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

1. 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 the heat conduction equation. The classical theory of the Cauchy problem and boundary-value problems for such equations and systems of equations is constructed in the works of I.G. Petrovskiy, S.D. Eidelman, S.D. Ivasyshen, M.I. Matiychuk, M.V. Zhitarashu, A. Friedman, S. Teklind, V.O. Solonnikov, and V.V. Krekhivskiy et al. The Cauchy problem with initial data from the spaces of generalized functions of the type of distributions and ultradistributions was studied by G.E. Shilov, B.L. Gurevich, M.L. Gorbachuk, V.I. Gorbachuk, O.I. Kashpirovskiy, Ya.I. Zhytomyrskiy, S.D. Ivasyshen, V.V. Gorodetskiy, and V.A. Litovchenko et al.

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 [mathematical expression not reproducible], where a is a function (symbol) that satisfies certain conditions and J ([J.sup.-1]) is the direct (inverse) Fourier or Bessel transform. The PDO includes differential operators, fractional differentiation and integration operators, convolution operators, and the Bessel operator [B.sub.v] = ([d.sup.2]/d[x.sup.2])+ (2v + 1)[x.sup.-1] (d/dx), > - (1/2), which contains the expression (1/x) in its structure and is formally represented as [mathematical expression not reproducible], where FB is the Bessel integral transformation.

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 [member of] (0, T), which refers to abstract equations of parabolic type, if and only if it is given as u (t) = [e.sup.-tA]f, f = u (0) [member of] 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'.sub.a] conjugate to the space [H.sub.a] of analytic vectors of the operator A; the role of A is played by the extension [??] of the operator A to the space [H'.sub.a], and the boundary value of u (t) at the point 0 exists in the space [H'.sub.a].

If A = [(I - [DELTA]).sup.1/2], [DELTA] = ([d.sup.2]/d[x.sup.2]), then A is a nonnegative self-adjoint operator in H = [L.sub.2] (R), since (id/dx) is a self-adjoint in L2(R) operator with the domain D (id/dx) = {[phi] [member of] [L.sub.2] (R): [there exists][phi]' [member of] [L.sub.2] (R)}. If [E.sub.[lambda]], [lambda] [member of] R, is the spectral function of the operator (id/dx), then, due to the basic spectral theorem for self-adjoint operators,

[mathematical expression not reproducible]. (1)

It is known (see, for example, [2]) that

[mathematical expression not reproducible]. (2)

It follows from this that d[E.sub.[lambda]][phi] (t) = (1/2[pi]) F[[phi]]([lambda])[e.sup.-it[lambda]][d.sub.[lambda]]. So,

[mathematical expression not reproducible]. (3)

Following [3], we call the operator A the Bessel operator of fractional differentiation. Therefore, A can be understood as a pseudodifferential operator constructed on the function symbol [(1 + [lambda]2).sup.1/2], [lambda] [member of] R. This allows us to interpret the function [e.sup.-tA]f that is a solution of the corresponding Cauchy problem as a convolution of the form [G(t, *).sup.*] f [4, 5], where [mathematical expression not reproducible].

In this paper, we give a similar depiction of the solution of a nonlocal multipoint time problem for the equation [mathematical expression not reproducible] when the initial condition [u|.sub.t=0] = f is replaced by the condition [mathematical expression not reproducible], and m [member of] N are fixed and [B.sub.1], ..., [B.sub.m] are pseudodifferential operators constructed of smooth symbols (if [[alpha].sub.0] = 1, [[alpha].sub.1] = ... = [[alpha].sub.m] = 0, [B.sub.0] = I, then obviously we have a Cauchy problem). This 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. The behavior of the solution at t [right arrow] + [infinity] (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, [6-11]).

2. Spaces of Test and Generalized Functions

Gelfand and Shilov introduced in [12] a series of spaces, which they called the spaces S. They consist of infinitely differentiable on R functions, which satisfy certain conditions on the decrease at infinity and the growth of derivatives. These conditions are given by the inequalities [mathematical expression not reproducible], where {[c.sub.km]} is some double sequence of positive numbers. If there are no restrictions on elements of the sequence {[c.sub.km]}, then obviously we have L. Schwartz's space S [equivalent to] S(R) of quickly descending at infinity functions. However, if the numbers [c.sub.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 [alpha], [beta] > 0, let us put

[mathematical expression not reproducible]. (4)

The introduced S spaces can also be described as in [12].

[S.sup.[beta].sub.[alpha]] consists of those infinitely differentiable on R functions [phi] (x) that satisfy the following inequalities:

[mathematical expression not reproducible], (5)

with some positive constants c, B, and a dependent only on the function f.

[mathematical expression not reproducible]. (6)

If 0 < [beta] < 1 and [alpha] [greater than or equal to] 1 -1, then [S.sup.[beta].sub.[alpha]] consists of only those [phi] [member of] [C.sup.[infinity]] (R), which admit an analytic extension into the complex plane C such that

[mathematical expression not reproducible]. (6)

The space [S.sup.l.sub.[alpha]] consists of functions [phi] [member of] [C.sup.[infinity]] (R), which can be analytically extended into some band [absolute value of (Imz)] < [delta] (dependent on [phi]) of the complex plane, so that the estimate

[mathematical expression not reproducible], (7)

is carried out.

The spaces [S.sup.[beta].sub.[alpha]], [alpha], [beta] > 0, are nontrivial if [alpha] + [beta] [greater than or equal to] 1, and they form dense sets in [L.sub.2] (R).

The topological structure in [S.sup.[beta].sub.[alpha]] is defined as follows. The symbol [S.sup.[beta],B.sub.[alpha],A], A,B > 0, denotes the set of functions [beta] [member of] [S.sup.[beta].sub.[alpha]] that satisfy the condition

[mathematical expression not reproducible]. (8)

This set is transformed into a complete, countably normed space, if the norms in it are defined by means of relations:

[mathematical expression not reproducible]. (9)

The specified norm system is sometimes replaced by an equivalent norm system:

[mathematical expression not reproducible]. (10)

If [mathematical expression not reproducible] is continuously embedded into [mathematical expression not reproducible], that is, [S.sup.[beta].sub.[alpha]]is endowed by the inductive limit topology of the spaces [S.sup.[beta],B.sub.[alpha],A] [12]. Therefore,

the convergence of a sequence {[[phi].sub.v], v [greater than or equal to] 1} [subset] [S.sup.[beta].sub.[alpha]] to zero in the space [S.sup.[beta].sub.[alpha]] is the convergence in the topology of some space [S.sup.[beta],B.sub.[alpha],A], to which all the functions [[phi].sub.v] belong. In other words (see [12]), [[phi].sub.v] [right arrow] 0 in [S.sup.[beta].sub.[alpha]] as v [right arrow] [infinity], if and only if for every n [member of] [Z.sub.+], the sequence [[phi].sup.(n).sub.v], v [greater than or equal to] 1, converges to zero uniformly on an arbitrary segment [a, b] [subset] R and for some c, a, B > 0 independent of y, and the inequality

[mathematical expression not reproducible]. (11)

holds.

In [S.sup.[beta].sub.[alpha]], the continuous translation operation [T.sub.x]: [phi]([xi]) [right arrow] [phi]([xi] + x) is defined. This operation is also differentiable (even infinitely differentiable [12]) in the sense that the limit relation ([phi](x + h) - [phi](x))[h.sup.-1] [right arrow] [phi]' (x), h [right arrow] 0, is true for every function [phi] [member of] [S.sup.[beta].sub.[alpha]] in the sense of convergence in the [S.sup.[beta].sub.[alpha]]-topology. In [S.sup.[beta].sub.[alpha]], the continuous differentiation operator is also defined. The 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, namely, the formula F[[S.sup.[beta].sub.[alpha]]] = [S.sup.[alpha].sub.[beta]], [alpha], [beta] > 0, is correct, where

[mathematical expression not reproducible]. (12)

Moreover, the operator F: [S.sup.[beta].sub.[alpha]] [right arrow] [S.sup.[alpha].sub.[beta]] is continuous.

Let ([S.sup.[beta].sub.[alpha]])' denote the space of all linear continuous functionals on [S.sup.[beta].sub.[alpha]] with weak convergence. Since the translation operator [T.sub.x] is defined in the space [S.sup.[beta].sub.[alpha]] of test functions, the convolution of a generalized function f [member of] ([S.sup.[beta].sub.[alpha]])' with the test function [phi] [member of] [S.sup.[beta].sub.[alpha]] is given by the following formula:

[mathematical expression not reproducible]. (13)

It follows from the infinite differentiability property of the argument translation operation in [S.sup.[beta].sub.[alpha]] that the convolution f * [phi] is a usual infinitely differentiable function on R.

We define the Fourier transform of a generalized function f [member of] ([S.sup.[beta].sub.[alpha]])' by the relation

<F[f], [phi]> = <f, F[[phi]]>, [for all][phi] [member of] [S.sup.[alpha].sub.[beta]], (14)

where the operator F: ([S.sup.[beta].sub.[alpha]])' [right arrow] ([S.sup.[alpha].sub.[beta]])' is continuous.

Let f [member of] ([S.sup.[beta].sub.[alpha]])'. If [mathematical expression not reproducible], and the convergence [[phi].sub.v] [right arrow] 0 in the [S.sup.[beta].sub.[alpha]]-topology as v [right arrow] [infinity] implies that f * [[phi].sub.v] [right arrow] 0 in the [S.sup.[beta].sub.[alpha]]-topology as [phi] [right arrow] [infinity], then the function f is called a convolutor in [S.sup.[beta].sub.[alpha]]. If f [member of] ([S.sup.[beta].sub.[alpha]])' is convolutor in [S.sup.[beta].sub.[alpha]], then for an arbitrary function [phi] [member of] [S.sup.[beta].sub.[alpha]], the formula F[f * [phi]] = F[f]F[[phi]] is valid, where F[f] is a multiplier in [S.sup.[beta].sub.[alpha]] [12].

Recall that the function g [member of] [C.sup.[infinity]] (R) is called a multiplier in [S.sup.[beta].sub.[alpha]] if g[phi] [member of] [S.sup.[beta].sub.[alpha]] for an arbitrary function [phi] [member of] [S.sup.[beta].sub.[alpha]], and the mapping [phi] [right arrow] g[phi] is continuous in the space [S.sup.[beta].sub.[alpha]].

3. Nonlocal Multipoint by Time Problem

Consider the function a([sigma]) = [(1 + [[sigma].sup.2]).sup.[omega]/2], [sigma] [member of] R, where [omega] [member of] [1,2) is a fixed parameter. Obviously, the function a ([sigma]) satisfies the inequality

a ([sigma]) [less than or equal to] [c.sub.[epsilon]] exp{[epsilon][[absolute value of ([sigma])].sup.w]}, [sigma] [member of] R, (15)

where [c.sub.[epsilon]] = [2.sup.[omega]/2] max{1, (1/[epsilon])] for an arbitrary [epsilon] > 0. Using direct calculations and the Stirling formula, we find that

[mathematical expression not reproducible], (16)

where [B.sub.0] = [B.sub.0] ([omega]) > 0, [B.sub.1] = [B.sub.1] ([omega]) > 0, and [c.sub.0], [c.sub.1] > 0. It follows from (15) and (16) that a(a) is a multiplier in [S.sup.1.sub.1/[omega]]. Indeed, let [phi] [member of] [S.sup.1.sub.1/[omega]], that is, the function f and its derivatives satisfy the inequality

[mathematical expression not reproducible], (17)

with some positive constants c, A, and a. Then, using the Leibniz formula for differentiating the product of two functions, as well as inequalities (15)--(17), we find that

[mathematical expression not reproducible]. (18)

Since [epsilon] > 0 is arbitrary, we can put [epsilon] = (a/2). Then,

[mathematical expression not reproducible], (19)

where [c.sub.2] = c([c.sub.[epsilon]] + [c.sub.1]) and [B.sub.2] = 2 max{A,[B.sub.1]} . It follows from (19) that a x [phi] is an element of [S.sup.1.sub.1/[omega]].

The operation of multiplying by a ([sigma]) is continuous in the space [S.sup.1.sub.1/[omega]]. In fact, let {[[phi].sub.n], n [greater than or equal to] 1} be a sequence of functions from [S.sup.1.sub.1/[omega]] convergent to 0 in this space. This means that [mathematical expression not reproducible] with some constants [A.sub.0], [B.sub.0] > 0 and

[mathematical expression not reproducible]. (20)

In other words, for an arbitrary [??] > 0, there exists a number [n.sub.0] = [n.sub.0] ([??]) such that, for n [greater than or equal to] [n.sub.0],

[mathematical expression not reproducible]. (21)

Using inequalities (15) and (16) (when putting in (15), [epsilon] = (a/2)(1 - [delta])), we get

[mathematical expression not reproducible], (22)

where [c.sub.3] = [c.sub.[epsilon]] + [c.sub.1] and [??] = 2 max{[B.sub.0], [B.sub.1]}. It follows from the last inequality that [mathematical expression not reproducible], that is, the sequence {a[[phi].sub.n], n [greater than or equal to] 1} converges to zero in the space [mathematical expression not reproducible] where [mathematical expression not reproducible]. This means that the sequence {a[[phi].sub.n], n [greater than or equal to] 1} converges to zero in the space [S.sup.1.sub.1/[omega]], which is what we needed to prove.

Remark 1. It follows from the proven property that the function a([sigma]) = [(1 + [[sigma].sup.2]).sup.[omega]/2], [sigma] [member of] R, is also a multiplier in every space [S.sup.[beta].sub.1/[omega]], where [beta] > 1. Therefore, in the space [S.sup.1/[omega].sub.[beta]], [beta] [greater than or equal to] 1, defined as a continuous linear pseudodifferential operator A, constructed by the function a ([sigma]):

[mathematical expression not reproducible], (23)

where A[phi] = [(I - [DELTA]).sup.[omega]/2] [phi], [DELTA] = ([d.sup.2]/d[x.sup.2]).

Let us consider the evolution equation with the operator A (the Bessel fractional differentiation operator):

[partial derivative]u/[partial derivative]t + Au = 0, (t, x) [member of] (0, +[infinity]) x R [equivalent to] [OMEGA]. (24)

By a solution of equation (24), we mean the function u(t, x), (t, x) [member of] [OMEGA], such that (1) u(t, *) [member of] [C.sup.1] (0, +[infinity]) for every x [member of] R; (2) u(*, x) [member of] [S.sup.1/[omega].sub.[beta]] for every t [member of] (0, +[infinity]); (3) u(t, x) is continuous at every point (0, x) of the boundary [[GAMMA].sub.[OMEGA]] = {0} x R of the region [OMEGA]; (4) [there exists]M: R [right arrow] [o, [infinity]), [for all]t [member of] (0, +[infinity]) : [absolute value of ([u.sub.t] (t, x))] [less than or equal to] M(x), [int.sub.R] M(x)dx < [infinity]; and (5) u(t, x), (t, x) [member of] [OMEGA] satisfies equation (24).

For equation (24), we consider the nonlocal multipoint by time problem of finding a solution of equation (24) that satisfies the condition

[mathematical expression not reproducible], (25)

where [mathematical expression not reproducible], [mathematical expression not reproducible] are fixed numbers, [mathematical expression not reproducible] are pseudodifferential operators in [S.sup.1/[omega].sub.[beta]] constructed by functions (characters) [g.sub.k]: R [right arrow] (0, +[infinity]), respectively, [B.sub.k] = [F.sup.-1] [[g.sub.k] ([sigma])F], k [member of] {1, ..., m}. The functions [g.sub.k] [member of] [C.sup.[infinity]] (R) and k [member of] {1, ..., m} satisfy the conditions

[mathematical expression not reproducible]. (26)

Note that the above properties of the functions [g.sub.k] imply that [g.sub.k], k [member of] {1, ..., m}, is a multiplier in [S.sup.1/[omega].sub.[beta]].

We are looking for the solution of problems (24) and (25) via the Fourier transform. Due to condition (19),

[mathematical expression not reproducible]. (27)

We introduce the notation F[u(t, x)] = v(t, [sigma]). Given the form of the operators A, [B.sub.1], ..., [B.sub.m], we get

[mathematical expression not reproducible]. (28)

So, for the function v: [OMEGA] [right arrow] R, we arrive at a problem with parameter [sigma]:

dv(t, [sigma])/dt + a ([sigma])v(t, [sigma]) = 0, (t, [sigma]) [member of] [OMEGA], (29)

[mathematical expression not reproducible], (30)

where [??] ([sigma]) = F[f]([sigma]). The general solution of equation (29) has the form

v(t, [sigma]) = cexp{-ta([sigma])}, (t, [sigma]) [member of] [OMEGA], (31)

where c = c ([sigma]) is determined by condition (30). Substituting (30) into (31), we find that

[mathematical expression not reproducible]. (32)

Now, put G(t, x) = [F.sup.-1] [Q(t, [sigma])] and

[mathematical expression not reproducible]. (3)

Then, thinking formally, we come to the relation

[mathematical expression not reproducible]. (34)

Indeed,

[mathematical expression not reproducible]. (35)

The 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. The properties of G are determined by the properties of Q because G = [F.sup.-1] [Q]. So, let us first examine the properties of the function Q(t, [sigma]) as a function of the variable [sigma].

Lemma 1. For derivatives of [Q.sub.1](t, [sigma]), (t, [sigma]) [member of] [OMEGA], the estimates

[mathematical expression not reproducible], (36)

are valid, where [gamma] = 0 if 0 < t [less than or equal to] 1 and [gamma] = 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:

[mathematical expression not reproducible], (37)

where the sum sign is applied to all the solutions in positive integers of the equation [mathematical expression not reproducible]. Put F = [e.sup.g] and g = -ta([sigma]). Then,

[mathematical expression not reproducible], (38)

where the symbol [LAMBDA] denotes the expression, and

[mathematical expression not reproducible]. (39)

Given estimate (16) is fulfilled, we find that

[mathematical expression not reproducible]. (40)

Using (40) and the Stirling formula, we arrive at the inequalities

[mathematical expression not reproducible], (41)

where [gamma] = 0 if 0 < t [less than or equal to] 1 and [gamma] = 1 if t > 1, and the values c > 1 and A > 0 do not depend on t.

Lemma is proved.

Remark 2. It follows from estimate (41) that [Q.sub.1] (t, *) [member of] [S.sub.1/[omega]] for every t > 0.

Lemma 2. The function [Q.sub.2] is a multiplier in [S.sup.2.sub.1/[omega]]

Proof. To prove the assertion, let us estimate the derivatives of [Q.sub.2]. For this we use formula (37) in which we put F = [[phi].sup.-1] and [phi] = R, where

[mathematical expression not reproducible]. (42)

Then, [Q.sub.2] = F ([phi]) = [R.sup.-1] and

[mathematical expression not reproducible]. (43)

Given the properties of [g.sub.1], ..., [g.sub.m] and inequality (41), we find that

[mathematical expression not reproducible], (44)

where we took into account that i! (j - i)! [less than or equal to] j!).

Let

[mathematical expression not reproducible], (45)

then

[mathematical expression not reproducible]. (46)

In addition, ([d.sup.p]/[dR.sup.p])[R.sup.1] = [(-1).sup.p] p![R.sup.-(p+1)] and

[mathematical expression not reproducible]. (47)

Since, by assumption, [mu] > [[summation].sup.m.sub.k=1] [[mu].sub.k] (assuming the properties of [g.sub.1], ..., [g.sub.m], and 0 < [t.sub.1] < [t.sub.2] < ... < [t.sub.m]). So,

[mathematical expression not reproducible]. (48)

It follows from the last inequality and boundedness of the function [Q.sub.2] ([sigma]) on R that [Q.sub.2] is a multiplier in [S.sup.2.sub.1/[omega]].

Corollary 1. For every t > 0, the function Q(t, [sigma]) = [Q.sub.1] (t, [sigma])[Q.sub.2] (a), [sigma] [member of] R, is an element of the space [S.sup.2.sub.1/[omega]], and the estimates

[mathematical expression not reproducible], (49)

are valid, where the constants [??] and [??] > 0 do not depend on t.

Taking into account the properties of the Fourier transform (direct and inverse) and the formula [F.sup.-1] [[S.sup.2.sub.1/[omega]]] = [S.sup.1/[omega].sub.2], we get that G (t, *) [member of] [S.sup.1/[omega].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

[mathematical expression not reproducible]. (50)

So,

[mathematical expression not reproducible]. (51)

Applying the Leibniz formula for differentiating the product of two functions and estimating the derivatives of the function Q(t, [sigma]), we find that

[mathematical expression not reproducible], (52)

where [m.sub.ks] = [k.sup.2k] [s.sup.s/w]. Taking into account the results in (see [12], p. 236-243), we find that this double sequence satisfies the inequality

[mathematical expression not reproducible]. (53)

Bearing in mind the last inequality and also that t > 1, we obtain

[mathematical expression not reproducible]

So,

[mathematical expression not reproducible]. (55)

Then,

[mathematical expression not reproducible], (56)

where the constants [c.sub.3], [bar.B], and [a.sub.0] > 0 do not depend on t; here, we used the well-known inequality (see [12], p. 204)

[mathematical expression not reproducible], (57)

in which [alpha] = 2 and L = [??]. Thus, such a statement is correct.

Lemma 3. The derivatives (by variable x) of the function G(t, x) for t > 1 satisfy the inequality

[mathematical expression not reproducible], (58)

where the constants [c.sub.3], [bar.B], and [a.sub.0] > 0 do not depend on t.

Remark 3. In what follows, we assume that in condition (25), [beta] = 2.

Here are a few more properties of the function G(t, x).

Lemma 4. The function G(t, *), t [member of] (0, +[infinity]), as an abstract function of t with values in the space [S.sup.1/[omega].sub.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.sup.2.sub.1/[omega]], is differentiable in t. In other words, it is necessary to prove that the boundary value relation

[mathematical expression not reproducible], (59)

is performed in the sense as follows:

(1) [mathematical expression not reproducible], uniformly on each segment [a, b] [subset] R

(2) [mathematical expression not reproducible], where the constants [bar.c], [bar.B], and [bar.a] > 0 do not depend on [DELTA]t if [DELTA]t is small enough

The function Q(t, [sigma]), (t, [sigma]) [member of] [OMEGA], is differentiable in t in the usual sense. Due to the Lagrange theorem on finite increments,

[mathematical expression not reproducible]. (60)

[mathematical expression not reproducible], (61)

[mathematical expression not reproducible]. (62)

Since

[mathematical expression not reproducible], (63)

it follows from this and estimate (49) that

[mathematical expression not reproducible], (64)

uniformly on an arbitrary segment [a, b] [subset] R. Then,

[mathematical expression not reproducible], (65)

uniformly on each segment [a, b] c R, too. Therefore, condition 1 is satisfied.

Taking into account (61), estimates (15), (16), and (49) for a ([sigma]), Q(t, [sigma]), and their derivatives, we find that

[mathematical expression not reproducible], (66)

where [epsilon] > 0 is arbitrary fixed. Take [epsilon] = (t/2), and then

[mathematical expression not reproducible], (67)

where [bar.B] = 2 max {[B.sub.1], [??][t.sup.[gamma]]}, [bar.a] = (t/2), and all the constants do not depend on [DELTA]t.

Corollary 2. The formula

[mathematical expression not reproducible], (68)

is correct.

Proof. According to the definition of a convolution of a generalized function with a test one, we have

[mathematical expression not reproducible]. (69)

Then,

[mathematical expression not reproducible]. (70)

Due to Lemma 4, the relation

[mathematical expression not reproducible], (71)

is performed in the sense of [S.sup.1/[omega].sub.2] topology. So,

[mathematical expression not reproducible]. (72)

The statement is proved.

Since

[mathematical expression not reproducible], (73)

we get the formula

[mathematical expression not reproducible], (74)

where a (0) = 1 is taken into account.

Lemma 5. In ([S.sup.1/[omega].sub.2])', the following relations are correct:

[mathematical expression not reproducible], (75)

where [delta] is the Dirac delta function.

Proof. (1) Taking into account the continuity property of the Fourier transform (direct and inverse) in spaces of S' type, it is sufficient to establish that

[mathematical expression not reproducible], (76)

in the space ([S.sup.2.sub.1/[omega]])'. To do this, we take an arbitrary function [phi] [member of] [S.sup.2.sub.1/[omega]], and using the fact that [Q.sub.2] is a multiplier in the space [S.sup.2.sub.1/[omega]] and the Lebesgue theorem on the limit passage under the Lebesgue integral sign, we find that

[mathematical expression not reproducible]. (77)

This leads us to statement (1) of Lemma 5.

(2) Given statement (1) and the form of the operators [B.sub.1], ..., [B.sub.m], we find that

[mathematical expression not reproducible]. (78)

Therefore, relation (75) is fulfilled in the space ([S.sup.1/[omega].sub.2])'. Lemma 5 is proved.

Remark 4. If [mu] = 1, [[mu].sub.1] = ... = [[mu].sub.m] = 0, then problems (24) and (25) degenerate into the Cauchy problem for equation (24). In this case, [Q.sub.2] ([sigma]) = 1, [for all][sigma] [member of] R, G(t, x) = [F.sup.1] [[e.sup.- ta([sigma])]], and G(t, *) [right arrow] [F.sup.-1] [1] = [delta] for t [right arrow] +0 in the space ([S.sup.1/[omega].sub.1])'.

Corollary 3. Let

[mathematical expression not reproducible], (79)

where ([S.sup.1/[omega].sub.2,*])' is a class of convolutors in [S.sup.1/[omega].sub.2]. Then, in ([S.sup.1/[omega].sub.2])', the following boundary relation holds:

[mathematical expression not reproducible]. (80)

Proof. Let us prove that the relation

[mathematical expression not reproducible] (81)

takes place in the space ([S.sup.2.sub.1/[omega]])'. Since f [member of] ([S.sup.1/[omega].sub.2,*])' and G (t, *) [member of] [S.sup.1/[omega].sub.2] for every t > 0, we have

F[-(t, *)] = F[f * G(t, *)] = F[f]F[G(t, *)] = F[f]Q(t, *). (82)

So, we need to prove that

[mathematical expression not reproducible], (83)

in ([S.sup.2.sub.1/[omega]])', when t [right arrow] + 0. Since Q (t, *) = [Q.sub.1] (t, *)[Q.sub.2] (*) [right arrow] [Q.sub.2] (*) for t [right arrow] + 0 in the space ([S.sup.2.sub.1/[omega])' (see the proof of assertion 1 and Lemma 5), the correlation

[mathematical expression not reproducible], (84)

is realized in the space ([S.sup.2.sub.1/[omega]])'. Thus, relation (81) and (80) are fulfilled in the corresponding spaces. The statement is proved.

The function G(t, *) satisfies equation (24) as t > 0. Indeed,

[mathematical expression not reproducible]. (85)

So,

[partial derivative]G(t, x)/[partial derivative]t + AG (t, x) = 0, (t, x) [member of] [OMEGA], (86)

which is what had to be proved.

It follows from Corollary 3 that the nonlocal multipoint by the time problem for equation (24) can be formulated in the following way: find a function u(t, x), (t, x) [member of] [OMEGA] that satisfies equation (24) and the condition

[mathematical expression not reproducible], (87)

where boundary relation (87) is considered in the space ([S.sup.1/[omega].sub.2])', and the constraints on the parameters [mu], [[mu].sub.1], ..., [[mu].sub.m], [t.sub.1], ..., [t.sub.m] are the same as in case of problems (24) and (25)).

Theorem 1. By nonlocal multipoint by time problem (24), (87) is solvable and its solution is given by the formula

u(t,x) = f * G(t, x), (t, x) [member of] [OMEGA], (88)

where u (t, *) [member of] [S.sup.1/[omega].sub.2] for every t > 0.

Proof. Make sure that the function u(t, x), (t, x) [member of] [OMEGA], satisfies equation (24). Indeed (see Corollary 2),

[mathematical expression not reproducible]. (89)

Since f is a convolutor in [S.sup.1/[omega].sub.2],

F[f * G(t, *)] = F[f]F[G(t, *)] = F[f]Q(t, *). (90)

So,

[mathematical expression not reproducible]. (91)

It follows from this that the function u(t, x), (t, x) [member of] [OMEGA], satisfies equation (24). From Corollary 3, we get that u (t, x) satisfies condition (87) in the defined sense. The theorem is proved.

Remark 5. If, in condition (87), [B.sub.1] = ... = [B.sub.m] = I (I is the identity operator), then we can prove that problems (24) and (87) are well posed and its solution is given by the formula

u(t,x) = f * G(t,x), f [member of] ([S.sup.1/[omega].sub.1,*])', (t, x) [member of] [OMEGA], (92)

where G(t, *) = [F.sup.-1] [Q(t, *)] [member of] [S.sup.1/[omega].sub.1] for every t > 0, and

[mathematical expression not reproducible]. (93)

Theorem 2. Suppose u(t, x), (t, x) [member of] [OMEGA], is the solution of problems (24) and (87). Then, u(t, *) [right arrow] 0 in the space ([S.sup.1/[omega].sub.2])' as t [right arrow] +[infinity].

Proof. Recall that the solution of problems (24) and (87) is given by the formula

[mathematical expression not reproducible].

Let us prove that <u (t, *), [phi]> [right arrow] 0 as t [right arrow] +[infinity] for an arbitrary function [phi] [member of] [S.sup.1/[omega].sub.2]. Put

[mathematical expression not reproducible]. (95)

In these notations, we prove that (a) for every t > 1 and R > 0, the function [[PSI].sub.t,R] ([xi]) belongs to the space [S.sup.1/[omega].sub.2]] and [[PSI].sub.t,R] (t) [right arrow] [[PSI].sub.t] ([xi]) in the space [S.sup.1/[omega].sub.2] as R [right arrow] [infinity]; (b) [[PSI].sub.t]([xi]) [member of] [S.sup.1/[omega].sub.2] for every t > 1. From this, we get

[mathematical expression not reproducible], (96)

where u (t, *) is interpreted as a regular generalized function from ([S.sup.1/[omega].sub.2])' for every t > 0.

So, let us set property (a). For fixed {k, m} [subset] [Z.sub.+], we have

[mathematical expression not reproducible]. (97)

Since [phi] [member of] [S.sup.1/[omega].sub.2], the inequality

[mathematical expression not reproducible], (98)

with some c, L, M > 0, is valid. It follows from this that for every [eta] [member of] R,

[mathematical expression not reproducible]. (99)

Next, we will use estimate (58). Then,

[mathematical expression not reproducible]. (100)

By the direct calculations, we find that

[mathematical expression not reproducible]. (101)

So,

[mathematical expression not reproducible], (102)

where [bar.c] = c[c.sub.3][c.sub.4][t.sup.1-(1/[omega])] and L = 2max{L, [??]t}. Therefore, [[PSI].sub.t,R] ([xi]) [member of] [S.sup.1/[omega].sub.2] for every t > 1 and an arbitrary R > 0. Furthermore, we make sure that [[PSI].sub.t,R] ([xi]) [right arrow] [[PSI].sub.t] ([xi]) as R [right arrow] [infinity], with all its derivatives uniformly in [xi] on each segment [a, b] [subset] R. In addition, the set of functions {[[xi].sup.k][D.sup.m.sub.[xi]][[PSI].sub.t,R] ([xi]), {k, m} [subset] [Z.sub.+], is uniformly bounded (by R) in the space [S.sup.1/[omega].sub.2] (this property follows from estimate (102) in which [bar.c], [bar.B], and [bar.L] > 0 do not depend on R). This means that condition (a) is fulfilled.

Condition (b) follows from (a) because, in a perfect space, every bounded set is compact.

Using properties (a) and (b), we obtain the equality

[mathematical expression not reproducible]. (103)

Since f [member of] ([S.sup.1/[omega].sub.2])' is a convolutor in [S.sup.1/[omega].sub.2], we have f * [??] [member of] [S.sup.1/[omega].sub.2]. Then, given estimate (58) (at s = 0), we find that

[mathematical expression not reproducible], (104)

for an arbitrary function [phi] [member of] [S.sup.1/[omega].sub.2], i.e., u (t, *) [right arrow] 0 in the space ([S.sup.1/[omega].sub.2]))' as t [right arrow] +[infinity]. The theorem is proved.

If the generalized function f in (87) is finite (i.e., its support supp f is a finite set in R), then we can say on uniform tending a solution u (t, x) of problems (24) and (87) to zero on R as t [right arrow] + [infinity]. Note that every finite generalized function is a convolutor in S spaces. This property follows from the general result concerning the theory of perfect spaces (see [12], p. 137): if [PHI] is a perfect space with differential translation operation, then every finite functional is a convolutor in [PHI]. Finite functionals form a fairly wide class. In particular, every bounded set F [subset] R is a support of some generalized function [12].

Theorem 3. Let u(t, x) be a solution of problems (24) and (87) with the boundary function f in condition (87), which is an element of ([S.sup.[beta].sub.2])' [subset] ([S.sup.1/[omega].sub.2])', [beta] > 1, with finite support in R. Then, u(t, x) [right arrow] 0 uniformly on R as t --> +to.

Here is a scheme for proving this statement. Let supp f [subset] [[a.sub.1], [b.sub.1]] [subset] [[a.sub.2], [b.sub.2]] [subset] R. Consider the function [phi] [member of] [S.sup.[beta].sub.2], [beta] > 1, such that [phi] (x) = 1, x [member of] [[a.sub.1], [b.sub.1]], and supp [phi] [subset] [[a.sub.2], [b.sub.2]]. This function exists when [beta] > 1 because [S.sup.[beta].sub.2] contains finite functions [12]. Consider the function

[mathematical expression not reproducible], (105)

where [gamma] = 1 - [phi]. Since supp ([gamma] ([xi])G (t, x - [xi])) n supp f = 0, we have

[mathematical expression not reproducible]. (106)

To prove the formulated above statement, it remains to establish that the set of functions [[PHI].sub.t,x] ([xi]) = [t.sup.1/[omega]] [phi]([xi])G(t, x - [xi]) is bounded in the space [S.sup.[beta].sub.1], [beta] > 1, for large t and x [member of] R.

For example, if, in condition (87), f = [delta], then [delta] is a convolutor in the space S, supp 8 = {0}, and u(t, x) = 8 * G(t, x) = G(t, x). It follows directly from estimate (58) that G(t, x) [right arrow] 0 uniformly on R as t [right arrow] +[infinity].

4. Conclusion

In this paper, the solvability of a nonlocal multipoint by the time problem for the evolutionary equation [mathematical expression not reproducible] is proved, herewith the operator [(I - ([d.sup.2]/d[x.sup.2])).sup.[omega]/2] is interpreted as a pseudodifferential operator in the space SO/10 constructed by function [(1 + [[sigma].sup.2]).sup.w/2], [sigma] [member of] R, while the nonlocal multipoint by time condition also contains pseudodifferential operators constructed on smooth symbols. The 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.sup.1/[omega].sub.2]). The behavior of the solution u(t, x), t [right arrow] + [infinity] in the space of generalized functions ([S.sup.1/[omega].sub.2])', is investigated. The conditions for the initial generalized function are found under which the solution is uniformly stabilized to zero on R. The method of research of a nonlocal multipoint by time problem offered in this paper allows to interpret differential-operator equations of a form ([partial derivative]u/[partial derivative]t) + [phi](i[partial derivative]/[partial derivative]x)u = 0 as an evolutionary equation with a pseudodifferential operator [phi](i[partial derivative]/[partial derivative]x) = [F.sup.-1] [[phi] x F] constructed by the function f 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 [phi]f(i[partial derivative]/[partial derivative]x)).

https://doi.org/10.1155/2020/1673741

Data Availability

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

Disclosure

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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V. V. Gorodetskiy [ID], R. S. Kolisnyk, and N. M. Shevchuk [ID]

Chernivtsi National University, Chernivtsi, Ukraine

Correspondence should be addressed to V. V. Gorodetskiy; v.gorodetskiy@chnu.edu.ua

Received 25 February 2020; Revised 31 May 2020; Accepted 10 June 2020; Published 18 July 2020

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Title Annotation:Research Article
Author:Gorodetskiy, V.V.; Kolisnyk, R.S.; Shevchuk, N.M.
Publication:International Journal of Differential Equations
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Date:Jul 31, 2020
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