On One Evolution Equation of Parabolic Type with Fractional Differentiation Operator in S Spaces

In the paper, we investigate a nonlocal multipoint by a time problem for the evolution equation with the operator A � (I − Δ)ω/2, Δ � (d2/dx2), and ω ∈ [1; − 2) is a fixed parameter. )e operator A is treated as a pseudodifferential operator in a certain space of type S. )e solvability of this problem is proved. )e representation of the solution is given in the form of a convolution of the fundamental solution with the initial function which is an element of the space of generalized functions of ultradistribution type. )e properties of the fundamental solution are investigated. )e behavior of the solution at t⟶ +∞ (solution stabilization) in the spaces of generalized functions of type S′ and the uniform stabilization of the solution to zero on R are studied.


Introduction
A rather wide class of differential equations with partial derivatives forms linear parabolic and B-parabolic equations, and the theory of which originates from the study of A formal extension of the class of parabolic type equations is the set of evolution equations with the pseudodifferential operator (PDO), which can be represented as where a is a function (symbol) that satisfies certain conditions and J(J − 1 ) is the direct (inverse) Fourier or Bessel transform. e PDO includes differential operators, fractional differentiation and integration operators, convolution operators, and the Bessel operator B ] � (d 2 /dx 2 ) + (2] + 1)x − 1 (d/dx), ] > − (1/2), which contains the expression (1/x) in its structure and is formally represented If A is a nonnegative self-adjoint operator in a Hilbert space H, then it is known [1] that a continuous on [0, T) function u(t) is continuously differentiable solution of the operator differential equation u′(t) + Au(t) � 0, t ∈ (0, T), which refers to abstract equations of parabolic type, if and only if it is given as u(t) � e − tA f, f � u(0) ∈ H. It turns out [1] that all continuously differentiable functions within the interval (0, T) solutions of this equation are described by the same formula where f is an element of the wider than H space H a ′ conjugate to the space H a of analytic vectors of the operator A; the role of A is played by the extension A of the operator A to the space H a ′ , and the boundary value of u(t) at the point 0 exists in the space H a ′ .
If A � (I − Δ) 1/2 , Δ � (d 2 /dx 2 ), then A is a nonnegative self-adjoint operator in H � L 2 (R), since (id/dx) is a selfadjoint in L 2 (R) operator with the domain D(id/dx) � φ ∈ L 2 (R): ∃φ ′ ∈ L 2 (R) . If E λ , λ ∈ R, is the spectral function of the operator (id/dx), then, due to the basic spectral theorem for self-adjoint operators, It is known (see, for example, [2]) that It follows from this that dE λ φ(t) � (1/2π) (3) Following [3], we call the operator A the Bessel operator of fractional differentiation. erefore, A can be understood as a pseudodifferential operator constructed on the function is allows us to interpret the function e − tA f that is a solution of the corresponding Cauchy problem as a convolution of the form G(t, ·) * f [4,5] In this paper, we give a similar depiction of the solution of a nonlocal multipoint time problem for the equation . . , α m ⊂ R, and m ∈ N are fixed and B 1 , . . . , B m are pseudodifferential operators constructed of smooth symbols (if α 0 � 1, α 1 � · · · � α m � 0, B 0 � I, then obviously we have a Cauchy problem).
is condition is interpreted in the classical or weak sense as if f is a generalized function (generalized element of the operator A) of ultradistribution type. Properties of the fundamental solution of the specified multipoint problem are investigated. e behavior of the solution at t ⟶ +∞ (solution stabilization) in the spaces of generalized functions of type S ′ and the uniform stabilization of the solution to zero on R are studied.
Note that the nonlocal multipoint time problem relates to nonlocal problems for operator differential equations and partial differential equations. Such problems arise when modeling many processes and problems of practice with boundary-value problems for partial differential equations, when describing correct problems for a particular operator and constructing a general theory of boundary-value problems (see, for example, [6][7][8][9][10][11]).

Spaces of Test and Generalized Functions
Gelfand and Shilov introduced in [12] a series of spaces, which they called the spaces S. ey consist of infinitely differentiable on R functions, which satisfy certain conditions on the decrease at infinity and the growth of derivatives. ese conditions are given by the inequalities where c km is some double sequence of positive numbers. If there are no restrictions on elements of the sequence c km , then obviously we have L. Schwartz's space S ≡ S(R) of quickly descending at infinity functions. However, if the numbers c km satisfy certain conditions, then the corresponding specific spaces are contained in S and they are called the spaces of S type. Let us define some of them.
For any α, β > 0, let us put e introduced S spaces can also be described as in [12]. S β α consists of those infinitely differentiable on R functions φ(x) that satisfy the following inequalities: with some positive constants c, B, and a dependent only on the function φ.
If 0 < β < 1 and α ≥ 1 − β, then S β α consists of only those φ ∈ C ∞ (R), which admit an analytic extension into the complex plane C such that , which can be analytically extended into some band |Imz| < δ (dependent on φ) of the complex plane, so that the estimate is carried out. e spaces S β α , α, β > 0, are nontrivial if α + β ≥ 1, and they form dense sets in L 2 (R). e topological structure in S β α is defined as follows. e symbol S β,B α,A , A, B > 0, denotes the set of functions φ ∈ S β α that satisfy the condition is set is transformed into a complete, countably normed space, if the norms in it are defined by means of relations: e specified norm system is sometimes replaced by an equivalent norm system: α,A , that is, S β α is endowed by the inductive limit topology of the spaces S β,B α,A [12]. erefore, 2 International Journal of Differential Equations the convergence of a sequence φ ] , ] ≥ 1 ⊂ S β α to zero in the space S β α is the convergence in the topology of some space S β,B α,A , to which all the functions φ ] belong. In other words (see [12]), φ ] ⟶ 0 in S β α as ] ⟶ ∞, if and only if for every n ∈ Z + , the sequence φ (n) ] , ] ≥ 1, converges to zero uniformly on an arbitrary segment [a, b] ⊂ R and for some c, a, B > 0 independent of ], and the inequality is operation is also differentiable (even infinitely differentiable [12]) in the sense that the limit relation is true for every function φ ∈ S β α in the sense of convergence in the S β α -topology. In S β α , the continuous differentiation operator is also defined. e spaces of type S are perfect [12] (that is, the spaces and all bounded sets of which are compact) and closely related to the Fourier transform, Moreover, the operator F : α is given by the following formula: It follows from the infinite differentiability property of the argument translation operation in S β α that the convolution f * φ is a usual infinitely differentiable function on R.
We define the Fourier transform of a generalized where the operator F:
We are looking for the solution of problems (24) and (25) via the Fourier transform. Due to condition (19), (27) (t, σ). Given the form of the operators A, B 1 , . . ., B m So, for the function v: Ω ⟶ R, we arrive at a problem with parameter σ: where c � c(σ) is determined by condition (30). Substituting (30) into (31), we find that (32) and Q(t, σ) � Q 1 (t, σ)Q 2 (σ), and thus, en, thinking formally, we come to the relation Indeed, e correctness of the transformations performed, the convergence of the corresponding integrals and, consequently, the correctness of formula (35) follow from the properties of the function G, which are given below. e properties of G are determined by the properties of Q are valid, where c � 0 if 0 < t ≤ 1 and c � 1 if t > 1, and the constants c > 1 and A > 0 do not depend on t.
Proof. To prove the statement, we use the Faa di Bruno formula for differentiation of a complex function: where the sum sign is applied to all the solutions in positive integers of the equation p 1 + 2p 2 + · · · + lp l � s, p 1 + · · · + p l � m. Put F � e g and g � − ta(σ). en, where the symbol Λ denotes the expression, and Given estimate (16) is fulfilled, we find that Using (40) and the Stirling formula, we arrive at the inequalities where c � 0 if 0 < t ≤ 1 and c � 1 if t > 1, and the values c > 1 and A > 0 do not depend on t. Lemma is proved.

Lemma 2.
e function Q 2 is a multiplier in S 2 1/ω .
Proof. To prove the assertion, let us estimate the derivatives of Q 2 . For this we use formula (37) in which we put F � φ − 1 and φ � R, where Given the properties of g 1 , . . . , g m and inequality (41), we find that where we took into account that In addition, Since, by assumption, μ > m k�1 μ k (assuming the properties of g 1 , . . . , g m , and 0 < t 1 < t 2 < · · · < t m ). So, International Journal of Differential Equations It follows from the last inequality and boundedness of the function Q 2 (σ) on R that Q 2 is a multiplier in S 2 are valid, where the constants c and A > 0 do not depend on t.
Taking into account the properties of the Fourier transform (direct and inverse) and the formula F − 1 [S 2 1/ω ] � S 1/ω 2 , we get that G(t, ·) ∈ S 1/ω 2 for every t > 0. We remove in the estimates of derivatives of the function G (in the variable x), the dependence on t, assuming t > 1. To do this, we use the relations So, (51) Applying the Leibniz formula for differentiating the product of two functions and estimating the derivatives of the function Q(t, σ), we find that where m ks � k 2k s s/ω . Taking into account the results in (see [12], p. 236-243), we find that this double sequence satisfies the inequality Bearing in mind the last inequality and also that t > 1, we obtain 6 International Journal of Differential Equations So, where the constants c 3 , B, and a 0 > 0 do not depend on t; here, we used the well-known inequality (see [12], p. 204) in which α � 2 and L � A. us, such a statement is correct.

Lemma 3. e derivatives (by variable x) of the function
where the constants c 3 , B, and a 0 > 0 do not depend on t.
Here are a few more properties of the function G(t, x).

Lemma 4.
e function G(t, ·), t ∈ (0, +∞), as an abstract function of t with values in the space S 1/ω 2 , is differentiable in t.
Proof. It follows from the continuity of the Fourier transform (direct and inverse) that, to prove the statement, it suffices to establish that the function F[G(t, ·)] � Q(t, ·), as a function of a parameter t with values in the space S 2 1/ω , is differentiable in t. In other words, it is necessary to prove that the boundary value relation is performed in the sense as follows: (1) D s σ Φ Δt (σ) ⟶ Δt ⟶ 0 D s σ (− a(σ)Q(t, σ)), s ∈ Z + , uniformly on each segment [a, b] ⊂ R (2) |D s σ Φ Δt (σ)| ≤ cB s s 2s exp − a|σ| ω { }, s ∈ Z + , where the constants c, B, and a > 0 do not depend on Δt if Δt is small enough e function Q(t, σ), (t, σ) ∈ Ω, is differentiable in t in the usual sense. Due to the Lagrange theorem on finite increments, constructed by function (1 + σ 2 ) ω/2 , σ ∈ R, while the nonlocal multipoint by time condition also contains pseudodifferential operators constructed on smooth symbols. e representation of the solution is given in the form of a convolution of the fundamental solution with the initial function which is an element of the space of generalized functions of the ultradistribution type (the coagulator in space S 1/ω 2 ). e behavior of the solution u(t, x), t ⟶ +∞ in the space of generalized functions (S 1/ω 2 ) ′ , is investigated. e conditions for the initial generalized function are found under which the solution is uniformly stabilized to zero on R. e method of research of a nonlocal multipoint by time problem offered in this paper allows to interpret differentialoperator equations of a form (zu/zt) + φ(i z/zx)u � 0 as an evolutionary equation with a pseudodifferential operator φ(i z/zx) � F − 1 [φ · F] constructed by the function φ acting in certain countable-normalized space of infinite-differential functions (the choice of space depends on the properties of the function which is the symbol of the operator φ(i z/zx)).

Data Availability
No data were used to support the findings of this study.

Disclosure
is study was conducted in the framework of scientific activity at Yuriy Fedkovych Chernivtsi National University.

Conflicts of Interest
e authors declare that they have no conflicts of interest.