Correct solvability of Volterra integrodifferential equations in Hilbert space

Correct solvability of Volterra integrodifferential equations in Hilbert space

Romeo Perez Ortiz

and Victor V. Vlasov

Lomonosov Moscow State University, Vorobievi Gori, Moscow, 119991, Russia Received 14 January 2016, appeared 1 June 2016

Communicated by Dimitri Mugnai

Abstract. Correct solvability of abstract integrodifferential equations of the Gurtin–

Pipkin type is studied. These equations represent abstract wave equations perturbed by terms that include Volterra integral operators.

Keywords: Volterra integral operators, integrodifferential equations, Sobolev space, Gurtin–Pipkin heat equation.

2010 Mathematics Subject Classification: 34D20, 47G20, 45D05, 35R09, 45K05.

1 Introduction

The paper is concerned with integrodifferential equations with unbounded operator coeffi- cients in a Hilbert space. The main part (d²u/dt²+A²u) of the equation under consideration is an abstract hyperbolic-type equation disturbed by terms involving Volterra operators. These equations can be looked upon as an abstract form of the Gurtin–Pipkin equation describing thermal phenomena and heat transfer in materials with memory or wave propagation in vis- coelastic media. A complete analysis and abundant examples of such equations in Banach and Hilbert spaces can be found in [1–3,8–11,23].

Consider the following class of second-order abstract models d²u

dt² +A²u+ku−

Z _t

0 K(t−s)A^2θu(s)ds= f(t), t∈_R+, (1.1) u(+0) = ϕ0, u⁽¹⁾(+0) = ϕ₁, (1.2) where A is a positive self-adjoint operator with domain dom(A) ⊂ H, H is a Hilbert space, and ϕ₀, ϕ₁, f(t)will be described later. The variable θ is a real number in [_{0, 1}]_, k is a non- negative constant and K is the kernel associated with the equation (1.1) (the Gurtin–Pipkin equation). This type of equations appear in various branches of mechanics and physics, for instance, in heat transfer with finite propagation speed [4], theory of viscoelastic media [2], kinetic theory of gases [5], and thermal systems with memory [24].

BEmail: cemees.romeo@gmail.com

*Email: vlasovvv@mech.math.msu.su

In heat theory and theory of viscoelastic media, the kernel K is determined empirically.

Properties of heat conduction with memory were studied, for example, in [15] with a smooth function as a kernel. In [6], the solutions of control problems with compactly supported boundary control and distributed control were studied.

In theory of viscoelasticity, a kernel K is also determined empirically. The curves thus obtained are often approximated by a finite sum of exponentials in the form:

K(t)≈

∑

c_kexp(−γ_kt).

Equations with structure and properties that are similar to the Gurtin–Pipkin equations also appear in the kinetic theory of gases. In this theory the equations of a solid medium are derived from the laws of pairwise interactions of molecules. A series of momentum equations can be derived from the Boltzmann equation. Here, the momenta represented by coordi- nates and velocities with respect to velocity variables with certain weights are average of the distribution function for gas molecules in thermal equilibrium. In particular, the ordinary components of the Navier–Stokes equations represented by velocity, pressure, and density can be represented as momenta in a series of momentum equations.

Phenomena like isotropic materials and ionized atmosphere can also be modeled by the Gurtin–Pipkin equation. For example, system (1.1)–(1.2) represents an isotropic viscoelastic model if θ = 1/2, k = 0 and A²u = −µ∆u−(λ+µ)∇(divu), where µ and λ are theLame coefficients. Similarly, system (1.1)–(1.2) represents a model of ionized atmosphere if θ = 0, k > 0 and A² = −_∆ (for more details, see [13,14]). Since the operators A² and A are both positive self-adjoint operators, the operatorA² is used instead ofA.

We want to make it clear that we do not study stability of solutions of abstract inte- grodifferential equation (1.1). We consider important to mention some results obtained in [3,12–14], because in the papers [16,17,19] were obtained some results associated with asymp- totic behavior of solutions for system (1.1)–(1.2) when θ ∈ [0, 1], k = 0. Results related to asymptotic behavior of solutions for systems with memory, for differentθ ∈ [0, 1], were ex- tensively studied in recent years (see [3,12–14] and the references given therein). In [12], for instance, Muñoz Rivera and co-authors showed that the solutions for system (1.1)–(1.2) with θ ∈ [0, 1/2)andk = 0 decay polynomially ast →+∞, even if the kernel K decays exponen- tially. Fabrizio and Lazzari in [3], assuming the exponential decay of kernel K, θ = 1/2 and k=0, proved the exponential decay of the solutions for system (1.1)–(1.2). In [13], for the case k > 0 and θ = 0, Muñoz Rivera and co-authors showed that for the ionized atmosphere the dissipation produced by the conductivity kernel alone is not enough to produce an exponen- tial decay of the solution of an integrodifferential equation. In [14], for the case k = 0 and θ = 1/2, Muñoz Rivera and Maria Naso proved that the solution of model (1.1)–(1.2) decays exponentially to zero if so does the kernelK.

Vlasov and co-authors [18–20] also studied model (1.1)–(1.2) for the case θ = 1 and k=0. They established the correct solvability of initial boundary value problems in weighted Sobolev space on the positive semi-axis and examined spectral properties of the operator- valued function L(λ) = λ²I+k+A²−Kb(λ)A^2θ, where Kb(λ) is the Laplace transform of K(t). The generalization of aforementioned results was obtained in the recent paper [21]. On the base of spectral analysis of operator function L(λ) Vlasov and Rautian [22] obtained the representation of solutions of model (1.1)–(1.2).

In the present work we obtain the correct solvability of system (1.1)–(1.2) in the casesk =0 andθ ∈ [_{0, 1}]. The correct solvability of initial boundary problems for the specified equations

is established in weighted Sobolev spaces on a positive semi-axis. Spectral properties of the operator-valued function L(λ)corresponding to system (1.2) for the casesθ ∈[0, 1]andk=0 were obtained in [16,17].

The paper is divided into four sections. Section 1 gives a brief introduction to the subject matter and describes applications of the Gurtin–Pipkin equation. The main results on the correct solvability for the caseθ ∈ [0, 1]andk =0 are formulated in the Section 2. The proof of these results are given in Section 3. Section 4 contains a comment concerning the spectra of operator-valued function L(λ).

Throughout the paper, the expressiona. ^bwill mean that a≤ Cb, C> 0, anda ≈ bwill be used to write that a.^b.^a.

2 Correct solvability

Let H be a separable Hilbert space and let A be a self-adjoint positive operator in H with compact inverse. We associate the domain dom(A^β)of the operator A^β,β>0, with a Hilbert space H_β by introducing on dom(A^β) the norm k · k_β = kA^β · k, which is equivalent to the graph norm of the operator A^β. We denote by{e_n}^∞_n=1 the orthonormal basis formed by the eigenvectors of A corresponding to its eigenvalues an such that Aen = anen, n ∈ _{N. The} eigenvalues a_nare arranged in increasing order and counted according to multiplicity; that is, 0< a₁ ≤a₂≤ · · · ≤a_n≤ · · ·_{, where}a_n→_∞asn→+_∞.

We denote byW_2,γⁿ (_R+,Aⁿ)the Sobolev space consisting of vector-functions on the semi- axisR+= (0,∞)with values in H; this space will be equipped with the norm

kuk_Wn

2,γ(_R+,Aⁿ)≡ _{Z ∞}

0 e^−2γt

ku⁽ⁿ⁾(t)k²_H+kAⁿu(t)k²_Hdt 1/2

, γ≥0.

For a complete description of the spaceW_2,γⁿ (_R+,Aⁿ) and some of its properties we refer to the monograph [7, Chapter I]. Forγ =0 we writeW_2,0ⁿ (_R+,Aⁿ)≡W₂ⁿ(_R+,Aⁿ). This space is endowed with the norm

kuk_Wn

2(_R+,Aⁿ)≡ Z _∞

0

ku⁽ⁿ⁾(t)k²_H+kAⁿu(t)k²_Hdt 1/2

.

For n = 0, setW_2,γ⁰ (_R+,A⁰) ≡ L2,γ(_R+,H), where L2,γ(_R+,H)denotes the space of measur- able vector functions with values in H, equipped with the norm

kfk_L2,γ(R+,H)≡ _{Z ∞}

0 e^−2γtkf(t)k²_Hdt 1/2

, γ≥0.

Let us consider the following system on the semi-axisR+ = (_0,∞)_: d²u

dt² +A²u−

Z _t

0 K(t−s)A^2θu(s)ds= f(t), θ ∈[0, 1], (2.1) u(+0) = ϕ₀, u⁽¹⁾(+0) = ϕ₁. (2.2) It is assumed that the vector-valued functionA^2−θf(t)belongs toL_2,ρ0(_R+,H)for someρ₀≥0, and the scalar function K(t) admits the representation K(t) = _∑^∞_j=1c_je^−γjt, where c_j > 0, γ_j+1 >γ_j >0,j∈_N,γ_j →+_∞(j→+_∞). Moreover, we assume that

a)

∑

c_j

γ_j <1, b)

∑

c_j <+_∞.

Note that ifa)holds, thenK∈ L₁(_R+)andkKk_L1(R+)<1. Ifa)andb)are both satisfied, then the kernelKbelongs to Sobolev spaceW₁¹(_R+).

Definition 2.1. A vector-valued functionuis called astrong solutionof system (2.1)–(2.2) if, for someγ≥0,u∈W_2,γ² (_R+,A²)satisfies the equation (2.1) almost everywhere on the semi-axis R+andualso satisfies the initial condition (2.2).

In [19, Theorem 1] it was shown the existence of a strong solution uand system (2.1)–(2.2) forθ = 1 was proved to be correctly solvable. In the present paper we establish the correct solvability of system (2.1)–(2.2) for θ ∈ [_{0, 1}]. The result obtained here is more general than that of [19, Theorem 1], although both results coincide forθ =1.

Theorem 2.2. Suppose that, for allθ ∈[0, 1]and for someρ₀≥0, A^2−θf(t)belongs to L2,ρ0(_R+,H). 1) If conditions a) and b) both hold, and ϕ₀ ∈ H₂, ϕ₁ ∈ H₁ for all θ ∈ [0, 1] then there is a ρe > ρ0 such that for any γ > ρ, system_e (2.1)–(2.2) has a unique solution in the Sobolev space W_2,γ² (_R+,A²)and this solution satisfies the estimate

kuk_W2

2,γ(_R+,A²)≤d

kA^2−θfk²_L

2,γ(_R+,H)+kA²ϕ₀k_H+kAϕ₁k_H, (2.3) where the constant d is independent of the vector-valued function f and the vectors ϕ₀,ϕ₁. 2) If condition a)is satisfied, but condition b)does not hold(i.e., K(t)∈/W₁¹(_R+))and ϕ₀ ∈ H₂+θ,

ϕ₁∈ H₁+θ for allθ ∈ (0, 1]then there is aeρ>ρ₀such that for anyγ> _eρsystem(2.1)–(2.2)has a unique solution in Sobolev space W_2,γ² (_R+,A²)and this solution satisfies the estimate

kuk_W2

2,γ(_R+,A²) ≤d

kA^2−θfk²_L

2,γ(_R+,H)+kA^2+θϕ₀k_H+kA^1+θϕ₁k_H, (2.4) where the constant d is independent of the vector-valued function f and the vectors ϕ₀,ϕ₁. From Theorem 2.2 we obtain the correct solvability of system (2.1)–(2.2) in the Sobolev space W₂²((0,T),A²), for every T > 0, where the space W₂²((0,T),A²) is equipped with the norm

kuk_W2

2((0,T),A²) ≡ Z _T

0

ku⁽²⁾(t)k²_H+kA²u(t)k²_Hdt 1/2

.

Corollary 2.3. Suppose that, for allθ∈ [0, 1]and for someρ₀≥0, A^2−θf(t)belongs to L_2,ρ0(_R+,H). 1) If condition1) of Theorem2.2is satisfied, then for an arbitrary T >0the following estimate of the

solution u is valid kuk_W2

2((0,T),A²)≤ de

kA^2−θfk²_L

2,γ(_R+,H)+kA²ϕ₀k_H+kAϕ₁k_H, (2.5) where the constantd is independent of the vector-valued function f and the vectorse ϕ₀,ϕ₁. 2) If condition2) of Theorem2.2is satisfied, then for an arbitrary T >0the following estimate of the

solution u is valid kuk_W2

2((0,T),A²)≤d^e

kA^2−θfk²_L

2,γ(_R+,H)+kA^2+θϕ₀k_H+kA^1+θϕ₁k_H, (2.6) where the constantd is independent of the vector-valued function f and the vectorse ϕ₀,ϕ₁.

3 Proof of Theorem 2.2

In the case of homogeneous initial conditions ϕ₀ = ϕ₁ = 0 we need to establish the correct solvability of the Cauchy problem for hyperbolic equations through the Laplace transform.

We give here some well-known facts that will be used later.

Definition 3.1. The Hardy space H2(Reλ > γ,H) is the class of holomorphic (or analytic) functions ˆf(λ) on the right half-plane {λ ∈ _C : Reλ > γ ≥ 0} values in H. The space H₂(Reλ>γ,H)is endowed with the norm

kf^ˆk_{H2(Reλ>γ,H)}= sup

Reλ>γ

Z +_∞

−_∞ kf^ˆ(x+iy)k²_H dy

!1/2

<+_∞, (λ=x+iy).

Let us formulate a well-known Paley–Wiener theorem about the Hardy space H2(Reλ>γ,H).

Theorem 3.2(Paley–Wiener).

1. The space H₂(Reλ>γ,H)coincides with the set of vector-valued functions (Laplace transforms), which admit the representation

fˆ(λ) = √¹ 2π

Z +_∞

0 e^−λtf(t)dt, (3.1)

where f(t)∈L_2,γ(_R+,H),λ∈_C,Reλ>γ≥0.

2. For any fˆ(λ) ∈ H₂(Reλ > γ,H) there is exactly one representation of the form (3.1), where f(t)∈ L_2,γ(_R+,H). Moreover, the following inversion formula holds:

f(t) = √¹ 2π

Z _+∞

−_∞

fˆ(γ+iy)e^(γ+iy)tdy, t∈_R+, γ≥0.

3. For fˆ(λ)∈ H₂(Reλ>γ,H)and f(t)∈ L_2,γ(_R+,H)as in(3.1), the following relation holds:

kf^ˆ(λ)k²_H

2(Reλ>γ,H) ≡ sup

Reλ>γ

Z +_∞

−_∞ kf^ˆ(x+iy)k²_Hdy

=

Z +_∞

0 e^−2γtkf(t)k²_Hdt≡ kf(t)k²_L

2,γ(_R+,H).

Proof. We begin with the proof of Theorem2.2 in the case of homogeneous initial conditions ϕ0 = ϕ₁=0. We note that the Laplace transform ˆu(λ)of any strong solution of equation (2.1) with the initial condition (2.2) has the form

ˆ

u(λ) =L⁻¹(λ)f^ˆ(λ), (3.2) where the operator-valued function L(λ)is the symbol of equation (2.1), which can be repre- sented as

L(λ) =λ²I+A²−Kb(λ)A^2θ :=λ²I+A²−

∑

k=1

c_k λ+γ_k

!

A^2θ, 0≤θ ≤_{1. (3.3)} Here the operator I is the identity operator in the Hilbert space H.

If we can prove that the vector-valued function of equation (3.2) is such that A²uˆ(λ)and λ²uˆ(λ)both belong to the Hardy space H2(Reλ > γ,H) for some γ > ρ₀ ≥ 0, then by the Paley–Wiener theorem we will be able to prove that A²u(t) and d²u(t)/dt² both belong to L_2,γ(_R+,H), which would imply that u(t) ∈ W_2,γ² (_R+,A²). Then, the solvability of system (2.1)–(2.2) in the Sobolev spaceW_2,γ² (_R+,A²)will be established.

With that idea in our mind, let us consider the projection ˆun(λ)of the vector-valued func- tion ˆu(λ)on the one-dimensional subspace spanned by the vectore_n:

ˆ

u_n(λ) =`⁻_n¹(λ)f^ˆ_n(λ)_{, (3.4)} where ˆf_n(λ) = (f^ˆ(λ)_,e_n)_and

`_n(λ):= (L(λ)e_n,e_n) =λ²+a²_n−a^2θ_n

∑

c_k λ+γ_k

! .

The restriction ofA²uˆ(λ)to the one-dimensional space spanned bye_nhas the form A²uˆ(λ),en

= ^a

θngˆn(λ)

`_n(λ) ^{, 0}≤θ ≤1, (3.5)

where ˆgn(λ)is the nth coordinate of the vector function ˆg(λ) = A^2−θfˆ(λ). According to the hypotheses of Theorem2.2, the vector-valued function g(t) = A^2−θf(t)belongs to the space L_2,ρ0(_R+,H). Consequently, the Laplace transform ˆg(λ)of the functiong(t)belongs to Hardy spaceH₂(Reλ>ρ₀,H).

In order to prove that A²uˆ(λ) belongs to H₂(Reλ > γ,H), it is enough to establish the estimate

sup

Reλ>γ n∈_N

a^θ_n

`_n(λ)

≤const, for allθ ∈[0, 1], (3.6)

which is uniform with respect toλ(_Reλ>γ)_andn∈_N.

For that purpose, we consider the functionm_n(λ) = ^`n(λ)

a²_n . We estimate this function from below by means of its real and imaginary parts:

Rem_n(λ) = ^x

2−y²

a²_n +1− ¹ a²_n^(1−θ)

∑

c_k(x+γ_k) (x+γ_k)²+y²

!

, λ= x+iy, Imm_n(λ) = ^2xy

a²_n + ^y a²_n^(1−θ)

∑

c_k (x+γ_k)²+y²

! .

First, we seek a lower bound for|Imm_n(λ)|with|y|> x, where x>γ≥ρ₁≥0:

|Imm_n(λ)|> ^2x|y|

a²_n + ¹

|y|a²_n^(1−θ)







∑

c_k

1+ ^γ_|^k

y|

2

+1





 > ^2γy

2+k0(γ)a^2θ_n

|y|a²_n , wherek₀(γ) = ^c1

(1+^γ_γ¹)²⁺1. Hence, for|y|>x withx> γ≥ρ₁ ≥_{0 we have} 1

|`_n(λ)| ≤ ¹

a²_n|Imm_n(λ)| < |y|

2γy²+k₀(γ)a^2θ_n < ¹ a^θ_np

2γ·k0(γ)^{. (3.7)}

Second, we estimate|Rem_n(λ)|from below with|y|< x, wherex>γ>ρ₁≥0. For fixed eit is possible to find aN= N(e)such that

∑

c_k x+γ_k <

∑

c_k γ_k < ^e

2, x>0. (3.8)

In turn for the finite sum ∑^Nk=1 c_k

x+γ_k it is possible to find aρ^∗ >0 such that forx >ρ^∗ >_0,

∑

c_k

x+γ_k < ^e

2. (3.9)

Hence from (3.8) and (3.9) we obtain the following inequality

∑

c_k

x+γ_k < e, x>ρ^∗>0. (3.10) We know that the sequence {a_n}^∞_n=1 is such that a_n → _∞ when n → _∞and the eigenvalues an are arranged in increasing order. Hence we can choose a sufficiently smalle> 0 such that e< a²₁^(1−θ)/2. Consequently for x>ρ^∗ >0 the following inequality holds

1 a²_n^(1−θ)

∑

c_k(x+γ_k)

(x+γ_k)²+y² < ^∑

∞k=1 ck

x+γk

a²_n^(1−θ)

< ¹

a²₁^(1−θ)

∑

c_k

x+γ_k < ¹

2. (3.11)

It follows that

|Rem_n(λ)|>

1− ¹ a²_n^(1−θ)

∑

c_k x+γ_k

>

1− ¹ a²₁^(1−θ)

∑

c_k x+γ_k

> ¹ 2. Therefore, for|y|<x, withx> _eρ=max(ρ₀,ρ₁,ρ^∗)>0, we have

1

|`_n(λ)| ≤ ¹

a²_n|Rem_n(λ)| < ¹ a²_n

1− ¹

a²₁^(1−θ)∑^∞k=1 c_k x+γ_k

< ²

a²_n. (3.12) From estimates (3.7) and (3.12) we obtain

a^θ_n

`_n(λ)

< _p ¹

2γ·k₀(γ)^, |y|>x >γ>ρ₁≥0,

a^θ_n

`_n(λ)

< ²

a²_n^−θ < ²

a²₁^−θ, |y|<x, x>ρ_e>0.

Therefore, forγ>max{ρ₁,ρe}we have sup

Reλ>γ n∈_N

a^θ_n

`_n(λ)

< ²

minp

2γ·k0(γ),a²₁^−θ

, for allθ ∈[0, 1]. (3.13)

Remark 3.3. Estimate (3.13) implies that sup

Reλ>γ

A^θL⁻¹(λ)≤const . (3.14)

The Hardy space H₂(Reλ > γ,H)is invariant with respect to multiplication of functions of the form _`^aθn

n(λ), since they are analytic and bounded in view of (3.13). Consequently, ˆg(λ) = A^2−θfˆ(λ)∈ H₂(Reλ> γ,H)implies that A²uˆ(λ)belongs to H₂(Reλ>γ,H).

Let us establish the estimate for the norm of the vector-valued function A²u(t) ∈ L_2,γ(_R+,H). From (3.2) it follows that

A²uˆ(λ) =A²L⁻¹(λ)f^ˆ(λ) =A^θL⁻¹(λ)A^2−θfˆ(λ). (3.15) This function can be represented in the form

A²uˆ(λ) =

∑

a^θ_k

`_k(λ)·a²_k^−θfˆ_k(λ)e_k.

According to the hypothesis of Theorem2.2, the vector-valued function A^2−θf(t)belongs to L_2,ρ0(_R+,H). Therefore, by the Paley–Wiener theorem,A^2−θfˆ(λ)∈ H₂(Reλ>ρ₀,H)and

kA^2−θfk_L2,ρ

0(_R+,H) =kA^2−θfˆk_{H2(Reλ>ρ0,H)}.

By (3.13) and Paley–Wiener theorem, the following relations hold forγ>_eρ:

kA²u(t)k²_L

2,γ(_R+,H) =kA²uˆ(λ)k²_H

2(Reλ>γ,H)=kA^θL⁻¹(λ)A^2−θfˆ(λ)k²_H

2(Reλ>γ,H). But

kA^θL⁻¹A^2−θfˆk²_H

2(Reλ>γ,H) = sup

Reλ>γ

Z _+∞

−_∞





∑

a^θ_k

`_k(λ)·a²_k^−θfˆ_k(λ)

2

dy

≤ sup

Reλ>γ k∈_N

a^θ_k

`_k(λ)

2

· sup

Reλ>γ +_∞ Z

−_∞

∑

a²_k^−θfˆ_k(λ)²

!

dy (3.16)

≤ sup

Reλ>γ k∈_N

a^θ_k

`_k(λ)

2

kA^2−θfˆ(λ)k²_H

2(Reλ>γ,H)

≤d²₁kA^2−θfk²_L

2,γ(_R+,H), where

d₁ = sup

Reλ>γ k∈_N

a^θ_k

`_k(λ) .

This shows thatA²u(t)belongs to L_2,γ(_R+,H), forγ>ρ, and the following inequality_e kA²uk_L2,γ(R+,H)≤d₁

kA^2−θfk_L2,γ(R+,H) (3.17) holds.

Now let us prove thatλ²uˆ also belongs toH₂(_Reλ>_γ,H)_{. We set} Kb(λ) =

∑

c_k λ+γ_k.

Note that for Reλ>γwe can write the identity operator I as I =λ²L⁻¹(λ) +I−Kb(λ)A^−2(1−θ)

A²L⁻¹(λ)_. Hence, for Reλ> γ, we obtain

fˆ(λ) =λ²L⁻¹(λ)f^ˆ(λ) +I−Kb(λ)A^−2(1−θ)

A²L⁻¹(λ)f^ˆ(λ)

= λ²uˆ(λ) +I−Kb(λ)A^−2(1−θ)

A^θL⁻¹(λ)A^2−θfˆ(λ). (3.18) Here, A^2−θfˆ(λ)∈ H₂(Reλ>γ,H)and sup_Reλ>γ

A^θL⁻¹(λ)≤ const. From (3.18) it follows that

λ²uˆ(λ) = f^ˆ(λ)−I−Kb(λ)A^−2(1−θ)

A^θL⁻¹(λ)A^2−θfˆ(λ). (3.19) Therefore, the functionλ²uˆ_n(λ)can be represented in the form

λ²uˆ_n(λ) = f^ˆ_n(λ)− 1− ^Kb(λ) a²_n^(1−θ)

! a^θ_n

`_n(λ)^gˆn(λ). (3.20) From the assumptions imposed on the sequences{c_k}^∞_k=1and{γ_k}^∞_k=1, we have that the func- tionKb(λ)is analytic and bounded on the right half-plane Reλ>0. From the inequality (3.11) forx >ρ_eand for alln∈_Nwe have

1− ^Kb(λ) a²_n^(1−θ)

< ³

2. (3.21)

Since the vector-function A^2−θf(t) ∈ L2,γ(_R+,H), it follows that its Laplace transform A^2−θfˆ(λ)belongs to Hardy space H₂(Reλ>γ,H). It implies that

kA^2−θfˆk²_H

2(Reλ>γ,H)= _sup

Reλ>γ

Z _+∞

−_∞

kA^2−θfˆ(x+iy)k²_Hdy< +_∞, λ=x+iy.

Hence we obtain the following relation kA^2−θfˆk²_H

2(Reλ>γ,H) = _sup

Reλ>γ n∈_N

Z _+∞

−_∞

∑

|a²_n^−θfˆ_n(x+iy)|²

! dy

> _sup

Reλ>γ n∈_N

a²₁^(2−θ) Z _+∞

−_∞

∑

|f^ˆ_n(x+iy)|²

! dy

=a²₁^(2−θ) sup

Reλ>γ

Z +_∞

−_∞ kf^ˆ(x+iy)k²_Hdy

=a²₁^(2−θ)kf^ˆ(λ)k²_H

2(Reλ>γ,H). (3.22) For that reason, ˆf(λ)∈ H2(Reλ>γ,H)and f(t)∈ L2,γ(_R+,H). Now, taking into account (3.13) and (3.22) we obtain the following estimate:

sup

Reλ>γ n∈_N

+_∞ Z

−_∞

|λ²uˆ_n(λ)|²dy< sup

Reλ>γ n∈_N

1 a²₁^(2−θ)

+4

a^θ_n

`_n(λ)

2!

· sup

Reλ>γ n∈_N

+_∞ Z

−_∞

|gˆ_n(λ)|²dy<+_∞.

Thereforeλ²uˆn(λ)∈ H2(Reλ>γ,C)and _dt^d22un(t)∈ L2,γ(_R+,C). By (3.13) and Paley–Wiener theorem we have

d² dt²u(t)

2

L2,γ(_R+,H)

=kλ²uˆ(λ)k²_H2

=

fˆ(λ)−1−Kb(λ)A^−2(1−θ)

A^θL⁻¹(λ)A^2−θfˆ(λ)

2 H₂

< ¹

a²₁^(2−θ)

kA^2−θfˆ(λ)k²_H

2 +4kA^θL⁻¹(λ)A^2−θfˆ(λ)k²_H

2

≤ ¹ a²₁^(2−θ)

kA^2−θfˆ(λ)k²_H

2 +4 sup

Reλ>γ k∈_N

a^θ_k

`_k(λ)

2

kA^2−θfˆ(λ)k²_H

2

<





 1

a²₁^−θ +2 sup

Reλ>γ k∈_N

a^θ_k

`_k(λ)







2

kA^2−θfˆ(λ)k²_H2

≤d²₂kA^2−θfk²_L

2,γ(_R+,H),

(3.23)

where H₂ := H₂(Reλ > γ,H) and d₂= ¹

a²₁^−θ +2d₁

. Hence, _dt^d2₂u(t) ∈ L_2,γ(_R+,H) and the inequality

d² dt²u(t)

L_2,γ(_R+,H)

≤d₂

kA^2−θfk_L2,γ(R+,H) (3.24) holds. Accordingly, combining estimates (3.17) and (3.24), we come to the desired inequality

ku(t)k_W2

2,γ(_R+,A²)≤d

kA^2−θfk_L

2,γ(_R+,H)

, (3.25)

where the constant d is independent of f. Consequently, this implies that the equation (2.1) has a solutionu(t), which belongs to Sobolev spaceW_2,γ² (_R+,A²).

Now, let us prove that the solution u(t) satisfies the initial conditions u(+0) = 0 and u⁽¹⁾(+₀) =_0.

Remark 3.4. If ϕ(λ)∈ H₂(Reλ > γ,C), then for anyη > γthere is a sequence{η_k}^∞_k=1 such that lim_k→∞η_k = +_∞and

klim→_∞ Z _η

γ

|ϕ(x±iη_k)|dx =0.

Proof. Indeed, for anyη>γandη_k >0, we have Z _ηk

−η_k

Z _η

γ

|ϕ(x±iη_k)|²dx

dy≤

Z _η

γ

Z _+∞

−_∞ |ϕ(x±iη_k)|²dy

dx<+_∞.

Therefore, for anyη>0 there is a sequence{η_k}^∞_k=1such that its limit tends to+_∞ask →_∞ and

klim→_∞ Z _η

γ

|ϕ(x±iη_k)|²dx=0.

Now it remains to use the Cauchy inequality.

The above argument shows that u(t) ∈ L_2,γ(_R+,H), which implies that ˆu(λ) belongs to space H2(Reλ > γ,H)and ˆun(λ) belongs to H2(Reλ > γ,C). Let us prove, moreover, that λuˆ_n(λ) ∈ H₂(Reλ > γ,C). Indeed, since λ²uˆ_n(λ) belongs to Hardy space H₂(Reλ > γ,C), we have

sup

Reλ>γ n∈_N

+_∞ Z

−_∞

|(Reλ+iy)uˆn(Reλ+iy)|²dy= sup

Reλ>γ n∈_N

+_∞ Z

−_∞

|(Reλ+iy)²uˆ_n(_Reλ+iy)|² (Reλ)²+y² dy

< ¹ γ sup

Reλ>γ n∈_N

+_∞ Z

−_∞

|λ²uˆn(λ)|²dy<+_∞.

By the Paley–Wiener theorem, ˆ

u_n(+0) = √¹ 2π lim

η_k→_∞ η_k

Z

−η_k

ˆ

u_n(x+iy)dy= √¹ 2πi lim

η_k→_∞ γ+iηk

Z

γ−iηk

ˆ

u_n(λ)dλ,

ˆ

u⁽_n¹⁾(+0) = √¹ 2π lim

η_k→_∞ ηk

Z

−η_k

(x+iy)uˆ_n(x+iy)dy= √¹ 2πi lim

η_k→_∞ γ+iηk

Z

γ−iηk

λuˆ_n(λ)dλ.

Since the functions ˆu_n(+0)and ˆu⁽_n¹⁾(+0)are analytic functions on the right half-plane Reλ>

γ≥0, it follows from Cauchy’s theorem that, for anyη> γ, Z _γ+iη_k

γ−iηk

ˆ

un(λ)dλ=

_{Z η−iη}

k

γ−iηk

−

Z _η+iη_k γ+iηk

+

Z _η+iη_k η−iηk

ˆ

un(λ)dλ

=

Z _η

γ

uˆn(x−iη_k)dx−

Z _η

γ

uˆn(x+iη_k)dx+i Z _ηk

−η_k

uˆn(η+iy)dy, and

Z _γ+iη_k γ−iηk

λuˆn(λ)dλ=

_{Z η−iη}

k

γ−iηk

−

Z _η+iη_k γ+iηk

+

Z _η+iη_k η−iηk

λuˆn(λ)dλ

=

Z _η

γ

(_x−iη_k)uˆn(_x−iη_k)_dx−

Z _η

γ

(_x+_iηk)uˆn(_x+_iηk)_dx +i

Z _ηk

−ηk

(η+iy)uˆn(η+iy)dy.

According to Remark3.4and the fact that λ²uˆ_n(λ)∈ H₂(Reλ> γ,C)we have

klim→_∞ Z _η

γ

|uˆ_n(x±iη_k)|²dx =0,

klim→_∞ Z _η

γ

|(x±iη_k)uˆ_n(x±iη_k)|²dx=0.

Thus, forη> γ,

|uˆ_n(+0)| ≤C₁ lim

η_k→_∞ Z _ηk

−η_k

|uˆ_n(η+iy)|dy=C₁ Z +_∞

−_∞

(η+iy)²uˆ_n(η+iy) (η+iy)²

dy

≤ C₁

_{Z +∞}

−_∞ |(η+iy)²uˆ_n(η+iy)|²dy

1/2_{Z +∞}

−_∞

dy (η²+y²)²

1/2

. ¹

η^3/2, (3.26)