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 type .
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 , which impose certain conditions for decreasing on infinity and increasing derivatives with increasing of order. These conditions are set using inequalities , , where is some sequence of positive numbers depending on the function . If sequence has a special form then we get a certain subclass of spaces of Schwartz space of rapidly decreasing functions on . In [1], the case where are fixed parameters is studied in detail; the corresponding spaces are called spaces of type and are denoted by the symbol . Functions from these spaces and all their derivatives decrease on the real axis as faster than 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 and spaces of the type that are topologically dual to the spaces 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 is a function , for every , this function is an element of the space as a function of ([7], p. 46) and the space of a space of the type .
If , where are some sequences of positive numbers, then we have generalized spaces of type , denoted by the symbol . The spaces (their topological structure, properties of functions, and basic operations in such spaces) were studied in [9]. Known spaces of type , 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 in the particular choice of sequences and (see [12]). From the results given in [9, 13], it follows that generalized spaces of type 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 (in particular, spaces ) with the corresponding topology are called spaces of type 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 .
2. The Spaces of Test Functions
Here, we dwell on the spaces , constructed by the sequences of the form , , where is the sequence of positive numbers, which has the following properties: (a)the sequence is monotonically decreasing(b)(c)(d)
The sequence , also has properties (a)–(d), with condition (b) having the form: . An example of a sequence with properties (a–d) can be a sequence , where is a fixed parameter. For example, let us check for the sequence of property (d). We have that where . If we take arbitrary and put , then we get the inequality , where . Note that condition (b) for this sequence is satisfied with the parameter .
We also consider that the parameters , in condition (b) for the sequences and are related by condition (e): .
By we denote a collection of functions , satisfying the condition
coincides with the union of the countably-normed spaces by all indices , where the symbol denotes a collection of those functions , that for arbitrary satisfy the inequalities the system of norms in is determined by formulas
It was established in [9] that the function belongs to the space , where , , if and only if it extends analytically into the complex plane to the whole function , which satisfies the condition: where
We note that is a continuously differentiable, even function in that increases monotonically over the interval . It follows from property (d) (see [9]) that
For example, if then . In addition, as it is proved in [9], is a convex function on in the sense that
The inequality follows from (8).
The function in (6) is related to the sequence , which constructs the sequence as follows [9]: where is the solution of the equation , The function is a nonnegative and continuous function on the interval ; herewith , , , i.e., . From properties of the function , it follows that increases faster than any linear function on the interval , as ; i.e., . Provided that , we obtain that possesses a unique solution . The sequence of solutions is increasing and it is unbounded. Indeed, suppose it is not, for example, , then we select a convergent subsequence , such that , ; so we obtain a contradiction, since and passing to the limit as we get .
Since , where , and , if then is a continuously differentiable, even function on that monotonically decreases on , . For example, if , then the following inequalities are true ([1], p. 204)
Function satisfies on the inequality [9].
It follows from the results given in [9], that the sequence converges to zero in this space if the functions and their derivatives of any order converge to zero uniformly on every segment and satisfy the inequality where the constants are independent of .
A function is called a multiplicator in the space if for any function and the mapping is a linear and continuous operator from into A function that admits an analytic extension onto the entire complex plane and satisfies the condition [9]: is a multiplicator in the space . The operators of multiplication by , all polynomials, operators of differentiation, shift, and extension are defined and continuous in the spaces [9].
By , we denote the collection of all even functions from the space . Since forms a subspace of , then the topology is naturally introduced in . This space with the corresponding topology is called a main space or a generalized space of type , and its elements are called test functions.
By , we denote the collection of functions that are extensions of functions from space into . According to the results obtained in [9], the space can be represented as a union of the countably-normed spaces , where consists of those functions for which inequality is true, where is any positive constant less than , is any constant greater than ; are constants of inequality (5). If for , we put then, these norms are equivalent to the corresponding norms in space . Therefore, the sequence of functions converges to zero if and only if the sequence of functions , converges to zero uniformly in every bounded domain of the complex plane , the inequalities are true, where constant are independent of [9].
Every integer even function satisfying condition (13) is the multiplicator in the space . An example of a multiplicator in is the normalized Bessel function , which is the solution of the equation , where is Bessel operator; is fixed parameter, provided that . Indeed, the normalized Bessel function is related to the ordinary Bessel function , of the first kind, so [15]:
It is known (see [15]) that the function admits an analytic extension into a complex plane , and the Poisson integral formula holds
It follows from relations (16) and (17) that the normalized Bessel function of the complex argument is an integer even function and for the integral image is correct:
In view of , and by using (18), we obtain estimate:
Since for any convex functions and and for any , the inequality is true, it follows that
It implies that is a multiplicator in space , , .
According to the results presented in [16], the direct and inverse Bessel transforms are defined in the spaces moreover, if the conditions (a)–(e) are satisfied for the sequences and , then the formula is true, moreover, operator is continuous [16]. The spaces are partial kind of spaces . The spaces consist of even functions of spaces with the same topology; accordingly, the formula is correct.
By , we denote the generalized shift operator corresponding to the Bessel operator [17]: where . Moreover, as it was proved in [18], the operation of a generalized shift is differentiable (even infinitely differentiable) in the space .
We define the convolution of two functions of space by a formula
The formula is true [18].
We note that the operation of multiplication of test functions is defined and continuous in the spaces .
The spaces form topological algebras with respect to the convolution of test functions.
Let us consider the pseudodifferential operator in the space . Provided that is a multiplicator in space , the operator is linear and continuous in space . It turns out that if we consider the operator in space , then it can be understood as a Bessel operator of “infinite order” in this space (see [18]):
3. The Space of Generalized Functions
We denote by 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 is determined by the formula
Every locally integrated even function on , which satisfies condition generates a regular generalized function : .
The following statement is correct: if locally integrated even functionsandonsatisfying the condition (28) do not coincide on the set of Lebesgue positive measure, then there exists a functionsuch that, i.e.,. On the contrary, ifthen the functionsanddo 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 generated by them from space . 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 , we define the convolution of a generalized function with a test function by the formula (the index in means that the functional acts upon the test function as a function of the argument ).
Let . If , and the relation as in the topology of the space implies that as in the topology of the space , then the functional is called a convolver in the space .
The Bessel transform of a generalized function is determined by the relation
In view of (31), the properties of the linearity and continuity of the functional and the properties of the Bessel transform of the test functions, the functional is linear and continuous in the space of the test functions . Thus, the Bessel transform of the generalized function defined on is a generalized function on the space .
If a generalized function is a convolver in the space , then for any function , the relation is true [18].
The following statement implies the following properties: (1) if the generalized function is a convolver in space then its Bessel transform is a multiplicator in the space ; (2) if the generalized function f is a multiplicator in the space then its Bessel transform is a convolver in the space .
4. A Nonlocal Multipoint in Time Problem
Let us consider an evolutionary equation where is a pseudodifferential operator in the , constructed according to a function , which is a multiplicator in the space and such that . Note that can be understood as a Bessel operator of infinite order of appearance (see Section 2). By , we denote the class of functions (symbols) 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 where , are fixed numbers and, moreover, We seek the solution of problem (33), (34) with the help of the Bessel transform specified as follows For the function , we get the following problem with parameter where The general solution of equation (35) has the form where is determined from condition (36). Substituting (37) in (36), we obtain
Then
Thus, the solution of problem (33), (34) has the form
We introduce the notation where
Hence, as a result of formal reasoning, we find
Indeed,
Since we have
The correctness of these transformations follows from the properties of the function presented in what follows. The properties of the function are connected with properties of the function , because . Thus, first of all, we study the properties of the function regarded as a function of the argument .
Since , therefore . Then (see Section 2) there exist numbers such that
Further, we assume that the constant in (45) satisfies the condition: , where is the parameter of the multipoint problem (33), (34). Then
Lemma 1. Let . The following estimates are true for the function and its derivatives (with respect to the variable ) on : where the constants are independent of , . The function belongs to the space for a fixed .
Proof. The inequality
is true for fixed . This property follows from the relation
where , and is a nonnegative, continuous function on , monotone increasing on Then
If , then the inequality is true. Then, and
where If , then On this
Let if is not integer and , if is an integer, Then, the inequality
is true for . Hence, it follows that for every
In the following considerations, we will use the estimate
In view of the integral Cauchy formula, we obtain
where is a circle of radius centered at a point . By using (56), we obtain the inequalities
where is a point of maximum of the function , . Since is an even function on that increases on the interval , then
In view the inequality , we prove that there exist constants such that
where Therefore,
We used the fact that , and the inequality of convexity for the function .
For any , the function is differentiable on and we obtain the following relations
from the properties of the function . Since , this function attains its infimum. Thus,
Lemma 1 is proved.
Lemma 2. The function is a multiplicator in the space .
Proof. In view of (46), the inequalities are true. Since moreover, hence, by using the polynomial formula, we get where . By using this result and (46), we obtain the inequalities where . Further, we use the formula Then where By using the last inequality and boundedness of the function on , we conclude that is a multiplicator in space . Lemma 2 is proved.
By using relations (56), (71) and the Leibniz formula of differentiation of the product of two functions, we obtain where the constants are independent of . By virtue of the last inequality, we conclude that the function (regarded as a function of ) is an element of the space (for any ).
In view of the relation , we conclude that for every .
In the estimates for the derivatives of the function (with respect to the variable ), we select the dependence on the parameter .
For this, we note (see [16]) that functions from space satisfy the condition
On the converse, if infinitely differentiable, even function on satisfies condition (73), then (see [16]) is an element of the space . In view of this observation, we estimate the function for fixed . To do this, we use the relation established in [16]: from the last relation implies that
We note also that for the function , the following formula is true where are coefficients dependent on , the functions , , are also elements of space . Then
We note that inequality is true, where , if is not integer and , if 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
In addition, where (here taken into account ) for every . 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 (see Section 2) there follows the inequality
Thus (here taken into account , see Section 2). By using (79) and the last inequality, we get
Similarly, by using (80), we have
This yields where , ,
Since then Moreover, we can assume that , i.e., So, where , The other additions in (77) are evaluated similarly. As a result, we obtain the inequality where the constants are independent of . Thus, it takes into account that , Hence where the constants are independent of . Thus, the following statement is true.
Lemma 3. The following inequalities are true for the function , for every : where the constants are independent of .
The function is differentiable with respect to on the interval . Indeed, since then, formally differentiating (93) under the sign of the integral, we obtain the function Since is a multiplicator in the space then
In addition, by (72), we obtain the estimate where , if is not integer and , if is integer. Hence,
It follows from the inequality of the convexity of the function that for . Since then the integrated function is a majorant for . Therefore, the integral of the derivative (with respect to the variable ) of the integrand in (93) converges uniformly on any interval and therefore the derivative with respect to under the sign of the integral in (93) can be applied at every point .
Lemma 4. The function , regarded as an abstract function of the parameter with values in the space , is differentiable respect to .
Proof. In view of the continuity of the direct and inverse Bessel transforms in spaces of the type , to prove the lemma, it is sufficient to show that the function , as an abstract function of the parameter with values in the space , is differentiable with respect to . In other words, it is necessary to that the limit relation
is true in a sense that:
(1) uniformly on every segment (2) where the constants are independent of if is sufficiently smallThe function , is differentiable with respect to 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 . Then
as uniformly on any segment . Thus, condition (1) from relation (99) is satisfied.
Since is a multiplicator in the space , then
Because of the Cauchy integral formula, we have that
where is a circle of radius centered at the point . Then, in view (106), we obtain inequalities
For sufficiently large values of , the inequality is true. Since the function increases monotonically for , then at the same values of For all values of , the following inequality is true:
Therefore, for Further, for a given , we assume that . Then
By using (112) and estimates for the derivatives of the functions , we find
where , if is not an integer; , if is an integer, , ; and Take . In view of the inequality of convexity for the function , we obtain
Thus
moreover, constants are independent of (for sufficiently small ). The case is proved similarly. Lemma is proved.
Corollary 5. The formula is true.
Proof. By the definition of a convolution of a generalized function with the test function, we find Then By virtue of Lemma 4, the limit relation is true in a sense of convergence in the topology of the space , thus Corollary is proved.
Lemma 6. In the space , the following limit relation is true where is the Dirac delta function.
Proof. In view of the continuity of the Fourier transform and the function regarded as an abstract function of the parameter with values in the space , we replace the relation (121) by the limit relation in the space . By using the representation of the function , we rewrite relation (122) in the form To prove relation (123), we take an arbitrary function , apply the theorem on the limit transition under the sign of Lebesgue integration, we find This implies that relation (123) is true in the space . Therefore, the relation (121) holds. Lemma is proved.
By , we denote a class of generalized functions from that are convolvers in the space .
Corollary 7. Let Then, the relation is true in the space .
Proof. Since is convolver in the space , we obtain ( is multiplicator in the space ). By using this fact and the property of the continuity of the Bessel transform, we write (125) in the form (the relation is considered in space ). In view (123), we obtain (125). The statement is proved.
The function is a solution of equation (33). Indeed is convolver in the space , we get
On the other hand (see Corollary 5)
This implies that the function satisfies equation (33) in the ordinary sense. By Corollary 7, the nonlocal -point in time problem for equation (33) can be formulated as follows: to find a solution of equation (33) satisfying the condition where the limit relation (130) is considered in space (the restriction imposed on the parameters are the same as in problem (33), (34)).
It follows from the above that the function , is a solution of (33). If then i.e., is also a solution of (33). By using this fact and relation (121), in what follows the function 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 for every .
Proof. The function satisfies (33). The solution continuously depends on the function in condition (130) in the sense that if and as in the space , then , in the space . 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
where is the restriction of operator adjoint to the operator to the space . We understand condition (132) in a weak sense. The Cauchy problem (131), (132) is solvable, moreover for every .
Let be an operator that associates a functional with a solution of problem (131), (132). The operator is linear and continuous, it is defined for any and such that and has the properties
(the limit is considered in the space ).
Let us consider a solution of problem (33), (130) understood as a regular functional from the space . We prove that problem (33), (130) may have only one solution in space . To this end, it suffices to prove that only the functional (for any ) can be a unique solution of Eq. (33) with the trivial boundary condition (130). We apply the functional to a function , where is an arbitrarily fixed element from the space . Further, differentiating with respect to and using Eq. (33), (131) we get
This implies that is constant. By using properties of abstract functions, we obtain the relation
at any point .Thus, if in (130) then
This implies that . Suppose it is not. For example, Then we get a relation where i.e., Since is any constant, and by are fixed parameters, and the obtained contradiction proves that . Similarly, we prove that Hence, for any , i.e., is a zero functional from the space . Since and is arbitrary then for all . Theorem is proved.
As an example, we consider equation (33) with the operator , constructed on the basic of the function . In this case, we obtain and equation (33) is the equation with the Bessel operator
The function is an element of the space . Indeed, , because
From the characteristic of the spaces and (139) imply that , where , , i.e., . In addition, the function , is a multiplicator in the space . In this case, the constant in inequality (45) is equal to one, i.e., the condition , is satisfied. By the above theorem, the nonlocal -point in time problem for equation (138) is correctly solvable if , herewith , where where . In particular, if (the case of two-point problem), then
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 in the case where the initial function is an element of the space of generalized functions of type is established in this paper. The properties of the fundamental solution this problem are investigated.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this article.
Acknowledgments
This study was conducted in the framework of postgraduate study in Yuriy Fedkovych Chernivtsi National University.