrum of operator-function L(λ) belongs to the set

Ω := λ ∈ C : Reλ < 0, |λ| < R0 ∪ λ ∈ C : α1 6 Reλ 6 α2where α1 = α0 − θ0, R0 > max(d2,−α0 + θ0),

α0 = −1

2sup||f ||=1

((K(0)A+Q(0)B) f, f)

((A+B)f, f), f ∈ D(A),

α2 = −1

2inf||f ||=1

((K(0)A+Q(0)B) f, f)((A+B + d2

2I)f, f) , f ∈ D(A).

Moreover, there exists such number γ0 > 0 that for any

λ ∈ λ ∈ C : Reλ < −R0 ∪ λ ∈ C : Reλ > γ0 the following estimate

operator-function L−1(λ) is valid ‖L−1(λ)‖ 6 const|λ||Reλ| .

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Remark

The quantity α0 in the statement of previous theorem can be estimated as

α0 > −1

2

∥∥∥A−1/20 (K(0)A+Q(0)B)A

−1/20

∥∥∥ .Theorem (4)

Let us suppose that the conditions of the previous theorem hold. Then the

nonreal spectrum of the operator-function L(λ) is symmetric with respect

to the real axis and consist of eigenvalues of nite algebraic multiplicity,

moreover for any ε > 0 in the domain

Ωε := Ω\ λ ∈ C : −d2 − ε < Reλ < 0, | Imλ| < ε

eigenvalues is isolated i.e., have no points of accumulation.

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See for more detail the following article and monograph:

1) Vlasov V. V., Rautian N. A., Spectral Analysis of Linear Models of

Viscoelasticity// Journal of Mathematical Science, 2017, V. 230:5, pp. 668672.

2) 1) Vlasov, V. V., Rautian, Well-posed solvability of volterra integro-

dierential equations in Hilbert space// Dierential Equations, 2016, V. 52:9,pp. 1123â-1132;

3) Vlasov V. V., Rautian N. A., Spectral analysis of functional-dierentialequations. M. MAKS Press, 2016, 488 p.

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In our previous works1) V. V. Vlasov, N. A. Rautian Spectral Analysis of Hyperbolic Volterra

Integro-Dierential Equations // Doklady Mathematics, 2015, V.92:2,pp.590593.2) V. V. Vlasov, N. A. Rautian, A. S. Properties of solutions of integro-

dierential equations arising in heat and mass transfer theory // Trans.Moscow Math. Soc., 2014, V.75, P. 185204.3) V. V. Vlasov, N. A. Rautian Spectral Analysis and Representations of

Solutions of Abstract Integro-dierential Equations in Hilbert Space //Operator Theory: Advances and Applications. Springer Basel AG, 2014,V.236, pp. 517535.we considered in detail the case when B = 0. In this case the equation(1) has the abstract form of Gurtin-Pipkin integro-dierential equation thatdescribe heat transfer in materials with memory with nite speed and has anumber of other applications.

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Statement of the problem with fractional-exponential kernel

We consider on the positive semiaxis R+ = (0,∞) the following initialproblem for the integro-dierential equation

d2u(t)

dt2+A2u(t)−

∫ t

0K(t− s)A2u(s)ds = f(t), t ∈ R+, (13)

u(+0) = ϕ0, u(1)(+0) = ϕ1, (14)

where A is a positive self-adjoint operator acting in the separable Hilbertspace H, having the bounded inverse operator.The operator A can be realized asA2y = −y′′(x), where x ∈ (0, π), y(0) = y(π) = 0, orA2y = −∆y with Dirichlet conditions in the bounded domain Q ⊂ Rn withsmooth boundary (H = L2(Q)) orA2y = −∆y − 1/3 · grad(divy), y = ~y(x) ∈ R3 with Dirichlet conditions inthe bounded domain Ω ⊂ R3 with smooth boundary (H = L2(Ω)).

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Kernels of integral operator

The following kernel functions are widely used in the problems of heatpropagation in media with memory and viscoelasticity:

K (t) =

∞∑j=1

cjRj (t) , (15)

where cj > 0, j ∈ N, functions Rj (t) are dened by one of the followingways

I) Rj (t) = e−βjt where βj+1 > βj > 0, j ∈ N, βj → +∞ (j → +∞).

II) Rj(t) = tα−1∞∑n=0

(−βj)ntnα

Γ[(n+ 1)α],

fractional-exponential functions, where 0 < α < 1, Γ(·) gamma-function,0 < βj < βj+1,, j ∈ N, βj → +∞, j → +∞.

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K(t) =

∫ ∞0

e−tτdµ(τ), (16)

where dµ is a positive measure. We identify this measure with its distributionfunction µ, so µ is increasing, continuous from the right, and the integral isinterpreted as a Stieltjes integral.In particular the kernel function (II) has the following representation

Rj(t) =1

2πilim

R→+∞

γ+iR∫γ−iR

eλtdλ

λα + βj=

sinπα

π

+∞∫0

e−tτdτ

τα + 2βj cosπα+ β2j τ−α ,

where λα (0 < α < 1) is the main branch of multivalue function f(λ) = λα,λ ∈ C with a cut on negative real semiaxis that is λα = |λ|αeiα arg λ,−π < arg λ < π.

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The results about correct solvability

Let us denote by Wn2,γ (R+, A

n) the Sobolev space of the vector-valuedfunctions on the positive semiaxis R+ = (0,∞) with the values in the spaceH equiped by the norm

‖u‖Wn2,γ(R+,An) ≡

(∫ ∞0

e−2γt

(∥∥∥u(n)(t)∥∥∥2

H+ ‖Anu(t)‖2H

)dt

)1/2

,

γ ≥ 0.

For more detail description of the space Wn2,γ (R+, A

n) see the monographJ. L. Lions and E. Magenes Nonhomogeneous Boundary-Value Problems andApplications // Springer-Verlag, Berlin-Heidelberg-New York. 1972, chapter1. For n = 0 we have W 0

2,γ

(R+, A

0)≡ L2,γ (R+, H), and for γ = 0 we

shall write Wn2,0 = Wn

2 .

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Denition

Vector-valued function u is called the strong solution of the problem (13),(14) if it belongs to the space W 2

2,γ(R+, A2) for some γ > 0, satises the

equation (13) almost everywhere on the semiaxis R+, and also initialconditions (14).

Denition

Vector-valued function u is called the weak (generalized) solution of theproblem (13), (14) if it belongs to the space W 1

2,γ(R+, A) for some γ > 0,satises the initial condition u(+0) = ϕ0 and also satises the followingidentity

〈A[u(t)−∫ t

0K(t− s)u(s)ds], Av(t)〉L2,γ(R+,H)−

−⟨u′(t), v′(t)

⟩L2,γ(R+,H)

+ 2γ⟨u′(t), v(t)

⟩L2,γ(R+,H)

=

= 〈f(t), v(t)〉L2,γ(R+,H) + (ϕ1, v(0))H

for every v(t) ∈W 12,γ(R+, A), satisfying the condition lim

t→+∞v(t)e−γt = 0.

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Let us convert the domain Dom(Aβ) of the operator Aβ , (β > 0) into theHilbert space Hβ , by introducing the norm ‖ · ‖β = ‖Aβ · ‖ on the spaceDom(Aβ) which is equivalent the graph norm of the operator Aβ .The following theorems present the results on the correct solvability of theproblem (13), (14).Denote by d the constant independent on vector-function f and vectors ϕ0

and ϕ1.

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In the rst case the kernel function K(t) can be represented in the followingform

K(t) =

∞∑j=1

cje−βjt, (17)

where the coecients cj > 0, βj+1 > βj > 0, j ∈ N, βj → +∞ (j → +∞)and the following condition

∞∑j=1

cjβj

< 1. (18)

is satised.Along with the condition (18) we shall use the following condition

K(0) =

∞∑j=1

cj < +∞. (19)

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Theorem (5)

Suppose that Af(t) ∈ L2,γ2 (R+, H) for some γ2 > 0 and the condition

(18) is satised. Then1) if the condition (19) is satised and ϕ0 ∈ H2, ϕ1 ∈ H1, then the

problem (13), (14) has the unique strong solution in the space

W 22,γ

(R+, A

2)for arbitrary γ > γ2. Moreover the following estimate is

valid

‖u‖W 22,γ(R+,A2) ≤ d

(‖Af(t)‖L2,γ(R+,H) +

∥∥A2ϕ0

∥∥H

+ ‖Aϕ1‖H)

; (20)

2) if the condition (19) is not satised and ϕ0 ∈ H3, ϕ1 ∈ H2, then the

problem (13), (14) has the unique strong solution in the space

W 22,γ

(R+, A

2)for arbitrary γ > γ2. Moreover the following estimate is

valid

‖u‖W 22,γ(R+,A2) ≤ d

(‖Af(t)‖L2,γ(R+,H) +

∥∥A3ϕ0

∥∥H

+∥∥A2ϕ1

∥∥H

). (21)

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Now we consider the kernel function K(t) of the following type

K (t) =

∞∑j=1

cjRj (t) , (22)

where cj > 0, j ∈ N,

Rj(t) = tα−1∞∑n=0

(−βj)ntnα

Γ[(n+ 1)α], (23)

where 0 < α < 1, Γ(·) gamma-function, 0 < βj < βj+1,, j ∈ N, βj →+∞, j → +∞.

∞∑j=1

cjβj

< 1, (24)

Along with the condition (24) we shall use the following condition

∞∑j=1

cj < +∞. (25)

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Theorem (6)

Suppose that Af (t) ∈ L2,γ0 (R+, H) for some γ0 > 0, the kernel K (t) hasthe form (22), (23) with constant α (0 < α < 1), and the condition (24) issatised. Moreover ϕ0 ∈ H3, ϕ1 ∈ H2. Then there exists γ1 > γ0 that the

problem (13), (14) has the unique strong solution in the space

W 22,γ

(R+, A

2)for arbitrary γ > γ1 and the following estimate is valid

‖u‖W 22,γ(R+,A2) 6 d

(‖Af‖L2,γ(R+,H) +

∥∥A3ϕ0

∥∥H

+∥∥A2ϕ1

∥∥H

).

Theorem (7)

Suppose that f (t) ∈ L2,γ0 (R+, H) for some γ0 > 0, the kernel K (t) hasthe form (22), (23) with constant α (0 < α < 1), and the condition (24) issatised. Moreover ϕ0 ∈ H2, ϕ1 ∈ H. Then there exists γ1 > γ0 that the

problem (13), (14) has the unique weak solution in the space

W 12,γ (R+, A) for arbitrary γ > γ1 and the following estimate is valid

‖u‖W 12,γ(R+,A) 6 d

(‖f‖L2,γ(R+,H) +

∥∥A2ϕ0

∥∥H

+ ‖Aϕ1‖H).

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Spectral Analysis

We shall suppose in what follows that operator A have a compact inverse.Let us denote by ej∞j=1 the orthonormal basis consisting of eigenvectorsof operator A corresponding to the eigenvalues aj : Aej = ajej , j ∈ N.The eigenvalues aj are numerated in increasing order 0 < a1 < a2 < ...;an → +∞ for n→ +∞.

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Applying the Laplace transform to the equation (13) with zero initialconditions we obtain the following operator-valued function

L (λ) = λ2I +A2 − K(λ)A2,

which are the symbol (analogue of the characteristic quasi-polynomial) ofthe equation (13) where K(λ) the Laplace transform of the function K(t),I is identity operator acting in the space H.

Denition

The set of values λ ∈ C is called the resolvent set R(L) of operator-valuedfunction L(λ) if there exists L−1(λ) is bounded for any λ ∈ R(L). The setσ(L) = λ ∈ C\R(L) |L(λ) exists is called the spectra of operator-valuedfunction L(λ).

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Let us consider the projection

ln (λ) = (L (λ) en, en) = λ2 + a2n

(1− K(λ)

).

of the operator-valued function L(λ) on the one-dimensional subspaceformed by the vector en, where Aen = anen, n ∈ N, an → +∞ forn → +∞. Thus we obtain the countable set of the meromorphic functionsln(λ), n ∈ N. Then the spectrum of the operator-valued function L(λ) isthe closure of the zeroes set of the functions ln(λ)∞n=1.

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We suppose the kernel function K(t) has the representation (17). Then

K(λ) =

∞∑j=1

cjλ+ βj

the Laplace transform of the function K(t).

Theorem (8)

Let us suppose that the conditions (18) and (19) are satised. Then the

complex eigenvalues λ±n , λ+n = λ−n of the vector valued function L(λ)

asymptotically represented in the form

λ±n = ±i(an +O

(1

an

))− 1

2K(0) +O

(1

a2n

), an → +∞. (26)

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Theorem (9)

Let us suppose the conditions (18) and (19). Then the spectrum of the

operator valued function L(λ) can be represented as follows

σ(L) := σR ∪ σI where σR and σI are the real and nonreal part of

spectrum of operator valued function L(λ), correspondently. Moreover the

real part of spectrum of the operator valued function L(λ) is the closure ofthe real zeroes set λk,n|k ∈ N, n ∈ N of the functions ln(λ)∞n=1 that

satisfy the inequalities

...− βk+1 < xk+1 < λk+1,n < −βk < ... < −β1 < λ1,n, k ∈ N, (27)

where xk are the real zeroes of the function K(λ), andλk,n = xk +O

(1/a2

n

).

σI =λ±n ∈ C\R, λ−n = λ+

n |n ∈ N,

where λ±n - nonreal eigenvalues of operator valued function L(λ) has therepresentation (26).

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Now we suppose the kernel function K(t) has the representation (22), (23).Then the Laplace transform of the function K(t) is follows

K(λ) =

∞∑j=1

cjλα + βj

, 0 < α < 1.

Here λα (0 < α < 1) is the main branch of multivalue function f(λ) = λα,λ ∈ C with a cut on negative real semiaxis that is λα = |λ|αeiα arg λ,−π < arg λ < π.

Theorem (10)

Suppose the condition (24) is satised. Then the spectra of

operator-valued function L(λ) lies on the left complex halfplane.

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Theorem (11)

Let us suppose the conditions (24) and cj = 0 for arbitrary j more thansome N ∈ N are satised. Then the spectrum of the operator valued

function L(λ) can be represented as follows

σ(L) :=λ±n ∈ C\R, λ−n = λ+

n |n ∈ N,

where the complex eigenvalues λ±n , λn+ = λ−n of the vector valued function

L(λ) asymptotically represented in the form

λ±n = − sin(πα

2

)a1−αn

Q

2± ian

(1− cos

(πα2

)a−αn

Q

2

)+ o

(a1−αn

),

n→ +∞, (28)

where Q =N∑j=1

cj .

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Remark

For α = 1 the asymptotic formula (28) comes to asymptotic formula (26)where the kernel function K(t) has the representation (17).

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See for more detail the following articles:

1) V. V. Vlasov, N. A. Rautian Well-Posedness and Spectral Analysis

of Volterra Integro-Dierential Equations with Singular Kernels// DokladyMathematics, 2018, V.98:2, pp.502505.

2) Vlasov, V. V., Rautian, Research of operator models arising in

viscoelasticity theory. (Russian)// Sovrem. Mat. Fundam. Napravl., 2018,V. 64:1, pp. 6073.

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Representation of the solutions

On the base of the spectral theorems we obtain the representation of thesolution of the problem (13), (14).

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Let us suppose the kernel function K(t) has the representation (17).

Theorem (12)

Let us suppose that f(t) = 0 for t ∈ R+, vector-function

u(t) ∈W 22,γ

(R+, A

2), γ > 0 is a strong solution of the problem (13), (14)

and the condition (18) is satised. Then, for arbitrary t ∈ R+ the solution

u(t) of the problem (13), (14) is represented in the following series

u(t) =

∞∑n=1

(ωn(t, λ+

n ) + ωn(t, λ−n ) +

∞∑k=1

ωn(t, λkn)

)en, (29)

that is convergent by the norm of the space H, where

ωn(t, λ) =(ϕ1n + λϕ0n) eλt

l(1)n (λ)

.

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Theorem (13)

Let us suppose vector-function f(t) ∈ C ([0, T ], H) for arbitrary T > 0,vector-function u(t) ∈W 2

2,γ

(R+, A

2), γ > 0 is a strong solution of the

problem (13), (14) and the conditions (18), ϕ0 = ϕ1 = 0 are satised.

Then, for arbitrary t ∈ R+ the solution u(t) of the problem (13), (14) isrepresented in the following series

u(t) =

∞∑n=1

(ωn(t, λ+

n ) + ωn(t, λ−n ) +

∞∑k=1

ωn(t, λkn)

)en, (30)

that is convergent by the norm of the space H, where

ωn(t, λ) =

t∫0

fn(τ)eλ(t−τ)dτ

l(1)n (λ)

.

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Let us suppose the kernel function K(t) has the representation (22).Denote by

Kn(τ) =a2n

(K− (−τ)− K+ (−τ)

)(τ2 + a2

n

(1− K+ (−τ)

))(τ2 + a2

n

(1− K− (−τ)

)) ,K± (−τ) =

N∑k=1

ckταe±iπα + βk

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Theorem (14)

Let us suppose that conditions of the theorem 11 are satised,

α ∈ (0, 1/2), f(t) ≡ 0, ϕ0 ∈ H3, ϕ1 ∈ H2. Then the strong solution of the

problem (13), (14) is represented in the following sum

u(t) = uI(t) + uR(t), t > 0,

uI(t) =

∞∑n=1

(ωn(t, λ+

n ) + ωn(t, λ−n ))en, ωn(t, λ) =

(ϕ1n + λϕ0n) eλt

l(1)n (λ)

,

(31)

uR(t) =

∞∑n=1

wn (t)en, wn (t) =

∫ ∞0

e−tτKn(τ) (−τϕ0n + ϕ1n)dτ,

(32)The series are convergent by the norm of the space H and λ±n are nonreal

eigenvalues of operator valued function L(λ), ϕkn = (ϕk, en), n ∈ N,k = 1, 2.

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Theorem (15)

Let us suppose that conditions of the theorem 11 are satised, α ∈(0, 1

2

),

ϕ0 = ϕ1 ≡ 0. Then the strong solution of the problem (13), (14) isrepresented in the following sum

u(t) = wI(t) + wR(t), t > 0,

wI(t) =

∞∑n=1

t∫0

[exp(λ+

n (t− τ))

l(1)n (λ+

n )+

exp(λ−n (t− τ))

l(1)n (λ−n )

]fn(τ)dτ

en,

wR(t) =

∞∑n=1

wn (t)en, wn (t) =

t∫0

∞∫0

exp(−p(t− τ))Kn(p)dp

fn(τ)dτ

The series are convergent by the norm of the space H, λ±n - are nonreal

eigenvalues of operator valued function L(λ), fn = (f, en).

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Denote by Pn the orthoprojector on the subspace, which is the linear coverof the eigenvectors ejnj=1, and denote by Qn the orthoprojector on thesubspace which is orthogonal to the subspace PnH. That is Qn = I − Pnand the space H is represented as the following sum H = PnH ⊕QnH.

Theorem (16)

Let us suppose that conditions of the theorem 14 are satised. Then for

any ε > 0 there exists n0 ∈ N and δ > 0, so that for vector-function uI (t),dened by (31), the following estimates are valid

‖Qn0uI (t)‖ 6 θ1

∥∥∥Qn0e−kA1−αtA2ϕ0

∥∥∥+ θ2

∥∥∥Qn0e−kA1−αtAϕ1

∥∥∥ , t > 0,

(33)

0 < k <1

2

(πα2

) N∑j=1

cj − ε,

‖Pn0uI (t)‖ 6 θ3e−δt ‖Pn0ϕ0‖+

∥∥Pn0A−1ϕ1

∥∥ , t > 0, (34)

with some positive constants δ, θ1, θ2, θ3 independent of the vectors ϕ0,ϕ1.

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Corollary

Let us suppose that vector-function uI(t) is dened by the formula (31),where λ±n for any suciently large n ∈ N, has the asymptotic (28), vectorsϕ0 ∈ Hp, ϕ1 ∈ Hp−1, p ∈ N. Then for any ε > 0 there exists such n0 ∈ N,that the following estimates are valid

‖ApQn0uI (t)‖ 6 θ4

∥∥∥Qn0e−kA1−αtApϕ0

∥∥∥+ θ5

∥∥∥Qn0e−kA1−αtAp−1ϕ1

∥∥∥ ,t > 0, (35)

0 < k =1

2sin(πα

2

) N∑j=1

cj − ε,

‖ApPn0uI (t)‖ 6 θ6e−δt ‖Pn0A

pϕ0‖+∥∥Pn0A

p−1ϕ1

∥∥ , t > 0, (36)

with some positive constants δ, θ4, θ5, θ6, independent on the vectors ϕ0,ϕ1.

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Theorem (17)

Let us suppose that conditions of the theorem 14 are satised. Then for

any ε > 0 vector-function uR (t), dened by the formula (32), has thefollowing estimate

‖uR (t)‖2 6 e−2εtk1

∥∥A−αϕ0

∥∥2+ k2

∥∥A−1−αϕ1

∥∥2

+

+ k3

ε2(2+α)

∥∥A−2ϕ0

∥∥2+ ε2(1+α)

∥∥A−2ϕ1

∥∥2, t > 0, (37)

with positive constants k1, k2, k3, independent on vectors ϕ0, ϕ1.

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Theorem (18)

Let us suppose that conditions of the theorem 11 are satised and

Af (t) ∈ L2,γ0 (R+, H) for some γ0 > 0. Then for any ε > 0 there exists

n0 ∈ N, that for solution of the problem (13), (14) the following estimate

is valid

‖Amu (t)‖ 6 d1t

∫ t

0

∥∥∥Qn0e−kA1−α(t−τ)Am−1f(τ)

∥∥∥2dτ+

+ d2t

∫ t

0e−2δ(t−τ)

∥∥Pn0Am−1f(τ)

∥∥2dτ+

+t

k1

∫ t

0e−2ε(t−τ)

∥∥∥Am−(1+α)f(τ)∥∥∥2dτ + k2ε

2(α+1)

∫ t

0

∥∥Am−2f(τ)∥∥2dτ

, t > 0,

(38)

where m = 0, 1, 2, positive constants d1, d2, k1, k2 independent on

vector-function f(t).

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Thank you very much for your attention.

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