# A Nonlocal in Time Problem for Evolutionary Singular Equations in Generalized Spaces of TypeS∘

Authors: Hanna Verezhak; Vasyl Gorodetskyi
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we establish the correct solvability of a nonlocal multipoint in time problem for the evolutionary equation of a parabolic type with the Bessel operator of infinite order in the case where the initial function is an element of the space of generalized functions of typeS∘′.


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## Body

## 1. Introduction

Singular parabolic equations with the Bessel operator are related to the equations with degeneracy in terms of the spatial variable operator (such equations degenerate on the boundary of domain), and they are close to evenly parabolic equations in terms of their internal properties. They are used in the study of temperature fields, in the construction of mathematical modes of diffusion processes, and in anisotropic media that describe the phenomena of heat and mass transfer, radial oscillations of waves, occur in crystallography, hydrodynamics, and problems of interaction.The theory of classical solutions of the Cauchy problem for such equations is constructed in the works of M.I. Matiychuk, V.V. Krekhivskyi, S.D. Ivasyshen, V.P. Lavrenchuk, I.I. Verenych, and others. The Cauchy problem for singular parabolic equations in the classes of distributions and ultra-distributions was studied by Ya.I. Zhytomyrskyi, V.V. Gorodetskyi, I.V. Zhytariuk, V.P. Lavrenchuk, O.V. Martyniuk, and others.Gel’fand and Shilov in monograph ([1], p. 203–211) proposed a method of constructing functional spaces of infinitely differentiable functions given on ℝ, which impose certain conditions for decreasing on infinity and increasing derivatives with increasing of order. These conditions are set using inequalities xkφnx≤ckn, k,n⊂ℤ+, where ckn is some sequence of positive numbers depending on the function φ. If sequence ckn has a special form then we get a certain subclass of spaces of Schwartz space S=Sℝ of rapidly decreasing functions on ℝ. In [1], the case where ckn=kkαnnβ,α>0,β>0 are fixed parameters is studied in detail; the corresponding spaces are called spaces of type S and are denoted by the symbol Sαβ. Functions from these spaces and all their derivatives decrease on the real axis as x⟶∞ faster than exp−ax1/α,a>0,x∈ℝ. Such spaces are often used in the study of the problem of the classes of uniqueness and the classes of correct solvability of the Cauchy problem for partial differential equations. In [2–8], it was established that spaces of type S and spaces of the type S′ that are topologically dual to the spaces S coincide with the sets of initial data for the Cauchy problem in broad classes of partial differential equations of finite and infinite orders for which the solutions are entire functions of the space variables. For example, the fundamental solution of the Cauchy problem for the heat-conduction equation ∂u/∂t=∂2u/∂x2 is a function Gt,x=2πt−1exp−x2/4t, for every t>0, this function is an element of the space S1/21/2 as a function of x ([7], p. 46) and the space S1/21/2 of a space of the type S.Ifckn=akbn, where ak,k∈ℤ+,bn,n∈ℤ+ are some sequences of positive numbers, then we have generalized spaces of type S, denoted by the symbol Sakbn. The spaces Sakbn (their topological structure, properties of functions, and basic operations in such spaces) were studied in [9]. Known spaces of type W, introduced by Gurevich [10] (see also ([11], p.7–17)), in which arbitrary convex functions are used to characterize the behavior of functions at infinity instead of power functions, are also embedded in spaces Sakbn in the particular choice of sequences ak and bn (see [12]). From the results given in [9, 13], it follows that generalized spaces of type S are a natural medium for the study of nonlocal multipoint in time problems for evolutionary pseudodifferential equations (in particular, for equations with operators of differentiation of infinite order), for evolutionary equations with generalized Gelfond–Leontiev differentiation operators of finite and infinite orders.Spaces consisting of even functions of spaces of typeS (in particular, spaces Sαβ) with the corresponding topology are called spaces of type S∘ and are used in the study of evolutionary singular equations of parabolic type with Bessel operators (see [6], [14]). The purpose of this work is to investigate a nonlocal multipoint time problem for evolutionary singular equations of infinite order in generalized spaces of type S∘.


## 2. The Spaces of Test Functions

Here, we dwell on the spacesSakbn, constructed by the sequences of the form bn=n!ρn,n∈ℤ+, ak=k!dk,k∈ℤ+, where ρn,ρ0=1 is the sequence of positive numbers, which has the following properties:
(a)
the sequence is monotonically decreasing(b)
∃cb>0∃γ1∈0,1∀n∈ℕ:ρn−1/ρn≤cb·nγ1(c)
limn→∞ρnn=0(d)
∀ε>0∃cε>0∀n∈ℤ+:ρn≥cε·εn/nnThe sequencedk,k∈ℤ+,d0=1, also has properties (a)–(d), with condition (b) having the form: ∃ca>0∃γ2∈0,1∀n∈ℕ:dk−1/dk≤ca·kγ2. An example of a sequence ρn with properties (a–d) can be a sequence ρn=nβ−nβenβ, where β∈0,1 is a fixed parameter. For example, let us check for the sequence ρn of property (d). We have that
(1)ρn=enβnβnβ=enβnβnβ·n1−βn1−βn1−βn1−β·en1−βen1−β=ennn1ββ1−β1−βn·n1−βn1−βen1−β=ennn1ωnsupλ≥0λnexpλ1/1−β,where ω=ββ1−β1−β<1. If we take arbitrary ε>0 and put λ=ε, then we get the inequality ρn≥cεεn/nn, where cε=exp−ε1/1−β. Note that condition (b) for this sequence is satisfied with the parameter γ1=β.We also consider that the parametersγ1, γ2 in condition (b) for the sequences ρn and dk are related by condition (e): γ1+γ2=θ≤1.BySakbn we denote a collection of functions φ∈C∞ℝ, satisfying the condition
(2)∃c,A,B>0∀k,n⊂ℤ+∀x∈ℝ:xkφnx≤cAkBnakbn.Sakbn coincides with the union of the countably-normed spaces Sak,Abn,B by all indices A,B⊂ℕ, where the symbol Sak,Abn,B denotes a collection of those functions φ∈Sakbn, that for arbitrary δ,ρ>0 satisfy the inequalities
(3)xkφnx≤cδρA+δkB+ρnakbn,k,n⊂ℤ+,x∈ℝ;the system of norms in Sak,Abn,B is determined by formulas
(4)φδρ=supx,k,nxkφnxA+δkB+ρnakbn,δ,ρ⊂1,12,13,⋯.It was established in [9] that the function φ∈C∞ℝ belongs to the space Sakbn, where ak=k!dk, bn=n!ρn, if and only if it extends analytically into the complex plane to the whole function φz,z∈ℂ, which satisfies the condition:
(5)∃a,b,c>0∀z=x+iy∈ℂ:φz≤cγaxρby,where
(6)γx=1,x<1,infkak/xk,x≥1,ρy=1,y<1,supnyn/bn,y≥1.We note thatρ is a continuously differentiable, even function in ℝ that increases monotonically over the interval 1,+∞. It follows from property (d) (see [9]) that
(7)∃c0,c>0∀y∈ℝ:ρy≥c0expcy.For example, ifbn=nnβ,0<β<1, then ρy~expy1/β. In addition, as it is proved in [9], lnρ is a convex function on 0,+∞ in the sense that
(8)∀y1,y2⊂0,+∞:lnρy1+lnρy2≤lnρy1+y2.The inequalitylnρy1−lnρy1+y2≤−lnρy2 follows from (8).The functionρ in (6) is related to the sequence ρn, which constructs the sequence bn=n!ρn as follows [9]:
(9)ρn=infω≥1ρω/ωn=νn−nρνn,where νn is the solution of the equation ωμω=n, n∈ℕ. The function μω=ρ′ω/ρω is a nonnegative and continuous function on the interval 0,+∞; herewith lnρω=∫0ωνξdξ, lnρω=μω~·ω, 0<ω~<ω, i.e., μω~=lnρω/ω. From properties of the function ρ, it follows that lnρω increases faster than any linear function on the interval 1;+∞, as ω⟶+∞; i.e., limω→+∞μω=+∞. Provided that ρ′2/ρ2=μ2>1, we obtain that ωμω=n possesses a unique solution νn<n+1,n∈ℕ. The sequence of solutions νn is increasing and it is unbounded. Indeed, suppose it is not, for example, supn∈ℕνn=c<∞, then we select a convergent subsequence νnk, k∈ℕ such that limk→∞νnk=a, a<+∞; so we obtain a contradiction, since νnk·μνnk=nk and passing to the limit as k⟶∞ we get a·μa=+∞.Sinceγx=1/γ~x, where γ~x=1,x<1, and γx=supkxk/ak, if x≥1 then γ is a continuously differentiable, even function on ℝ that monotonically decreases on 1,+∞, 0<γx≤1,x∈ℝ. For example, if ak=kkα,α∈0,1, then the following inequalities are true ([1], p. 204)
(10)exp−αex1/α≤γx≤cexp−αex1/α,c=expαe2.Functionlnγ satisfies on 0,+∞ the inequality [9].
(11)lnγx1+lnγx2≥lnγx1+x2,x1,x2⊂0,+∞.It follows from the results given in [9], that the sequence φν,ν≥1⊂Sakbn converges to zero in this space if the functions φν and their derivatives of any order converge to zero uniformly on every segment a,b⊂ℝ and satisfy the inequality
(12)xkφνnx≤cAkBnakbn,k,n⊂ℤ+,x∈ℝ,where the constants c,A,B>0 are independent of ν.A functiong is called a multiplicator in the space Sakbn, if gψ∈Sakbn for any function ψ∈Sakbn and the mapping ψ⟶gψ is a linear and continuous operator from Sakbn into Sakbn. A function g∈C∞ℝ that admits an analytic extension onto the entire complex plane and satisfies the condition [9]:
(13)∀ε>0∃cε>0:gz≤cεγεx−1ρεy,z=x+iy∈ℂ,is a multiplicator in the space Sakbn,ak=k!dk,bn=n!ρn. The operators of multiplication by x, all polynomials, operators of differentiation, shift, and extension are defined and continuous in the spaces Sakbn,ak=k!dk,bn=n!ρn, [9].ByS∘akbn, we denote the collection of all even functions from the space Sakbn. Since S∘akbn forms a subspace of Sakbn, then the topology is naturally introduced in S∘akbn. This space with the corresponding topology is called a main space or a generalized space of type S∘, and its elements are called test functions.ByS∘akbnℂ, we denote the collection of functions that are extensions of functions φ from space S∘akbn into ℂ. According to the results obtained in [9], the space S∘akbnℂ can be represented as a union of the countably-normed spaces S∘ak,Abn,Bℂ, where S∘ak,Abn,Bℂ consists of those functions φ∈S∘akbnℂ for which inequality φx+iy≤cγa¯xρb¯y,z=x+iy∈ℂ is true, where a¯ is any positive constant less than a, b¯ is any constant greater than b; a,b>0 are constants of inequality (5). If for φ∈S∘ak,Abn,Bℂ, we put
(14)φpω=supz∈ℂφzγa1−1/pxρb+ωy,p∈2,3,⋯,ω∈ℕ,then, these norms are equivalent to the corresponding norms in space S∘ak,Abn,B. Therefore, the sequence of functions φνx,ν≥1⊂S∘akbn,x∈ℝ, converges to zero if and only if the sequence of functions φνz,ν≥1,z∈ℂ, converges to zero uniformly in every bounded domain of the complex plane ℂ, the inequalities
(15)φνz≤cγaxρby,z=x+iy∈ℂ,are true, where constant c,A,B>0 are independent of ν [9].Every integer even function satisfying condition (13) is the multiplicator in the space S∘akbn. An example of a multiplicator in S∘akbn is the normalized Bessel function jν,ν>−1/2, which is the solution of the equation Bνu+λu=0, where Bν is Bessel operator; Bν=d2/dx2+2ν+1/xd/dx,ν>−1/2 is fixed parameter, provided that u0=1,u′0=0. Indeed, the normalized Bessel function is related to the ordinary Bessel function Jν,ν>−1/2, of the first kind, so [15]:
(16)jνx=2νΓν+1xνJνx.It is known (see [15]) that the function Jν admits an analytic extension into a complex plane ℂ, and the Poisson integral formula holds
(17)Jνz=2πΓν+1/2z2ν∫0π/2coszcostsin2νtdt.It follows from relations (16) and (17) that the normalized Bessel function jν of the complex argument z is an integer even function and for jν the integral image is correct:
(18)jνz=2Γν+1πΓν+1/2∫0π/2coszcostsin2νtdt.In view ofcosz=1/2eiz+e−iz, z=x+iy∈ℂ and by using (18), we obtain estimate:
(19)jνz≤cνey,z=x+iy∈ℂ,cν=πΓν+1Γν+1/2,ν>−12.Since for any convex functionslnγ¯x and lnρy and for any ε>0, the inequality
(20)y≤lnγ~εx+lnρεy+c,c>0,is true, it follows that
(21)jνz≤cν,εelnγ~εx+lnρεy≡cν,εγεx−1ρεy.It implies thatjνx is a multiplicator in space S∘akbn, ak=k!dk, bn=n!ρn.According to the results presented in [16], the direct and inverse Bessel transforms
(22)ψσ≡FBνφσ=∫0+∞φxjνσxx2ν+1dx,σ∈ℝ,φx≡FBν−1ψx=122νΓ2ν+1∫0+∞ψσjνσxσ2ν+1dσ,x∈ℝ,are defined in the spaces S∘akbn; moreover, if the conditions (a)–(e) are satisfied for the sequences ρn and dk, then the formula FBνS∘akbn=S∘bkan is true, moreover, operator FBν is continuous [16]. The spaces S∘αβ are partial kind of spaces S∘akbn. The spaces S∘αβ consist of even functions of spaces Sαβ with the same topology; accordingly, the formula FBνS∘αβ=S∘βα is correct.ByTxξ, we denote the generalized shift operator corresponding to the Bessel operator [17]:
(23)Txξφx=bν∫0πφx2+ξ2−2xξcosωsin2νωdω,φ∈S∘akbn,where bν=Γν+1/Γ1/2Γν+1/2,ν>−1/2. Moreover, as it was proved in [18], the operation of a generalized shift is differentiable (even infinitely differentiable) in the space S∘akbn.We define the convolution of two functions of spaceS∘akbn by a formula
(24)φ∗ψx=∫0∞Txξφxψξξ2ν+1dξ,φ,ψ⊂S∘akbn.The formula(25)FBφ∗ψ=FBφ·FBψ,∀φ,ψ⊂S∘akbn,is true [18].We note that the operation of multiplication of test functions is defined and continuous in the spacesS∘akbn.The spacesS∘akbn form topological algebras with respect to the convolution of test functions.Let us consider the pseudodifferential operatorAφ=FB−1φFB in the space S∘akbn. Provided that φ is a multiplicator in space S∘bkan, the operator Aφ is linear and continuous in space S∘akbn. It turns out that if we consider the operator Aφ in space S∘bkbn, then it can be understood as a Bessel operator of “infinite order” in this space (see [18]):
(26)Aφ=∑k=0∞c2k−Bνk,φσ=∑k=0∞c2kσ2k.


## 3. The Space of Generalized FunctionsS∘akbn′

We denote byS∘akbn′ the space of all linear continuous functionals over the corresponding space of test functions with weak convergence. Linear continuous functionals are called regular generalized functions or regular functionals. The action of such functionals upon the test functions φ∈S∘akbn is determined by the formula
(27)f,φ=∫0∞fxφxx2ν+1dx.Every locally integrated even functionf on ℝ, which satisfies condition
(28)∀ε>0∃cε>0∀x∈ℝ:fx≤cεγεx−1,generates a regular generalized function Ff∈S∘akbn′: Ff,φ=∫0∞fxφxx2ν+1dx,∀φ∈S∘akbn.The following statement is correct:if locally integrated even functionsfandgonℝsatisfying the condition (28) do not coincide on the set of Lebesgue positive measure, then there exists a functionφ0∈S∘akbnsuch thatf,φ0≠g,φ0, i.e.,Ff≠Fg. On the contrary, ifFf≠Fgthen the functionsfandgdo not coincide on set of the Lebesgue positive measure. The proof of this statement is analogous to the proof of the corresponding theorem in [19].The formulated statement allows us to identify locally integrated functions with the generalized functionsFf generated by them from space S∘akbn′. It follows from the properties of the Lebesgue integral that the embedding
(29)S∘akbn∋f⟶Ff∈S∘akbn′,is continuous.Since the operation of a generalized shift of the argument is defined in spaceS∘akbn, we define the convolution of a generalized function f∈S∘akbn′ with a test function by the formula
(30)f∗φx=fξ,Txξφx=fξ,Tξxφξ,(the index ξ in fξ means that the functional f acts upon the test function Tξxφξ as a function of the argument ξ).Letf∈S∘akbn′. If f∗φ∈S∘akbn, ∀φ∈S∘akbn and the relation φν⟶0 as ν⟶+∞ in the topology of the space S∘akbn implies that f∗φν⟶0 as ν⟶+∞ in the topology of the space S∘akbn, then the functional f is called a convolver in the space S∘akbn.The Bessel transform of a generalized functionf∈S∘akbn′ is determined by the relation
(31)FBf,φ=f,FBφ,∀φ∈S∘bkan.In view of (31), the properties of the linearity and continuity of the functional f and the properties of the Bessel transform of the test functions, the functional FBf is linear and continuous in the space of the test functions S∘bkan. Thus, the Bessel transform of the generalized function f defined on S∘akbn is a generalized function on the space S∘bkan.If a generalized functionf∈S∘akbn′ is a convolver in the space S∘akbn, then for any function φ∈S∘akbn, the relation
(32)FBf∗φ=FBfFBφ,is true [18].The following statement implies the following properties: (1) if the generalized functionf is a convolver in space S∘akbn then its Bessel transform is a multiplicator in the space S∘bkan; (2) if the generalized function f is a multiplicator in the space S∘akbn then its Bessel transform is a convolver in the space S∘bkan.


## 4. A Nonlocal Multipoint in Time Problem

Let us consider an evolutionary equation(33)∂udt=Aφut,x,t,x∈0,T×ℝ≡ΠT,where Aφ=FBσ→x−1φσFBx→σ is a pseudodifferential operator in the S∘bkbn, constructed according to a function φ, which is a multiplicator in the space and such that eφ∈S∘bkbn. Note that Aφ can be understood as a Bessel operator of infinite order of appearance Aφ=∑k=0∞c2k−1kBνk (see Section 2). By P∘bkbn, we denote the class of functions (symbols) φ satisfying the conditions formulated above. For equation (33) we define a nonlocal multipoint in time problem as follows: to find the solution of equation (33) that satisfies the condition
(34)μut,·t=0−μ1ut,·t=t1−⋯−μmut,·t=tm=f,where m∈ℕ,μ,μ1,⋯,μm⊂0,+∞,t1,⋯,tm⊂0,T, 0<t1<⋯<tm≤T are fixed numbers and, moreover, μ>m∑k=0mμk,f∈S∘bkbn. We seek the solution of problem (33), (34) with the help of the Bessel transform specified as follows ut,x=FBν−1vt,σx,t,x∈ΠT. For the function v:ΠT⟶ℝ, we get the following problem with parameter σ:(35)dvt,σdt=φσvt,σ,t,σ∈ΠT,(36)μvt,σt=0−∑k=1mμkvt,σt=tk=f~σ,σ∈ℝ,where f~σ=FBνfσ. The general solution of equation (35) has the form
(37)vt,σ=cexptφσ,t,x∈ΠT,where c=cσ is determined from condition (36). Substituting (37) in (36), we obtain
(38)c=f~σμ−∑k=1mμkexptkφσ−1.Then(39)vt,σ=f~σexptφσμ−∑k=1mμkexptkφσ−1,t,x∈ΠT.Thus, the solution of problem (33), (34) has the form
(40)ut,x=cν∫0∞vt,σjνσxσ2ν+1dσ,t,x∈ΠT.We introduce the notationGt,x=FBν−1Qt,σ, where
(41)Qt,σ=exptφσμ−∑k=1mμkexptkφσ−1.Hence, as a result of formal reasoning, we find(42)ut,x=∫0∞TxξGt,xfξξ2ν+1dξ=Gt,x∗fx,t,x∈ΠT.Indeed,(43)ut,x=cν∫0∞Qt,σ∫0∞fξjνσξξ2ν+1dξjνσxσ2ν+1dσ.Sincejνσξjνσx=Txξjνσx, we have
(44)ut,x=∫0∞cν∫0∞Qt,σTxξjνσξσ2ν+1dσfξξ2ν+1dξ=∫0∞TxξGt,xfξξ2ν+1dξ=Gt,x∗fx,t,x∈ΠT.The correctness of these transformations follows from the properties of the functionG presented in what follows. The properties of the function G are connected with properties of the function Q, because G=FBν−1Q. Thus, first of all, we study the properties of the function Qt,σ regarded as a function of the argument σ.Sinceφ∈P∘bkbn, therefore eφ∈S∘bkbn. Then (see Section 2) there exist numbers c0,a,b>0 such that
(45)eφz≤c0e−lnγ~aσ+lnρbτ,γ~=1/γ=ρ,z=σ+iτ∈ℂ.Further, we assume that the constantc0>0 in (45) satisfies the condition: c0≤m, where m is the parameter of the multipoint problem (33), (34). Then
(46)etφz=eφzt≤c0exp−lnγ~aσ+lnρbτt≤cexp−tlnγ~aσ+tlnρbτ,c=max1,c0T.Lemma 1.
Letφ∈P∘bkbn. The following estimates are true for the function
(47)Qt,s=exptφs,t∈0,T,s=σ+iτ∈ℂ,and its derivatives (with respect to the variable s) on ℝ:
(48)DσnQ1t,σ≤c~b~nn!ρnexp−tlnγ~a~σ,n∈ℤ+,where the constants c~,a~,b~>0 are independent of t, ρn=infτρτ/τn. The function Qt,s belongs to the space S∘bkbnℂ for a fixed t∈0,T.Proof.
The inequality(49)tlnγ~aσ≥lnγ~taσ,is true for fixed t∈0,1. This property follows from the relation
(50)lnγ~taσ=∫0taσμξdξ=t∫0aσμtydy≤t∫0aσμydy=tlnγ~aσ,where μξ=γ~′ξ/γ~ξ, and μ is a nonnegative, continuous function on ℝ, monotone increasing on 0,∞. Then
(51)−tlnγ~aσ≤−lnγ~taσ,e−tlnγ~aσ+tlnρbτ≤e−lnγ~taσ+lnρbτ=e−lnγ~a1σ+lnρbτ,a1=at,t∈0,1.
Ift>1, then the inequality tlnρbτ≤lnρtbτ is true. Then, t=t+t and
(52)e−lnγ~aσ=e−tlnγaσe−tlnγ~aσ≤e−tlnγ~aσ≤e−lnγ~a2σ,where a2=at,(53)etlnρbτ≤elnρtbτ=elnρb1τ,b1=bt.
Ift=n, n∈ℕ then t=1+n−1. On this
(54)e−lnγ~aσ=e−lnγaσe−n+1lnγ~aσ≤e−lnγ~aσ,
Leta¯=minat,at=at, if t is not integer and a¯=a, if t is an integer, b¯=maxb,bT. Then, the inequality
(55)Q1t,s≤ce−lnγ~a¯σ+lnρb¯τ,is true for t∈0,T. Hence, it follows that Q1t,s∈S∘bkbnℂ for every t∈0,T.
In the following considerations, we will use the estimate(56)Q1t,s≤ce−tlnγ~aσ+lnρb¯τ,s=σ+iτ∈ℂ.
In view of the integral Cauchy formula, we obtain(57)DσnQ1t,σ=n!2πi∫ΓRQ1t,ss−σn+1ds,n∈ℤ+,where ΓR is a circle of radius R centered at a point σ∈ℝ. By using (56), we obtain the inequalities
(58)DσnQ1t,σ≤n!Rnmaxs∈ΓRQ1t,s≤n!Rne−tlnγ~aσ0+lnρb¯R,where σ0 is a point of maximum of the function exp−tlnγ~aξ, ξ∈σ−R,σ+R. Since lnγ~aξ is an even function on ℝ that increases on the interval 0,+∞, then
(59)σ0=0,forσ≤R,σ+R,forσ≤−R,σ−R,forσ≥R.
In view the inequality−lnγ~σ1+σ2+lnγ~σ1≤−lnγ~σ2, σ1,σ2>0, we prove that there exist constants a~,a2>0,a~≤a such that
(60)∀σ≥0,∀R>0:exp−tlnγ~aσ0≤exp−tlnγ~a~σexptlnγ~a2R≤exp−tlnγ~aσexplnγ~a¯2R,where a¯2=maxa2,a2T. Therefore,
(61)DσnQ1t,σ≤cn!Rnexp−tlnγ~a~σexptlnγ~a2Rexplnρb¯R≤cn!Rnexp−tlnγ~a~σexplnρb~R,b~=b¯+a¯2.
We used the fact thatγ~=ρ, and the inequality of convexity for the function lnρ:lnρb¯R+lnρa¯2R≤lnρb¯+a¯2R.
For anyn∈ℤ+, the function gn,tR=R−nexplnρb~R=R−nρb~R is differentiable on 0,+∞ and we obtain the following relations
(62)limR→+∞gn,tR=+∞,n∈ℤ+;limR→+0gn,tR=+∞,n∈ℕ,1,n=0.from the properties of the function ρ. Since gn,tR>0,R∈0,+∞, this function attains its infimum. Thus,
(63)DσnQ1t,σ≤cn!infRgn,tRexp−tlnγ~a~σ=cn!b~ninfρb~Rb~Rnexp−tlnγ~a~σ=c~b~nn!ρnexp−tlnγ~a~σ,c~=c.
Lemma1 is proved.Lemma 2.
The function(64)Q2σ=μ−∑k=1mμkexptkφσ−1=μ−∑k=1mμkQ1tk,σ−1,σ∈ℝ,is a multiplicator in the space S∘bkbn.Proof.
In view of (46), the inequalities
(65)Q1tk,σ≤ce−tklnγ~aσ≤c,k∈1,⋯,m,are true. Since
(66)μ−∑k=1mμkQ1tk,σ=μ1−1μ∑k=1mμkexptkφσ,moreover,
(67)1μ∑k=1mμkexptkφσ≤cμ∑k=1mμk<mμ∑k=1mμk<1,hence, by using the polynomial formula, we get
(68)Q2σ=1μ1−1μ∑k=1mμkQ1tk,σ−1=1μ∑r=0∞μ−r∑k=1mμketkφσr=∑r=0∞μ−r+1∑r1+⋯+rm=rr!r1!⋯rm!μ1et1φσr1⋯μmetmφσrm=∑r=0∞μ−r+1∑r1+⋯+rm=rr!r1!⋯rm!μ1r1⋯μmrmQ1λ,σ,where λ≔t1r1+⋯+tmrm,Q1λ,σ=eλφσ. By using this result and (46), we obtain the inequalities
(69)DσnQ2σ≤cb~nn!ρn∑r=0∞μ−r+1∑r1+⋯+rm=rr!r1!⋯rm!μ0rexp−λlnγ~aσ≤cb~nn!ρn∑r=0∞μ−r+1μ0r∑r1+⋯+rm=rr!r1!⋯rm!,n∈ℕ,where μ0=maxμ1,⋯μm. Further, we use the formula
(70)∑r1+⋯+rm=rr!r1!⋯rm!mr.
Then(71)DσnQ2σ≤cb~nn!ρn∑r=0∞μ~r=c′b~nn!ρn,n∈ℕ,where μ~=μ−1μ0m<1,c′=μ−1∑r=0∞μ~r=μ−11−μ~−1. By using the last inequality and boundedness of the function Q2 on ℝ, we conclude that Q2 is a multiplicator in space S∘bkbn. Lemma 2 is proved.By using relations (56), (71) and the Leibniz formula of differentiation of the product of two functions, we obtain
(72)DσnQt,σ=∑l=0nCnlDσlQ1t,σDσn−lQ2σ≤cc′∑l=0nCnlb~ll!ρlb~n−lρn−le−tlnγ~aσ≤c1b1nn!ρnexp−tlnγ~aσ≡c1b1nbnexp−tlnγ~aσ,σ∈ℝ,where c1=cc′,b1=2b~,bn=n!ρn, the constants c1,b>0 are independent of t. By virtue of the last inequality, we conclude that the function Qt,σ (regarded as a function of σ) is an element of the space S∘bkbn (for any t∈0,T).In view of the relationFBν−1S∘bkbn=S∘bkbn, we conclude that Gt,·=FBν−1Qt,·∈S∘bkbn for every t∈0,T.In the estimates for the derivatives of the functionGt,x (with respect to the variable x), we select the dependence on the parameter t.For this, we note (see [16]) that functions from space S∘bkbn satisfy the condition
(73)∃c~=cφ>0∃A~=A~φ>0∃B~=B~φ>0∀x∈0,∞∀k,q⊂ℤ+:x2kBνqφx≤c~A~2kB~2qb2kb2q,φ∈S∘bkbq.On the converse, if infinitely differentiable, even functionφ on ℝ satisfies condition (73), then (see [16]) φ is an element of the space S∘bkbq. In view of this observation, we estimate the function σ2qBνkGt,σ,σ∈ℝ, for fixed k,q⊂ℤ+. To do this, we use the relation established in [16]:
(74)σ2qBνkFbνφσ=∫0∞Bνqx2kφxjνσxx2ν+1dx,∀φ∈S∘bkbq,from the last relation implies that
(75)σ2qBνkGt,σ=∫0∞Bνqx2kQt,xjνσxx2ν+1dx.We note also that for the functionφ∈S∘bkbq, the following formula is true
(76)Bνqφx=∑i=0qciνφ2q−ixxi,q∈ℤ+,where ciν are coefficients dependent on ν, the functions φ2q−i/xi, i∈0,1,⋯,q, are also elements of space S∘bkbq. Then
(77)Bνqx2kQt,x=c0νx2kQt,x2q+c1νx2kQt,x2q−1x+c2νx2kQt,x2q−2x2+⋯+cqνx2kQt,xqxq.We note that inequality(78)exp−tlnγ~aσ=exp−t2lnγ~aσ·exp−t2lnγ~aσ≤exp−tlnγ~a¯σ·exp−t2lnγ~aσ=γ~a¯σexp−t2lnγ~aσis true, where a¯=at/2, if t/2 is not integer and a¯=a, if t/2 is integer (see the proof of Lemma 1). By using (78) and (72), we get that in addition to inequalities (72), the following inequalities are true
(79)x2kDx2qQt,x≤c1b12qb2qinfkb2ka¯x2kx2kexp−t2lnγ~ax≤c1a12kb12qb2qb2kexp−t2lnγ~ax,a1=1a¯.In addition,(80)∃c~1,A1,B1>0∀x∈ℝ∀l,n⊂ℤ+x2l−1Dx2n−1Qt,x=x2l−2xDx2n−1Qt,x≤c~1A1a12l−1B12nb2l−1b2nexp−t2lnγ~ax,where a1=1/a (here taken into account xDx2n−1Qt,x∈S∘bkbn) for every t>0. According to Leibniz formula of differentiation of the product of two functions
(81)x2kQt,x2q=∑i=02qC2qix2kiQ2q−it,x.Let us present the right part (81) as the sum of two terms
(82)I1≔∑i=02qC2q2ix2k2iQ2q−2it,x,I2≔∑i=02qC2q2i−1x2k2i−1Q2q−2i+1t,x.From condition (b) for the sequenceρk (see Section 2) there follows the inequality
(83)b2k−2b2k≤cbk−21−γ1,cb>0.Thus(84)b2k−2b2q−2b2kb2q≤cb2kγ1qγ1kq2≤cb2maxk,q2γ1kq2≤cb2k+q2γ1kq2≤γk+qkq2,γ=cb2,(here taken into account 2γ1≤1, see Section 2). By using (79) and the last inequality, we get
(85)I1≤c1a12kb12qb2kb2q1+12!∏i=012k−i2q−i1a1b12b2k−2b2q−2b2kb2q+14!∏i=032k−i2q−i1a1b14·b2k−4b2q−4b2k−2b2q−2·b2k−2b2q−2b2kb2q+⋯exp−t2lnγ~ax≤c1a12kb12qb2kb2q1+12!4γk+qa1b12+14!4γk+qa1b14+⋯exp−t2lnγ~ax≤c1a12kb12qb2kb2qch4γk+qa1b1exp−t2lnγ~ax.Similarly, by using (80), we have
(86)I2≤c~1a1A12kB12qb2kb2q2k2q1a1A12B10b2k−2b2qb2kb2q+13!∏i=022k−i2q−i1a1A14B12·b2k−4b2q−2b2k−2b2q−2·b2k−2b2q−2b2kb2q+⋯exp−t2lnγ~ax≤c~1B1a1A1a1A12kB12qb2kb2q11!4γk+qa1A1B1+13!4γk+qa1A1B13+⋯×exp−t2lnγ~ax≤c~1B1a1A1a1A12kB12qb2kb2qsh4γk+qa1A1B1exp−t2lnγ~ax.This yields(87)I1+I2≤c~0cB~02qb2kb2qexp4γdk+qexp−t2lnγ~ax;where c~0=maxc1,c~1B1/a1A1,A~0=maxa1,a1A1=a1·max1,A1=1/amax1,A1, B~0=maxB1,B2, d=mina1b1,a1A1B1=a1minb1,A1B1.Sincea1=1/a then 1/a=a¯minb1,A1B1−1≤aminb1,A1B1−1≡a·α. Moreover, we can assume that A≥1, i.e., A~0=1/aA1. So,
(88)x2kDx2qQt,x≤c~La¯2kM2qb2kb2qexp−t2lnγ~ax,k,q⊂ℤ+,where L=A1exp4γα, M=B~0exp4γα. The other additions in (77) are evaluated similarly. As a result, we obtain the inequality
(89)Bνqx2kQt,x≤c2A2a¯2kB22qb2kb2qexp−t2lnγ~ax≤c2A2a¯2kB22qb2kb2qexp−a0tx,where the constants c2,A2,B2,a0>0 are independent of t. Thus,
(90)σ2qBνkGt,σ=∫0∞Bνqx2kQt,xjνσxx2ν+1dx≤Aν∫0∞Bνqx2kQt,x∣x2ν+1dx≤c3A2a¯2kB22qb2kb2q∫0∞x2ν+1e−a0txdx=c4t−2ν+1A2a¯2kB22qb2kb2q;it takes into account that jνσx≤Aν, Aν=πΓν+1/Γν+1/2,σ,x⊂ℝ. Hence
(91)BνkGt,σ≤c4A2a¯2kb2k·t−2ν+1infqb2qσB2−12q=c4t−2ν+1A2a¯2kb2kγd0σ=c4t−2ν+1A2a¯2kb2kexp−lnγ~d0σ,where d0=B2−1, the constants c4,A2,d0>0 are independent of t. Thus, the following statement is true.Lemma 3.
The following inequalities are true for the functionBνkGt,x, Gt,·∈S∘bkbn for every t∈0,T:
(92)BνkGt,x≤L0t−2ν+1A0a¯2kb2kexp−lnγ~d0x,k∈ℤ+,where the constants L0,A0,d0>0 are independent of t.The functionGt,x is differentiable with respect to t on the interval 0,T. Indeed, since
(93)Gt,x=cν∫0+∞Qt,σjνσxσ2ν+1dσ;then, formally differentiating (93) under the sign of the integral, we obtain the function Λt,σ≔φσQt,σjνσxσ2ν+1. Since φ is a multiplicator in the space S∘bkbn then
(94)∀ε>0∃cε>0∀σ∈ℝ:φσ≤cεelnγ~εσ.In addition, by (72), we obtain the estimate
(95)Qt,σ≤ce−t0lnγ~aσ≤ce−lnγ~a¯σ,σ∈0,+∞,t∈t0,T⊂0,T,where a¯=at0, if t0 is not integer and a¯=a, if t0 is integer. Hence,
(96)Λt,σ≤celnγ~εσ−lnγ~a¯σσ2ν+1,σ∈0,+∞,t∈t0,T⊂0,T.It follows from the inequality of the convexity of the functionlnγ~ that
(97)lnγ~εσ−lnγ~a¯σ≤−lnγ~a¯−εσ=−lnγ~a¯¯σ,a¯¯=a¯−ε=a¯/2>0,for ε=a¯/2. Since
(98)∃d>0∀σ∈0,+∞:exp−12lnγ~a¯σσ2ν+1≤d;then the integrated function exp−t/2lnγ~a¯¯σ is a majorant for Λt,σ,t∈t0,T,σ∈0,+∞. Therefore, the integral of the derivative (with respect to the variable t) of the integrand in (93) converges uniformly on any interval t0,T⊂0,T and therefore the derivative with respect to t under the sign of the integral in (93) can be applied at every point t∈0,T.Lemma 4.
The functionGt,·,t∈0,T, regarded as an abstract function of the parameter t with values in the space S∘bkbn, is differentiable respect to t.Proof.
In view of the continuity of the direct and inverse Bessel transforms in spaces of the typeS∘, to prove the lemma, it is sufficient to show that the function FBνGt,·=Qt,·, as an abstract function of the parameter t with values in the space FBνS∘bkbn=S∘bkbn, is differentiable with respect to t. In other words, it is necessary to that the limit relation
(99)ΦΔtσ≔1ΔtQt+Δt,σ−Qt,σ⟶∂∂tQt,σ,Δt⟶0,is true in a sense that:
(1)
DσsΦΔtσ⟶Δt→0DσsφσQt,σ,s∈ℤ+, uniformly on every segment a,b⊂ℝ(2)
DσsΦΔtσ≤c¯B¯sbse−lnγ~a¯σ,s∈ℤ+, where the constants c¯,a¯,b¯>0 are independent of Δt if Δt is sufficiently small
The functionQt,σ, t,σ∈0,T×ℝ, is differentiable with respect to t in the ordinary sense. Hence, by the Lagrange theorem on finite spaces,
(100)ΦΔtσ=φσQt+θΔt,σ,0<θ<1,t+θΔt≤T.
Thus,(101)DσsΦΔtσ=∑l=0sCslDσlφσDσs−lQt+θΔt,σ,and
(102)DσsΦΔtσ−∂∂tQt,σ=∑l=0sCslDσlφσDσs−lQt+θΔt,σ−Dσs−lQt,σ.
Since(103)Dσs−lQt+θΔt,σ−Dσs−lQt,σ=Dσs−l+1Qt+θ1Δt,σθΔt,0<θ1<1,in view of estimates (72), we obtain that
(104)Dσs−l+1Qt+θ1Δt,σθΔt⟶0,Δt⟶0,uniformly on any segment a,b⊂ℝ. Then
(105)DσsΦΔtσ⟶Dσs∂∂tQt,σ,as Δt⟶0 uniformly on any segment a,b⊂ℝ. Thus, condition (1) from relation (99) is satisfied.
Sinceφ is a multiplicator in the space S∘bkbn, then
(106)∀ε>0∃cε>0∀z=σ+iτ∈ℂ:φz≤cεelnγ~εσ+lnρετ,γ~=1/γ=ρ.
Because of the Cauchy integral formula, we have that(107)φnσ=n!2πi∫ΓRφzz−σn+1dz,n∈ℤ+,where ΓR is a circle of radius R centered at the point σ∈ℝ. Then, in view (106), we obtain inequalities
(108)φnσ≤n!Rnmaxz∈ΓRφz≤cεn!Rnelnγ~εσ+R+lnρεR≤cεn!infRρεRRnelnγ~εσ+R,σ≥0.
For sufficiently large values ofσ>0, the inequality εσ+R≤ε+Rσ is true. Since the function lnγ~ increases monotonically for σ≥0, then at the same values of σ(109)lnγ~εσ+R≤lnγ~ε+Rσ.
For all values ofσ≥0, the following inequality is true:
(110)lnγ~εσ+R≤lnγ~ε+Rσ+cR.
Therefore, forσ≥0(111)elnγ~εσ+R≤c~Relnγ~ε+Rσ.
Further, for a givenε>0, we assume that R=ε. Then
(112)φnσ≤c~εεnn!ρnelnγ~2εσ,n∈ℤ+.
By using (112) and estimates for the derivatives of the functions Qt,σ, we find
(113)DσsΦΔtσ≤c~~ε∑l=0sCslεlblb1s−lbs−lelnγ~2εσ−t+θΔtlnγ~aσ≤c¯B¯sbselnγ~2εσ−tlnγ~aσ≤c¯B¯sbselnγ~2εσ−lnγ~a¯σ,where a¯=at, if t is not an integer; a¯=a, if t is an integer, c¯=c~~ε, B¯=2maxε,b1; and t+θΔt>0. Take ε=a¯/4. In view of the inequality of convexity for the function lnγ~, we obtain
(114)lnγ~2εσ−lnγ~a¯σ≤−lnγ~a¯−2εσ≡−lnγ~a¯¯σ,a¯¯=a¯−2ε=a/2>0.
Thus(115)DσsΦΔtσ≤c¯B¯sbse−lnγ~a¯σ,σ≥0,moreover, constants c¯,B¯,a¯>0 are independent of Δt (for sufficiently small Δt). The case σ<0 is proved similarly. Lemma is proved.Corollary 5.
The formula(116)∂∂tf∗Gt,x=f∗∂Gt,x∂t,∀f∈S∘bkbn′,t∈0,T,is true.Proof.
By the definition of a convolution of a generalized function with the test function, we find(117)f∗Gt,x=fξ,TxξGt,ξ=fξ,TξxGt,ξ.
Then(118)∂∂tf∗Gt,·=limΔt→01Δtf∗Gt+Δt,x−f∗Gt,x=limΔt→0fξ,1ΔtTxξGt+Δt,x−TxξGt,x.
By virtue of Lemma4, the limit relation
(119)1ΔtTxξGt+Δt,x−TxξGt,x⟶Δt→0∂∂tTxξGt,x,is true in a sense of convergence in the topology of the space S∘bkbn, thus
(120)∂∂tf∗Gt,x=fξ,limΔt→01ΔtTxξGt+Δt,x−TxξGt,x=fξ,∂∂tTxξGt,x=fξ,Txξ∂∂tGt,x=f∗∂∂tGt,x.
Corollary is proved.Lemma 6.
In the spaceS∘bkbn, the following limit relation is true
(121)μlimt→+0Gt,·−∑l=1mμllimt→tlGt,·=δ,where δ is the Dirac delta function.Proof.
In view of the continuity of the Fourier transform and the functionGt,· regarded as an abstract function of the parameter t with values in the space S∘bkbn, we replace the relation (121) by the limit relation
(122)μlimt→+0FBνGt,·−∑l=1mμllimt→tlFBνGt,·=FBνδ,in the space S∘bkbn′. By using the representation of the function G, we rewrite relation (122) in the form
(123)μlimt→+0Qt,·−∑l=1mμllimt→tlQt,·=1.
To prove relation (123), we take an arbitrary function ψ∈S∘bkbn, apply the theorem on the limit transition under the sign of Lebesgue integration, we find
(124)μlimt→+0Qt,·,ψ−∑l=1mμllimt→tlQt,·,ψ=μlimt→+0∫ℝQt,σψσdσ−∑l=1mμllimt→tl∫ℝQt,σψσdσ=∫ℝμQ0,σ−∑l=1mμlQtk,σψσdσ=∫ℝμμ−∑k=1mμkQ1tk,σ−∑l=1mμlQ1tl,σμ−∑k=1mμkQ1tk,σψσdσ=∫ℝμ−∑l=1mμlQ1tl,σμ−∑k=1mμkQ1tk,σψσdσ=∫ℝψσdσ=1,ψ.
This implies that relation (123) is true in the space S∘bkbn′. Therefore, the relation (121) holds. Lemma is proved.ByS∘bkbn,∗′, we denote a class of generalized functions from S∘bkbn′ that are convolvers in the space S∘bkbn.Corollary 7.
Letωt,x=f∗Gt,x,f∈S∘bkbn∗′,t,x∈ΠT. Then, the relation
(125)μlimt→+0ωt,·−∑k=1mμklimt→tkωt,·=f,is true in the space S∘bkbn′.Proof.
Sincef is convolver in the space S∘bkbn, we obtain
(126)FBνf∗Gt,·=FBνf·FBνGt,·=FBνf·Qt,·.
(FBνf is multiplicator in the space S∘bkbn). By using this fact and the property of the continuity of the Bessel transform, we write (125) in the form
(127)μlimt→+0FBνωt,·−∑k=1mμklimt→tkFBνωt,·=FBνfμlimt→+0Qt,·−∑k=1mμklimt→tkQt,·=FBνf,(the relation is considered in space S∘bkbn′). In view (123), we obtain (125). The statement is proved.The functionωt,· is a solution of equation (33). Indeed f is convolver in the space S∘bkbn, we get
(128)Aφut,x=FBν−1φσFBνf∗Gt,·=FBν−1φσFBνfQt,σ=FBν−1∂∂tQt,σFBνf=FBν−1FBν∂∂tGt,··FBνf=FBν−1FBνf∗∂Gt,·∂t=f∗∂Gt,·∂t.On the other hand (see Corollary5)
(129)∂∂tf∗Gt,·=f∗∂Gt,·∂t.This implies that the functionωt,· satisfies equation (33) in the ordinary sense. By Corollary 7, the nonlocal m-point in time problem for equation (33) can be formulated as follows: to find a solution of equation (33) satisfying the condition
(130)μlimt→+0ut,·−∑k=1mμklimt→tkut,·=f,f∈S∘bkbn∗′,where the limit relation (130) is considered in space S∘bkbn′ (the restriction imposed on the parameters μ,μ1,⋯,t1,⋯ are the same as in problem (33), (34)).It follows from the above that the functionut,x=f∗Gt,x, f∈S∘bkbn∗′, is a solution of (33). If f=δ∈S∘bkbn∗′, then f∗Gt,x=Gt,x i.e., Gt,· is also a solution of (33). By using this fact and relation (121), in what follows the function G is called the fundamental solution of problem (33), (130).Theorem 8.
The problem (33), (130) is correctly solvable. Its solution is given by the formula ut,x=f∗Gt,x,t,x∈ΠT,ut,·∈S∘bkbn for every t∈0,T.Proof.
The functionf∗Gt,x satisfies (33). The solution ut,x continuously depends on the function f in condition (130) in the sense that if f,fn,n≥1⊂S∘bkbn∗′ and fn⟶f as n⟶∞ in the space S∘bkbn′, then unt,·=fn∗Gt,·⟶ut,·=f∗Gt,·, n⟶∞ in the space S∘bkbn′. This property follows from the property of continuity of convolution.
It remains to show that problem (33), (130) possesses a unique solution. To this end, we consider the Cauchy problem
(131)∂v∂t+Aφ∗v=0,t,x∈0,t0×ℝ,0≤t<t0≤T,(132)vt,·t=t0=g,g∈S∘bkbn∗′,where Aφ∗ is the restriction of operator adjoint to the operator A to the space S∘bkbn⊂S∘bkbn′. We understand condition (132) in a weak sense. The Cauchy problem (131), (132) is solvable, moreover vt,·∈S∘bkbn for every t∈0,t0.
LetQt0t:S∘bkbn∗′⟶S∘bkbn be an operator that associates a functional g∈S∘bkbn∗′ with a solution of problem (131), (132). The operator Qt0t is linear and continuous, it is defined for any t and t0 such that 0≤t<t0≤T and has the properties
(133)∀g∈S∘bkbn∗′:dQt0tgdt+Aφ∗Qt0tg=0,limt→t0Qt0tg=g,(the limit is considered in the space S∘bkbn′).
Let us consider a solutionut,x,t,x∈ΠT of problem (33), (130) understood as a regular functional from the space S∘bkbn∗′⊃S∘bkbn. We prove that problem (33), (130) may have only one solution in space S∘bkbn∗′. To this end, it suffices to prove that only the functional ut,x≡0 (for any t∈0,T) can be a unique solution of Eq. (33) with the trivial boundary condition (130). We apply the functional u to a function Qt0tg∈S∘bkbn, where g is an arbitrarily fixed element from the space S∘bkbn⊂S∘bkbn∗′. Further, differentiating with respect to t and using Eq. (33), (131) we get
(134)∂∂tu,Qt0tg=∂u∂t,Qt0tg+u,∂Qt0tg∂t=Aφu,Qt0tg−u,Aφ∗Qt0tg=Aφu,Qt0tg−Aφu,Qt0tg=0,t∈0,t0.
This implies thatut,·,Qt0tg is constant. By using properties of abstract functions, we obtain the relation
(135)limt→t0ut,·,Qt0tg=ut0,·,g=const≡c,at any point t0∈0,T.Thus, if f=0 in (130) then
(136)μlimt→+0ut,·,g−∑k=1mμklimt→tkut,·,g=μc0−∑k=1mμkck=0.
This implies thatc0=c1=⋯=cm=0. Suppose it is not. For example, c0≠0. Then we get a relation μ−∑k=1mμkαk=0, where αk=ck/c0, i.e., μ=∑k=1mμkαk. Since αk is any constant, and by μ,μ1,⋯,μm are fixed parameters, and μ>∑k=1mμk, the obtained contradiction proves that c0=0. Similarly, we prove that c1=c2=⋯=cm=0. Hence, ut0,·,g=0 for any g∈S∘bkbn, i.e., ut0,x is a zero functional from the space S∘bkbn∗′. Since t0∈0,T and t0 is arbitrary then ut,x=0 for all t∈0,T. Theorem is proved.As an example, we consider equation (33) with the operator Aφ, constructed on the basic of the function φx=−x2,x∈ℝ. In this case, we obtain
(137)Aφ=Bν=d2udx2+2ν+1−1x−1ddx,ν>−1/2,and equation (33) is the equation with the Bessel operator
(138)∂u∂t=∂2u∂x2+2ν+1x∂u∂x,ν>−12,t,x∈ΠT.The functionφx=−x2 is an element of the space P∘1/21/2≡P∘kk/2nn/2≡P∘bkbn. Indeed, e−x2∈S∘1/21/2, because
(139)e−z2=e−x+iy2=e−x2+y2,z=x+iy∈ℂ.From the characteristic of the spacesS∘αβ and (139) imply that e−x2∈S∘αβ, where α=1/2, 1/1−β=2, i.e., β=1/2. In addition, the function −x2, x∈ℝ is a multiplicator in the space S∘1/21/2. In this case, the constant c0 in inequality (45) is equal to one, i.e., the condition c0≤m, m∈ℕ is satisfied. By the above theorem, the nonlocal m-point in time problem for equation (138) is correctly solvable if f∈S∘1/21/2,∗′, herewith ut,x=f∗Gt,x, where
(140)Gt,x=2−νΓ−1ν+1∑r=0∞1μr+1∑r1+⋯+rm=rr!μ1r1⋯μmrmr1!⋯rm!2λt,r−v+1×exp−x2/4λt,r,where λt,r=t1r1+⋯+tmrm+t. In particular, if f=δ∈S∘1/21/2,∗′,m=1,t1=T (the case of two-point problem), then
(141)ut,x=Gt,x=2−2ν+1μ−1Γ−1ν+1∑r=0∞μ1μr1t+rTν+1exp−x24t+rT.


## 5. Conclusion

The correct solvability of a nonlocal multipoint in time problem for the evolutionary equation of a parabolic type with the Bessel operator of infinite order of appearance∑k=0∞c2k−B0k in the case where the initial function is an element of the space of generalized functions of type S∘′ is established in this paper. The properties of the fundamental solution this problem are investigated.



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*Source: 6873414-2020-06-15.xml*