A Nonlocal in Time Problem for Evolutionary Singular Equations in Generalized Spaces of Type S∘

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 type S∘′.


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 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 jx k φ ðnÞ ðxÞj ≤ c kn , fk, ng ⊂ ℤ + , where fc kn g is some sequence of positive numbers depending on the function φ. If sequence fc kn g has a special form then we get a certain subclass of spaces of Schwartz space S = SðℝÞ of rapidly decreasing functions on ℝ. In [1], the case where c kn = k kα n nβ , 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 S β α . Functions from these spaces and all their derivatives decrease on the real axis as jxj ⟶ ∞ faster than exp f−a jxj 1/α g, 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][3][4][5][6][7][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 = ∂ 2 u/∂x 2 is a function Gðt, xÞ = ð2 ffiffiffiffi ffi πt p Þ −1 exp f−x 2 /ð4tÞg, for every t > 0, this function is an element of the space S 1/2 1/2 as a function of x ( [7], p. 46) and the space S 1/2 1/2 of a space of the type S. If c kn = a k b n , where fa k , k ∈ ℤ + g, fb n , n ∈ ℤ + g are some sequences of positive numbers, then we have generalized spaces of type S, denoted by the symbol S b n a k . The spaces S b n a k (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 S b n a k in the particular choice of sequences fa k g and fb n g (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 type S (in particular, spaces S β α ) 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 ∘ .

The Spaces of Test Functions
Here, we dwell on the spaces S b n a k , constructed by the sequences of the form fb n = n!ρ n , n ∈ ℤ + g, fa k = k!d k , k ∈ ℤ + g, where fρ n g, ρ 0 = 1 is the sequence of positive numbers, which has the following properties: (a) the sequence is monotonically decreasing The sequence fd k , k ∈ ℤ + g,d 0 = 1, also has properties (a)-(d), with condition (b) having the form: ∃c a > 0 ∃γ 2 ∈ ð0, 1Þ∀n ∈ ℕ :d k−1 /d k ≤ c a · k γ 2 . An example of a sequence fρ n g with properties (a-d) can be a sequence ρ n = ðnβÞ −nβ e nβ , where β ∈ ð0, 1Þ is a fixed parameter. For example, let us check for the sequence ρ n of property (d). We have that where ω = β β ð1 − βÞ 1−β < 1. If we take arbitrary ε > 0 and put λ = ε, then we get the inequality ρ n ≥ c ε ε n /n n , where c ε = exp f−ε 1/ð1−βÞ g. Note that condition (b) for this sequence is satisfied with the parameter γ 1 = β. We also consider that the parameters γ 1 , γ 2 in condition (b) for the sequences fρ n g and fd k g are related by condition (e): γ 1 + γ 2 = θ ≤ 1.
By S b n a k we denote a collection of functions φ ∈ C ∞ ðℝÞ, satisfying the condition S b n a k coincides with the union of the countably-normed spaces S b n ,B a k ,A by all indices fA, Bg ⊂ ℕ, where the symbol S b n ,B a k ,A denotes a collection of those functions φ ∈ S b n a k , that for arbitrary δ, ρ > 0 satisfy the inequalities It was established in [9] that the function φ ∈ C ∞ ðℝÞ belongs to the space S b n a k , where a k = k!d k , b n = n!ρ n , if and only if it extends analytically into the complex plane to the whole function φðzÞ, z ∈ ℂ, which satisfies the condition: where We note that ρ is a continuously differentiable, even function in ℝ that increases monotonically over the interval ½1, +∞Þ. It follows from property (d) (see [9]) that For example, if b n = n nβ , 0 < β < 1, then ρðyÞ~exp fjyj 1/β g. In addition, as it is proved in [9], ln ρ is a convex function on ð0, +∞Þ in the sense that ∀ y 1 , y 2 f g⊂ 0, +∞ ð Þ: ln ρ y 1 ð Þ + ln ρ y 2 ð Þ ≤ ln ρ y 1 + y 2 ð Þ: ð8Þ 2 Journal of Function Spaces The inequality ln ρðy 1 Þ − ln ρðy 1 + y 2 Þ ≤ −ln ρðy 2 Þ follows from (8).
Provided that ρ ′ ð2Þ/ρð2Þ = μð2Þ > 1, we obtain that ωμðωÞ = n possesses a unique solution ν n < n + 1, n ∈ ℕ. The sequence of solutions fν n g is increasing and it is unbounded. Indeed, suppose it is not, for example, sup n∈ℕ ν n = c < ∞, then we select a convergent subsequence ν n k , k ∈ ℕ such that lim k→∞ ν n k = a, a < +∞; so we obtain a contradiction, since ν n k · μðν n k Þ = n k and passing to the limit as k ⟶ ∞ we get a · μðaÞ = +∞.
It follows from the results given in [9], that the sequence fφ ν , ν ≥ 1g ⊂ S b n a k converges to zero in this space if the functions φ ν and their derivatives of any order converge to zero uniformly on every segment ½a, b ⊂ ℝ and satisfy the inequality where the constants c, A, B > 0 are independent of ν. A function g is called a multiplicator in the space S b n a k , if gψ ∈ S b n a k for any function ψ ∈ S b n a k and the mapping ψ ⟶ gψ is a linear and continuous operator from S b n a k into S b n a k : A function g ∈ C ∞ ðℝÞ that admits an analytic extension onto the entire complex plane and satisfies the condition [9]: is a multiplicator in the space S b n a k , a k = k!d k , b n = n!ρ n . The operators of multiplication by x, all polynomials, operators of differentiation, shift, and extension are defined and continuous in the spaces S b n a k , a k = k!d k ,b n = n!ρ n , [9]. By S ∘ b n a k , we denote the collection of all even functions from the space S b n a k . Since S ∘ b n a k forms a subspace of S b n a k , then the topology is naturally introduced in S ∘ b n a k . 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.
By S ∘ b n a k ðℂÞ, we denote the collection of functions that are extensions of functions φ from space S ∘ b n a k into ℂ. According to the results obtained in [9], the space S ∘ b n a k ðℂÞ can be represented as a union of the countably-normed spaces S then, these norms are equivalent to the corresponding norms in space S ∘ b n ,B a k ,A . Therefore, the sequence of functions fφ ν ðxÞ, ν ≥ 1g ⊂ S ∘ b n a k , x ∈ ℝ, converges to zero if and only if the sequence of functions fφ ν ðzÞ, ν ≥ 1g, z ∈ ℂ, converges to zero uniformly in every bounded domain of the complex plane ℂ, the inequalities 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 ∘ b n a k . An example of a multiplicator in S ∘ b n a k is the normalized Bessel function j ν , ν > −1/2, which is the solution of the equation B ν u + λu = 0, where B ν is Bessel operator; B ν = ðd 2 /dx 2 Þ + ð2ν + 1/xÞðd/dxÞ, ν > −1/2 is fixed parameter, provided that uð0Þ = 1, u ′ ð0Þ = 0. Indeed, the normalized Bessel function is related to the ordinary Bessel function J ν , ν > −1/2, of the first kind, so [15]: It is known (see [15]) that the function J ν admits an analytic extension into a complex plane ℂ, and the Poisson integral formula holds It follows from relations (16) and (17) that the normalized Bessel function j ν of the complex argument z is an integer even function and for j ν the integral image is correct: In view of cos z = 1/2ðe iz + e −iz Þ, z = x + iy ∈ ℂ and by using (18), we obtain estimate: Since for any convex functions ln γðxÞ and ln ρðyÞ and for any ε > 0, the inequality is true, it follows that It implies that j ν ðxÞ is a multiplicator in space S According to the results presented in [16], the direct and inverse Bessel transforms are defined in the spaces S ∘ b n a k ; moreover, if the conditions (a)-(e) are satisfied for the sequences fρ n g and fd k g, then the formula F B ν ½S ∘ b n a k = S ∘ a n b k is true, moreover, operator F B ν is continuous [16]. The spaces S ∘ β α are partial kind of spaces S ∘ b n a k . The spaces S ∘ β α consist of even functions of spaces S β α with the same topology; accordingly, the formula By T ξ x , we denote the generalized shift operator corresponding to the Bessel operator [17]: where b ν = Γðν + 1Þ/ðΓð1/2ÞΓðν + 1/2ÞÞ, 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 ∘ b n a k . We define the convolution of two functions of space S The formula is true [18]. We note that the operation of multiplication of test functions is defined and continuous in the spaces S ∘ b n a k . The spaces S ∘ b n a k form topological algebras with respect to the convolution of test functions.
Let us consider the pseudodifferential operator Provided that φ is a multiplicator in space S ∘ a n b k , the operator A φ is linear and continuous in space S ∘ b n a k . It turns out that if we consider the operator A φ in space S ∘ b n b k , then it can be understood as a Bessel operator of "infinite order" in this space (see [18]): Every locally integrated even function f on ℝ, which satisfies condition Journal of Function Spaces generates a regular generalized function The following statement is correct: if locally integrated even functionsf andgonℝsatisfying the condition (28) do not coincide on the set of Lebesgue positive measure, then there exists On the contrary, ifF f ≠ F g then the functionsf andgdo 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 functions F f generated by them from space ðS ∘ b n a k Þ′. It follows from the properties of the Lebesgue integral that the embedding is continuous.
Since the operation of a generalized shift of the argument is defined in space S ∘ b n a k , we define the convolution of a generalized function f ∈ ðS ∘ b n a k Þ′ with a test function by the formula (the index ξ in f ξ means that the functional f acts upon the test function T x ξ φðξÞ as a function of the argument ξ).
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 F B ½ f is linear and continuous in the space of the test functions S ∘ a n b k . Thus, the Bessel transform of the generalized function f defined on S ∘ b n a k is a generalized function on the space S ∘ a n b k . If a generalized function f ∈ ðS ∘ b n a k Þ ′ is a convolver in the space S ∘ b n a k , then for any function φ ∈ S ∘ b n a k , the relation is true [18].
The following statement implies the following properties: (1) if the generalized function f is a convolver in space S ∘ b n a k then its Bessel transform is a multiplicator in the space S ∘ a n b k ; (2) if the generalized function f is a multiplicator in the space S ∘ b n a k then its Bessel transform is a convolver in the space S ∘ a n b k .

Journal of Function Spaces
Thus, the solution of problem (33), (34) has the form We introduce the notation Gðt, Hence, as a result of formal reasoning, we find Indeed, Since j ν ðσξÞj ν ðσxÞ = T ξ x j ν ðσxÞ, we have The correctness of these transformations follows from the properties of the function G presented in what follows. The properties of the function G are connected with properties of the function Q, because G = F −1 B ν ½Q. Thus, first of all, we study the properties of the function Qðt, σÞ regarded as a function of the argument σ. Since Then (see Section 2) there exist numbers c 0 , a, b > 0 such that Further, we assume that the constant c 0 > 0 in (45) satisfies the condition: c 0 ≤ m, where m is the parameter of the multipoint problem (33), (34). Then The following estimates are true for the function and its derivatives (with respect to the variable s) on ℝ: where the constantsc,ã,b > 0 are independent of t, ρ n = inf τ ðρðτÞ/jτj n Þ. The function Qðt, sÞ belongs to the space Proof. The inequality is true for fixed t ∈ ð0, 1Þ. This property follows from the relation where μðξÞ =γ′ðξÞ/γðξÞ, and μ is a nonnegative, continuous function on ℝ, monotone increasing on ½0, ∞Þ: Then If t > 1, then the inequality t ln ρðbτÞ ≤ ln ρðtbτÞ is true. Then, t = ½t + ftg and where a 2 = aftg, If t = n, n ∈ ℕ then t = 1 + n − 1: On this Let a = min fat, aftgg = aftg, if t is not integer and a = a, if t is an integer, b = max fb, bTg: Then, the inequality is true for t ∈ ð0, T. Hence, it follows that Q 1 ðt, sÞ ∈ S ∘ b n b k ðℂÞ for every t ∈ ð0, T: 6 Journal of Function Spaces In the following considerations, we will use the estimate In view of the integral Cauchy formula, we obtain where Γ R is a circle of radius R centered at a point σ ∈ ℝ. By using (56), we obtain the inequalities where σ 0 is a point of maximum of the function exp f−t lñ γðaξÞg, ξ ∈ ½σ − R, σ + R. Since lnγðaξÞ is an even function on ℝ that increases on the interval ½0, +∞Þ, then In view the inequality −lnγðσ 1 + σ 2 Þ + lnγðσ 1 Þ ≤ −lñ γðσ 2 Þ, σ 1 , σ 2 > 0, we prove that there exist constantsã, a 2 > 0,ã ≤ a such that where a 2 = max fa 2 , a 2 Tg: Therefore, We used the fact thatγ = ρ, and the inequality of convexity for the function ln ρ : ln ρð bRÞ + ln ρð a 2 RÞ ≤ ln ρðð b + a 2 ÞRÞ.

Lemma 2. The function
is a multiplicator in the space S Proof. In view of (46), the inequalities are true. Since moreover, hence, by using the polynomial formula, we get 7 Journal of Function Spaces where λ ≔ t 1 r 1 + ⋯ + t m r m , Q 1 ðλ, σÞ = e λφðσÞ . By using this result and (46), we obtain the inequalities where μ 0 = max fμ 1 , ⋯μ m g. Further, we use the formula Then whereμ = μ −1 μ 0 m < 1, c ′ = μ −1 ∑ ∞ r=0μ r = μ −1 ð1 −μÞ −1 : By using the last inequality and boundedness of the function Q 2 on ℝ, we conclude that Q 2 is a multiplicator in space S ∘ b n b k . Lemma 2 is proved.
By using relations (56), (71) and the Leibniz formula of differentiation of the product of two functions, we obtain where c 1 = cc ′ , b 1 = 2b, b n = n!ρ n , the constants c 1 , b > 0 are independent of t. By virtue of the last inequality, we conclude that the function Qðt, σÞ (regarded as a function of σ) is an element of the space S ∘ b n b k (for any t ∈ ð0, T). In view of the relation F −1 , T. In the estimates for the derivatives of the function Gðt, xÞ (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 ∘ b n b k satisfy the condition On the converse, if infinitely differentiable, even function φ on ℝ satisfies condition (73), then (see [16]) φ is an element of the space S ∘ b q b k . In view of this observation, we estimate the function σ 2q B k ν Gðt, σÞ,σ ∈ ℝ, for fixed fk, qg ⊂ ℤ + . To do this, we use the relation established in [16]: from the last relation implies that We note also that for the function φ ∈ S ∘ b q b k , the following formula is true where c i ðνÞ are coefficients dependent on ν, the functions φ ð2q−iÞ /x i , i ∈ f0, 1, ⋯, qg, are also elements of space S We note that inequality is true, where a = aft/2g, 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), 8 Journal of Function Spaces we get that in addition to inequalities (72), the following inequalities are true In addition, where a 1 = 1/a (here taken into account xD 2n−1 x Qðt, xÞ ∈ S ∘ b n b k ) for every t > 0. According to Leibniz formula of differentiation of the product of two functions Let us present the right part (81) as the sum of two terms From condition (b) for the sequence fρ k g (see Section 2) there follows the inequality (here taken into account 2γ 1 ≤ 1, see Section 2). By using (79) and the last inequality, we get Similarly, by using (80), we have This yields wherec 0 = max fc 1 ,c 1 B 1 /a 1 A 1 g,Ã 0 = max fa 1 , a 1 A 1 g = a 1 · max f1, A 1 g = 1/a max f1, A 1 g,B 0 = max fB 1 , B 2 g, d = min fa 1 b 1 , a 1 A 1 B 1 g = a 1 min fb 1 , A 1 B 1 g: Since a 1 = 1/a then 1/a = aðmin fb 1 , A 1 B 1 gÞ −1 ≤ a ðmin fb 1 , A 1 B 1 gÞ −1 ≡ a · α: Moreover, we can assume that A ≥ 1, i.e.,Ã 0 = ð1/aÞA 1 : So, Journal of Function Spaces where L = A 1 exp f4γαg, M =B 0 exp f4γαg: The other additions in (77) are evaluated similarly. As a result, we obtain the inequality where the constants c 2 , A 2 , B 2 , a 0 > 0 are independent of t. Thus, it takes into account that jj ν ðσxÞj ≤ A ν , A ν = ffiffiffi π p Γðν + 1Þ/ Γðν + 1/2Þ, fσ, xg ⊂ ℝ: Hence where d 0 = B −1 2 , the constants c 4 , A 2 , d 0 > 0 are independent of t. Thus, the following statement is true.
where the constants L 0 , A 0 , d 0 > 0 are independent of t.
The function Gðt, xÞ is differentiable with respect to t on the interval ð0, T. Indeed, since then, formally differentiating (93) under the sign of the integral, we obtain the function Λðt, σÞ ≔ φðσÞQðt, σÞj ν ðσxÞσ 2ν+1 : Since φ is a multiplicator in the space S In addition, by (72), we obtain the estimate where a = aft 0 g, if t 0 is not integer and a = a, if t 0 is integer. Hence, It follows from the inequality of the convexity of the function lnγ that then the integrated function exp f−ðt/2Þ lnγð aσÞg is a majorant for Λðt, σÞ,t ∈ ½t 0 , T,σ ∈ ½0, +∞Þ. Therefore, the integral of the derivative (with respect to the variable t) of the integrand in (93) converges uniformly on any interval ½t 0 , T ⊂ ð0, T and therefore the derivative with respect to t under the sign of the integral in (93) can be applied at every point t ∈ ð0, T.
Lemma 4. The function Gðt, ·Þ, t ∈ ð0, T, regarded as an abstract function of the parameter t with values in the space S Proof. In view of the continuity of the direct and inverse Bessel transforms in spaces of the type S ∘ , to prove the lemma, it is sufficient to show that the function F B ν ½Gðt, ·Þ = Qðt, ·Þ, as an abstract function of the parameter t with values in the space In other words, it is necessary to that the limit relation is true in a sense that: (1) D s σ Φ Δt ðσÞ⟶ Δt→0 D s σ ðφðσÞQðt, σÞÞ, s ∈ ℤ + , uniformly on every segment ½a, b ⊂ ℝ

10
Journal of Function Spaces (2) jD s σ Φ Δt ðσÞj ≤ c B s b s e −lnγð aσÞ , s ∈ ℤ + , where the constants c, a, b > 0 are independent of Δt if Δt is sufficiently small The function Qðt, σÞ, ðt, σÞ ∈ ð0, T × ℝ, is differentiable with respect to t in the ordinary sense. Hence, by the Lagrange theorem on finite spaces, Thus, and Since in view of estimates (72), we obtain that uniformly on any segment ½a, b ⊂ ℝ. Then as Δt ⟶ 0 uniformly on any segment ½a, b ⊂ ℝ. Thus, condition (1) from relation (99) is satisfied.

Corollary 5. The formula
is true.