Abstract
In the paper, we investigate a nonlocal multipoint by a time problem for the evolution equation with the operator , , and is a fixed parameter. The operator is treated as a pseudodifferential operator in a certain space of type . The solvability of this problem is proved. 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 ultradistribution type. The properties of the fundamental solution are investigated. The behavior of the solution at (solution stabilization) in the spaces of generalized functions of type and the uniform stabilization of the solution to zero on are studied.
1. Introduction
A rather wide class of differential equations with partial derivatives forms linear parabolic and -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 , , , where is a function (symbol) that satisfies certain conditions and is the direct (inverse) Fourier or Bessel transform. The PDO includes differential operators, fractional differentiation and integration operators, convolution operators, and the Bessel operator , , which contains the expression in its structure and is formally represented as , where is the Bessel integral transformation.
If is a nonnegative self-adjoint operator in a Hilbert space , then it is known [1] that a continuous on function is continuously differentiable solution of the operator differential equation , , which refers to abstract equations of parabolic type, if and only if it is given as , . It turns out [1] that all continuously differentiable functions within the interval solutions of this equation are described by the same formula where is an element of the wider than space conjugate to the space of analytic vectors of the operator ; the role of is played by the extension of the operator to the space , and the boundary value of at the point 0 exists in the space .
If , , then is a nonnegative self-adjoint operator in , since is a self-adjoint in operator with the domain . If , , is the spectral function of the operator , then, due to the basic spectral theorem for self-adjoint operators,
It is known (see, for example, [2]) that
It follows from this that . So,
Following [3], we call the operator the Bessel operator of fractional differentiation. Therefore, can be understood as a pseudodifferential operator constructed on the function symbol , . This allows us to interpret the function that is a solution of the corresponding Cauchy problem as a convolution of the form [4, 5], where .
In this paper, we give a similar depiction of the solution of a nonlocal multipoint time problem for the equation , when the initial condition is replaced by the condition , where , , , and are fixed and are pseudodifferential operators constructed of smooth symbols (if , , , then obviously we have a Cauchy problem). This condition is interpreted in the classical or weak sense as if is a generalized function (generalized element of the operator ) of ultradistribution type. Properties of the fundamental solution of the specified multipoint problem are investigated. The behavior of the solution at (solution stabilization) in the spaces of generalized functions of type and the uniform stabilization of the solution to zero on 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 . They consist of infinitely differentiable on functions, which satisfy certain conditions on the decrease at infinity and the growth of derivatives. These conditions are given by the inequalities , , , where is some double sequence of positive numbers. If there are no restrictions on elements of the sequence , then obviously we have L. Schwartz’s space of quickly descending at infinity functions. However, if the numbers satisfy certain conditions, then the corresponding specific spaces are contained in and they are called the spaces of type. Let us define some of them.
For any , , let us put
The introduced spaces can also be described as in [12].
consists of those infinitely differentiable on functions that satisfy the following inequalities:with some positive constants , , and dependent only on the function .
If and , then consists of only those , which admit an analytic extension into the complex plane such that
The space consists of functions , which can be analytically extended into some band (dependent on ) of the complex plane, so that the estimateis carried out.
The spaces , , are nontrivial if , and they form dense sets in .
The topological structure in is defined as follows. The symbol , , denotes the set of functions that satisfy the condition
This set is transformed into a complete, countably normed space, if the norms in it are defined by means of relations:
The specified norm system is sometimes replaced by an equivalent norm system:
If , , then is continuously embedded into and , that is, is endowed by the inductive limit topology of the spaces [12]. Therefore, the convergence of a sequence to zero in the space is the convergence in the topology of some space , to which all the functions belong. In other words (see [12]), in as , if and only if for every , the sequence , converges to zero uniformly on an arbitrary segment and for some , independent of , and the inequalityholds.
In , the continuous translation operation : is defined. This operation is also differentiable (even infinitely differentiable [12]) in the sense that the limit relation , , is true for every function in the sense of convergence in the -topology. In , the continuous differentiation operator is also defined. The spaces of type 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 , , is correct, where
Moreover, the operator : is continuous.
Let denote the space of all linear continuous functionals on with weak convergence. Since the translation operator is defined in the space of test functions, the convolution of a generalized function with the test function is given by the following formula:
It follows from the infinite differentiability property of the argument translation operation in that the convolution is a usual infinitely differentiable function on .
We define the Fourier transform of a generalized function by the relationwhere the operator : is continuous.
Let . If , , and the convergence in the -topology as implies that in the -topology as , then the function is called a convolutor in . If is a convolutor in , then for an arbitrary function , the formula is valid, where is a multiplier in [12].
Recall that the function is called a multiplier in if for an arbitrary function , and the mapping is continuous in the space .
3. Nonlocal Multipoint by Time Problem
Consider the function , , where is a fixed parameter. Obviously, the function satisfies the inequalitywhere for an arbitrary . Using direct calculations and the Stirling formula, we find thatwhere , , and . It follows from (15) and (16) that is a multiplier in . Indeed, let , that is, the function and its derivatives satisfy the inequalitywith some positive constants , , and . Then, using the Leibniz formula for differentiating the product of two functions, as well as inequalities (15)–(17), we find that
Since is arbitrary, we can put . Then,where and . It follows from (19) that is an element of .
The operation of multiplying by is continuous in the space . In fact, let be a sequence of functions from convergent to 0 in this space. This means that with some constants , and
In other words, for an arbitrary , there exists a number such that, for ,
Using inequalities (15) and (16) (when putting in (15), ), we getwhere and . It follows from the last inequality that , , that is, the sequence converges to zero in the space where and . This means that the sequence converges to zero in the space , which is what we needed to prove.
Remark 1. It follows from the proven property that the function , , is also a multiplier in every space , where . Therefore, in the space , , defined as a continuous linear pseudodifferential operator , constructed by the function :where , .
Let us consider the evolution equation with the operator (the Bessel fractional differentiation operator):By a solution of equation (24), we mean the function , , such that (1) for every ; (2) for every ; (3) is continuous at every point of the boundary of the region ; (4) : : , ; and (5) , satisfies equation (24).
For equation (24), we consider the nonlocal multipoint by time problem of finding a solution of equation (24) that satisfies the conditionwhere , , , , and are fixed numbers, , , and are pseudodifferential operators in constructed by functions (characters) : , respectively, , . The functions and satisfy the conditionsNote that the above properties of the functions imply that , , is a multiplier in .
We are looking for the solution of problems (24) and (25) via the Fourier transform. Due to condition (19),We introduce the notation . Given the form of the operators , , , , we getSo, for the function , we arrive at a problem with parameter :where . The general solution of equation (29) has the formwhere is determined by condition (30). Substituting (30) into (31), we find thatNow, put and , and thus,Then, thinking formally, we come to the relationIndeed,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 , which are given below. The properties of are determined by the properties of because . So, let us first examine the properties of the function as a function of the variable σ.
Lemma 1. For derivatives of , , the estimatesare valid, where if and if , and the constants and do not depend on .
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 , . Put and . Then,where the symbol denotes the expression, andGiven estimate (16) is fulfilled, we find thatUsing (40) and the Stirling formula, we arrive at the inequalitieswhere if and if , and the values and do not depend on .
Lemma is proved.
Remark 2. It follows from estimate (41) that for every .
Lemma 2. The function is a multiplier in .
Proof. To prove the assertion, let us estimate the derivatives of . For this we use formula (37) in which we put and , whereThen, andGiven the properties of and inequality (41), we find thatwhere we took into account that ).
LetthenIn addition, andSince, by assumption, (assuming the properties of , and ). So,It follows from the last inequality and boundedness of the function on that is a multiplier in .
Corollary 1. For every , the function , , is an element of the space , and the estimatesare valid, where the constants and do not depend on .
Taking into account the properties of the Fourier transform (direct and inverse) and the formula , we get that for every . We remove in the estimates of derivatives of the function (in the variable ), the dependence on , assuming . To do this, we use the relations
So,
Applying the Leibniz formula for differentiating the product of two functions and estimating the derivatives of the function , we find thatwhere . 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 , we obtain
So,
Then,where the constants , , and do not depend on ; here, we used the well-known inequality (see [12], p. 204)in which and . Thus, such a statement is correct.
Lemma 3. The derivatives (by variable ) of the function for satisfy the inequalitywhere the constants , , and do not depend on .
Remark 3. In what follows, we assume that in condition (25), .
Here are a few more properties of the function .
Lemma 4. The function , , as an abstract function of with values in the space , is differentiable in .
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 , as a function of a parameter with values in the space , is differentiable in . In other words, it is necessary to prove that the boundary value relationis performed in the sense as follows:(1), , uniformly on each segment (2), where the constants do not depend on if is small enoughThe function , , is differentiable in in the usual sense. Due to the Lagrange theorem on finite increments,So,Sinceit follows from this and estimate (49) thatuniformly on an arbitrary segment . Then,uniformly on each segment , too. Therefore, condition 1 is satisfied.
Taking into account (61), estimates (15), (16), and (49) for , , and their derivatives, we find thatwhere is arbitrary fixed. Take , and thenwhere , , and all the constants do not depend on .
Corollary 2. The formulais correct.
Proof. According to the definition of a convolution of a generalized function with a test one, we haveThen,Due to Lemma 4, the relationis performed in the sense of topology. So,The statement is proved.
Sincewe get the formulawhere is taken into account.
Lemma 5. In , the following relations are correct:where is the Dirac delta function.
Proof. (1) Taking into account the continuity property of the Fourier transform (direct and inverse) in spaces of type, it is sufficient to establish thatin the space . To do this, we take an arbitrary function , and using the fact that is a multiplier in the space and the Lebesgue theorem on the limit passage under the Lebesgue integral sign, we find thatThis leads us to statement (1) of Lemma 5.
(2) Given statement (1) and the form of the operators , we find thatTherefore, relation (75) is fulfilled in the space .
Lemma 5 is proved.
Remark 4. If , , then problems (24) and (25) degenerate into the Cauchy problem for equation (24). In this case, , , , and for in the space .
Corollary 3. Letwhere is a class of convolutors in . Then, in , the following boundary relation holds:
Proof. Let us prove that the relationtakes place in the space . Since and for every , we haveSo, we need to prove thatin , when . Since for in the space (see the proof of assertion 1 and Lemma 5), the correlationis realized in the space . Thus, relation (81) and (80) are fulfilled in the corresponding spaces. The statement is proved.
The function satisfies equation (24) as . Indeed,So,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 , that satisfies equation (24) and the conditionwhere boundary relation (87) is considered in the space , and the constraints on the parameters , , , , , , 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 formulawhere for every .
Proof. Make sure that the function , , satisfies equation (24). Indeed (see Corollary 2),Since is a convolutor in ,So,It follows from this that the function , , satisfies equation (24). From Corollary 3, we get that satisfies condition (87) in the defined sense. The theorem is proved.
Remark 5. If, in condition (87), ( is the identity operator), then we can prove that problems (24) and (87) are well posed and its solution is given by the formulawhere for every , and
Theorem 2. Suppose , , is the solution of problems (24) and (87). Then, in the space as .
Proof. Recall that the solution of problems (24) and (87) is given by the formulaLet us prove that as for an arbitrary function . PutIn these notations, we prove that (a) for every and , the function belongs to the space and in the space as ; (b) for every . From this, we getwhere is interpreted as a regular generalized function from for every .
So, let us set property (a). For fixed , we haveSince , the inequalitywith some , , , is valid. It follows from this that for every ,Next, we will use estimate (58). Then,By the direct calculations, we find thatSo,where and . Therefore, for every and an arbitrary . Furthermore, we make sure that as with all its derivatives uniformly in on each segment . In addition, the set of functions , , is uniformly bounded (by ) in the space (this property follows from estimate (102) in which , , and do not depend on ). 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 equalitySince is a convolutor in , we have . Then, given estimate (58) (at ), we find thatfor an arbitrary function , i.e., in the space as . The theorem is proved.
If the generalized function in (87) is finite (i.e., its support is a finite set in ), then we can say on uniform tending a solution of problems (24) and (87) to zero on as . Note that every finite generalized function is a convolutor in spaces. This property follows from the general result concerning the theory of perfect spaces (see [12], p. 137): if is a perfect space with differential translation operation, then every finite functional is a convolutor in . Finite functionals form a fairly wide class. In particular, every bounded set is a support of some generalized function [12].
Theorem 3. Let be a solution of problems (24) and (87) with the boundary function in condition (87), which is an element of , , with finite support in . Then, uniformly on as .
Here is a scheme for proving this statement. Let . Consider the function , , such that , , and . This function exists when because contains finite functions [12]. Consider the functionwhere . Since , we have
To prove the formulated above statement, it remains to establish that the set of functions is bounded in the space , , for large and .
For example, if, in condition (87), , then is a convolutor in the space , , and . It follows directly from estimate (58) that uniformly on as .
4. Conclusion
In this paper, the solvability of a nonlocal multipoint by the time problem for the evolutionary equation , is proved, herewith the operator is interpreted as a pseudodifferential operator in the space constructed by function , , 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 ). The behavior of the solution , in the space of generalized functions , is investigated. The conditions for the initial generalized function are found under which the solution is uniformly stabilized to zero on . The method of research of a nonlocal multipoint by time problem offered in this paper allows to interpret differential-operator equations of a form as an evolutionary equation with a pseudodifferential operator constructed by the function 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 ).
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.