Modified Cauchy Problem with Impulse Action for Parabolic Shilov Equations

For parabolic Shilov equations with continuous coefficients, the problem of finding classical solutions that satisfy a modified initial condition with generalized data such as the Gelfand and Shilov distributions is considered. (is condition arises in the approximate solution of parabolic problems inverse in time. It linearly combines the meaning of the solution at the initial and some intermediate points in time. (e conditions for the correct solvability of this problem are clarified and the formula for its solution is found. Using the results obtained, the corresponding problems with impulse action were solved.


Introduction
In the medium R n , we consider a certain process whose evolution u(t; x) during the time (0; T] is described by the partial differential equation: z t u(t; x) � P t; iz x u(t; x), (t; x) ∈ Π (0;T] ≔ (0; T] × R n , (1) in which differential expression of order p > 1 with continuous coefficients a k (·) on the set [0; T]. Here, i is an imaginary unit, z k x is a partial derivative of a variable x of order k, and R n is a real Euclidean space of dimension n with a scalar product (·, ·) and norm ‖x‖ ≔ (x, x) 1/2 .
We consider that equation (1) on the set Π [0;T] is a uniformly parabolic Shilov equation with the index of parabolicity h, 0 < h ≤ p, i.e., such that ∃δ > 0∃δ 0 ≥ 0∀t ∈ [0; T]∀ξ ∈ R n : ReP(t; ξ) ≤ − δ‖ξ‖ h + δ 0 . (3) A simpler example (1) is the classical thermal conductivity equation z t u(t; x) � a 2 Δu(t; x), (t; x) ∈ Π (0;T] , with the Laplace operator Δ, which describes the process of heat propagation or diffusion in the medium R n . is equation is also related to the random Wiener process. Its fundamental solution is the density distribution of the probability of motion of microscopic particles in a liquid or gas under the action of medium molecules. If the initial state f(x), x ∈ R n , is known, this process at time t � 0, i.e., the initial condition is set, then to find its evolution u(t; x) we get the Cauchy problem (1) and (5). e study of the Cauchy problem (1) and (5) has been presented in many research works. In particular, authors in [1,2] describe the classes of unity and correctness of this problem. e stabilization properties of the solutions of equation (1) under special Λ-conditions were studied in [3][4][5]. In [6][7][8], the alternative methods of the fundamental solution study are offered, which allow to avoid the notion of the equation kind (1) and difficulties associated with its location. e abstract theory of the Cauchy problem (1) and (5) in Banach spaces is developed in [9,10]. e works [11][12][13][14][15][16] are devoted to the construction of the theory of the Cauchy problem for equation (1) with variable coefficients. e results of these studies naturally complement and generalize the classical theory of the Cauchy problem for parabolic Petrovsky equations [17][18][19].
e Cauchy problem is a so-called direct problem. However, in many cases, there is a situation when it is necessary to study the evolution of the u(t; x) process described by equation (1), based only on the known information about its final (or intermediate) state in time: for example, astrophysical problems concerning the celestial bodies study and some problems of thermophysics, diffusion, demography, and mathematical biology (see [20][21][22][23]).
Problem (1) and (6) is called an inverse problem. Usually, such a problem is incorrectly set by Hadamard [24]. Among the existing methods for solving the inverse problem of thermal conductivity, the method of quasireversibility should be noted [25]. However, this method is associated with an increase in the order of the original differential equation, which leads to significant difficulties in its numerical implementation. e Tikhonov regularization method for solving the equivalent integral equation [24] is an alternative to the quasireversibility method. In [26], developing Tikhonov's idea, a method for finding an approximate solution of the inverse thermal conductivity problem is proposed. It is based on the replacement of this problem by a corresponding problem with a nonlocal by time condition, in which a partial shift of the condition is carried out at t � T at the initial moment t � 0: where ] is a known parameter. For parabolic Petrovsky equations, i.e., equation (1), in which p � h � 2b, problems with condition (7), and those with the conditions of a more general form were considered in [27][28][29][30]. Here various questions concerning the correctness of such problems and methods of solving them under certain conditions on the input data are considered. In this case, similar problems for parabolic Shilov equations with p ≠ h still remain in the state of expectation.
It should be noted that (1) and (6) is the problem of studying the evolution of a process in the past on the interval (0; T) based on the data available about it f(x), x ∈ R n , at the moment of time t * � T. However, if the information about the evolution of this process is also important for us in the near future, then we should already consider the problem for equation (1) with the condition for t * ≤ T. In this regard, the corresponding condition (7) takes the form u(t; ·) |t�0 + ]u(t; ·) |t�t * � f.
Most of the observed processes are influenced by impulses that are not taken into account by the differential equation of the corresponding mathematical model. erefore, these effects should be reflected in the form of additional conditions to achieve the desired compliance of the model with the real process. Problems with impulse action for differential equations have been studied in many works, in [31][32][33][34][35][36][37][38][39] in particular. In [31,32], the basics of the results of the theory of systems of differential equations with impulse action are presented. In [34], sufficient conditions are established for the controllability of a class of semilinear impulsive integrodifferential systems with nonlocal initial conditions in Banach spaces. e internal approximate controllability of the semilinear impulse deterministic thermal equation is set in [35]. In [36], this issue is already clarified for the semilinear impulse stochastic equation of thermal conductivity with delay. e Cauchy problem with impulse action for parabolic Petrovsky equations was studied in [37][38][39].
e abovementioned works and the references given in them refer to direct problems with impulse action, while inverse problems of this type have escaped the attention of researchers.
Let us return to the inverse problem (1) and (8). Suppose that we somehow managed to find out that at a certain point in time t i , t i ≠ t * , the process in question (1) was (or will be) subjected to a momentum effect of χ units. en, model (1) and (8) must be supplemented by an additional impulse condition where u(t ± i ; ·) ≔ lim t⟶t i + 0u(t; ·) and lim t⟶t i − 0u(t; ·). In this case, if t i < t * , then problem (1), (8), and (10) is naturally called an inverse problem with an impulse precursor, in other words, an inverse problem with an impulse after-effect.
In this research, we study the modified Cauchy problem (1) and (9), which is generated by the inverse problem (1) and (8). By reducing to the corresponding Cauchy problem (1) and (5) in combination with the Fourier transform method, the correct solvability of this problem in a wide class of generalized initial data, such as the Gelfand and Shilov distributions, is established. At the same time, the explicit formulae of its classical solutions are found and the smoothness of these solutions by temporal and spatial variables is clarified. e results obtained are applied to solving the original problem (1) and (9) with the available time impulse (10), which can occur both before the time t * and after it. e structure of the work is as follows. Section 2 provides the necessary information about the spaces of basic and generalized functions, which will serve as an environment for the study of the modified problem. e information about the correct solvability of the Cauchy problem for parabolic Shilov equations (1) is also presented here. In Section 3, the classical solutions of the modified Cauchy problem (1) and (9) with generalized initial data f are found and their uniqueness is substantiated. Problem (1) and (9) with impulse after-effect and preeffect is solved in Sections 4 and 5, respectively. Section 6 presents the conclusions.

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Preliminary Information
Let C ∞ (R n ) be a class of all functions infinitely differentiable on R n and S be the space of L. Schwartz elements from C ∞ (R n ) rapidly fall to infinity, and S ′ is the corresponding space of Schwartz distributions [40]. e set of all n-dimensional multi-indices is denoted by Z n + . And let |l| � l 1 + · · · + l n , z l � z e sets S α , S β , and S β α with corresponding topologies [40] are countably normalized complete perfect spaces, which are called Gelfand and Shilov spaces of the type S.
with positive constants c, B, and δ, dependent on the function φ only [40].
In spaces of the type S, continuous addition, multiplication, convolution, and the operator F of the Fourier transform are defined, and the following topological equations are satisfied [40]: For α 1 ≤ α 2 and β 1 ≤ β 2 , the following continuous inclusions are satisfied [40]: e Fourier transform of the generalized function f ∈ Φ ′ and the convolution f * g of the elements f, g ⊂ Φ ′ are determined by the relations [40]: where the angle brackets 〈, 〉 indicate the effect of the generalized function on the basic one. e convolution operation of the generalized function f ∈ Φ ′ with the main function φ ∈ Φ is determined by the equality e convolution (f * φ)(·) is a classic function from the class C ∞ (R n ) [40].
It is obvious that the convolution operation f * g in the space Φ ′ will exist if the generalized function g is a convoluter in the space Φ, i.e., it is such that e following criterion of the multiplier [6,7] is correct: , it is necessary and important for the function μ(·) ∈ C ∞ (R n ) that for each fixed t, . In [6], the properties of the function θ t τ (·) are studied, in particular, the belonging of θ t τ (·) to the space S for each fixed t > τ, and the following estimates are obtained: International Journal of Mathematics and Mathematical Sciences with positive constants c, δ, δ 0 , and A. Here, We consider the Cauchy problem for equation (1) with the initial condition (5), in which f is a functional from the space Φ ′ .

Definition 1.
e solution of the Cauchy problem (1) and (5) on the set Π (0;T] is the function u, which on Π (0;T] satisfies equation (1) in the usual sense and the initial condition (5)in the sense of convergence in the space Φ ′ : e fundamental solution of the Cauchy problem for equation (1) is a function Obviously, the solution G(t, 0; ·) ∈ S (1/h) (p− 1/h) for each fixed t ∈ (0; T]. e next statement is correct. Theorem 1 (see [6]). Let f be a real functional from the space Φ ′ , then the corresponding Cauchy problem (1) and (5) on the set Π (0;T] is correctly solvable and its solution u(t; x) is differentiable with respect to the variable t and infinitely differentiable with respect to the variable x, for which the following conditions are satisfied: ese results will help us to determine the correct solvability of the corresponding modified Cauchy problem (1) and (9), which will be considered in Section 3.
At the end of this section, we comment on conditions (1) and (2) of eorem 1 for a better understanding of our further considerations. Since G(t, 0; ·) ∈ Φ, t ∈ (0; T], then according to equation (15), the solution of the Cauchy problem (1) and (5) is determined by the formula with while the equality is considered in the space Φ ′ .

The Modified Cauchy Problem
We arbitrarily fix the real-valued functional f from the space Φ ′ , and for equation (1), we set the modified initial condition (9), which, considering the differentiability of the function u at the point t * , we regard as a weak convergence in Φ ′ : We shall solve the obtained problem (1) and (9) by the Fourier transform method.
To do this, we write the corresponding dual Fourier problem: where According to the classical Cauchy theorem, all solutions of equation (23) are described by the formula v(t; ·) � c(·)θ t 0 (·) with an arbitrary function c(·). Hence, for 1 + ]θ t * 0 (ξ) ≠ 0, ξ ∈ R n , we arrive at the equivalence of the condition (24) to the following initial condition: us, for 1 + ]θ t * 0 (ξ) ≠ 0, ξ ∈ R n , the dual Fourier problem (23) and (24) is equivalent to the Cauchy problem (23) and (25), and the only solution to these problems is the function en, taking into account the statement of eorem 1, to prove the correct solvability of the original problem (1) and (9), it will be enough to substantiate the belonging of the functional g � F − 1 [(1 + ]θ t * 0 (ξ)) − 1 ] * f to the space Φ ′ and its real-valuedness. For this, it would obviously be enough to show that the function μ ] (·) � (1 + ]θ t * 0 (·)) − 1 is a multiplier in the space F[Φ]. Definition 2. We assume that for problems (1) and (9), the following condition holds: In particular, condition (27) for problem (1) and (9) will be satisfied if ] ∈ R n is such that where δ 0 is the corresponding constant from the condition of parabolicity (3). Indeed, directly from (3), we come to the estimate en, according to (28), we have

International Journal of Mathematics and Mathematical Sciences
If the symbol P(t; ·) of equation (1) for each t ∈ [0; T] acquires only valid values on R n , then condition (27) will already be satisfied for all ] > − e − δ 0 t * .

Lemma 1. Suppose that condition (27) is satisfied for problem (1) and (9), then the corresponding function μ ] (·) is a multiplier in the space F[Φ].
Proof. To simplify the calculations, we present a scheme of proof for the case n � 1.
According to the well-known Faa di Bruno formula of the differentiation of a composite function where the sign of the sum extends to all integer nonnegative solutions of the equation k � q + 2j + · · · + lm, and r � q + j + · · · + m, we obtain that Hence, according to Stirling's formula and the estimates (16), we find out the existence of positive constants c and A, such that for all k ∈ Z + and ξ ∈ R, the following inequality is fulfilled: which in combination with the estimate (16) ensures that the product (μ ] θ t 0 )(·) belongs to the space S (1/h) . Now that we have relation (13), we obtain the statement of the original lemma. e lemma is proved. We summarize the previous reflections in the form of the following statement. (27) is satisfied and f is a real-valued functional from the space Φ′, then the corresponding modified problem (1) and (9) on the set Π (0;T] is correctly solvable. Its solution u is determined by the following formula: (36) In this case, u(t; x) is a classical function on Π (0;T] , which is once differentiable with respect to the variable t and infinitely differentiable with respect to the variable x, for which equality (21) on the set (0; T] is correct.

Theorem 2. Suppose that condition
We now turn to consider the modified Cauchy problem (1) and (9) with a single impulse action that occurs at time t i ∈ (0; T). In this case, equation (1) will be considered on the set Π (0;T]\ t i { } with an additional impulse condition (10), in which χ ∈ R\ 0 { }. We assume that the solution u at the point t i is continuous on the left. en we understand condition (10) in the following limiting limiting sense: Taking into account the specificity of condition (9), for t * ≠ T there are two possible cases: t * < t i (impulse after-effect) and t i < t * (impulse preeffect).
We consider each of these cases separately.

The Problem with Impulse After-Effect
Let us consider here the situation when the impulse occurred after the "measurement" evolution u(t; ·) of the considered process, i.e., when t * < t i . In this case, on the set Π (0;T]\ t i { } , we have problem (1), (9), and (10) for t * < t i .
We shall solve this problem thinking as follows.
If we take into account the continuity on the left of the function u(t; ·) at the point t i , then on the time interval (0; t i ] the initial problem (1), (9), and (10) obviously reduces to problem (1) and (9) with T � t i . en, according to eorem 2, under the appropriate conditions, the desired solution can be written as (38) with this solution being unique on the set Π (0;t i ] .

International Journal of Mathematics and Mathematical Sciences
Further, the impulse condition (10) takes the form where ϱ � χ + f * F − 1 [μ ] ] * G(t i , 0; ·)-regular functionality from the space Φ ′ . erefore, finding the solution u of problem (1), (9), and (10) on the interval (t i ; T] is reduced to solving the Cauchy problem (1) and (39) on the set Π (t i ;T] . Using the statement of eorem 1, we find that and this solution is also unique on Π (t i ;T] . e use of the Heaviside function enables to write the found solution on the set Π (0;T]\ t i { } in the form Hence, according to the convolution formula we get erefore, the following statement is correct. (27) is satisfied and f is a realvalued functional from the space Φ ′ , then for t * < t i the corresponding problem (1), (9), and (10)  which is also the only one. is solution allows images

Theorem 3. If condition
is is easy to see, if we take into account equation (18), the properties of the function θ t τ (·), statement of Lemma 1, and the known formula (56) Further, if t ∈ (0; t i ], then and, if t ∈ (t i ; T], then However, each of equations (57) and (58) determines the solution of the Cauchy problem (1) and (5) with the corresponding initial function on the corresponding time interval.
erefore, according to eorem 1, the found function u is a classical solution of equation (1) on the set Π (0;T]\ t i { } , besides this u(t; ·) ∈ C ∞ (R n ), (0; T]\ t i , and the equality is satisfied Now applying the convolution formula (43) to equation (55), we come to the image of the solution of problem (1), (9), and (10) in the form (45). e theorem is proved. In conclusion, we demonstrate the obtained results on the example of the classical equation of thermal conductivity (4). e fundamental solution of the Cauchy problem for (4) is a function International Journal of Mathematics and Mathematical Sciences Since P(t; ξ) � − a 2 ‖ξ‖ 2 ∈ R, ξ ∈ R n , we choose ] > − 1. is choice holds condition (27) for the corresponding task (4) and (9).
In a similar way from formula (45), we obtain the solution of the modified Cauchy problem (4), (9), and (10) with impulse preeffect on the set Π (0;+∞)\ t i { } : is solution is an element of the class C ∞ (R n ) for each t ∈ (0; +∞)\ t i .

Conclusions
is research deals with a modified Cauchy problem for parabolic Shilov equations with variable coefficients, which arises at approximately solving the time-inverse parabolic problem. e correct solvability of this problem in a wide class of generalized initial data such as the Gelfant and Shilov distributions is determined. At that, the method of reducing the modified problem to the classical Cauchy problem is applied.
is method allows obtaining important results about the solvability of the new problem in the form of corresponding consequences from the known statements about the Cauchy problem; this significantly simplifies the research process. Also, the correct solvability of this problem with the available single impulse is found. At the same time, the cases with impulse after-effect and preeffect in relation to the moment of "measurement" of evolution of the considered process are separately considered. For parabolic Shilov equations and in the case of the problem with impulse action, for parabolic Petrovsky equations as well, the results obtained here are new. e formulae of classical solutions with generalized boundary values of these problems found here can be applied for virtual visualization of these processes using computer technology. ey are also appropriate for 8 International Journal of Mathematics and Mathematical Sciences numerical analysis using modern application packages. On the other hand, these results are important for further studies of parabolic equations with nonlocal and impulse conditions of a more general structure.
Data Availability e data used in the research to support the findings of this study are purely bibliographic and from scientific publications, which are included in the article with their respective citations.

Conflicts of Interest
e author declares that he has no conflicts of interest.