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. This 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. The 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.

1. Introduction

In the medium ℝⁿ, we consider a certain process whose evolution u(t; x) during the time (0; T] is described by the partial differential equation:

(1)

in which

(2)

differential expression of order p > 1 with continuous coefficients a_k(·) on the set [0; T]. Here, i is an imaginary unit,

is a partial derivative of a variable x of order k, and ℝⁿ 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

(3)

A simpler example (1) is the classical thermal conductivity equation

(4)

with the Laplace operator Δ, which describes the process of heat propagation or diffusion in the medium ℝⁿ. This 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 ∈ ℝⁿ, is known, this process at time t = 0, i.e., the initial condition is set,

(5)

then to find its evolution u(t; x) we get the Cauchy problem (1) and (5).

The 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. The stabilization properties of the solutions of equation (1) under special Λ-conditions were studied in [3–5]. In [6–8], 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. The abstract theory of the Cauchy problem (1) and (5) in Banach spaces is developed in [9, 10]. The works [11–16] are devoted to the construction of the theory of the Cauchy problem for equation (1) with variable coefficients. The results of these studies naturally complement and generalize the classical theory of the Cauchy problem for parabolic Petrovsky equations [17–19].

The 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:

(6)

for example, astrophysical problems concerning the celestial bodies study and some problems of thermophysics, diffusion, demography, and mathematical biology (see [20–23]). 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. The 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:

(7)

where ν 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 [27–30]. 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 ∈ ℝⁿ, 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

(8)

for t_∗ ≤ T. In this regard, the corresponding condition (7) takes the form

(9)

Most of the observed processes are influenced by impulses that are not taken into account by the differential equation of the corresponding mathematical model. Therefore, 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 [31–39] 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. The 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. The Cauchy problem with impulse action for parabolic Petrovsky equations was studied in [37–39]. The 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. Then, model (1) and (8) must be supplemented by an additional impulse condition

(10)

where

and

. 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. The 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.

The 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. The 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.

2. Preliminary Information

Let ℂ^∞(ℝⁿ) be a class of all functions infinitely differentiable on ℝⁿ and S be the space of L. Schwartz elements from ℂ^∞(ℝⁿ) rapidly fall to infinity, and S^′ is the corresponding space of Schwartz distributions [40]. The set of all n-dimensional multi-indices is denoted by . And let |l| = l₁ + ⋯+l_n, , if z = (z₁; …; z_n) ∈ ℝⁿ, .

For α > 0 and β > 0,

(11)

The sets S_α, S^β, and with corresponding topologies [40] are countably normalized complete perfect spaces, which are called Gelfand and Shilov spaces of the type S.

The correct topological relationship is [41].

The space

is nontrivial for α + β ≥ 1 and consists only of those functions φ ∈ ℂ^∞(ℝⁿ) that satisfy the inequality

(12)

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]: F[S_α] = S^α, F[S^β] = S_β, and .

We shall denote by Φ^′ the space topologically conjugate to the space , where h > 0 and p > 1.

For α₁ ≤ α₂ and β₁ ≤ β₂, the following continuous inclusions are satisfied [40]:

(13)

The 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]:

(14)

where the angle brackets 〈, 〉 indicate the effect of the generalized function on the basic one.

The convolution operation of the generalized function f ∈ Φ^′ with the main function φ ∈ Φ is determined by the equality

(15)

The convolution (f∗φ)(·) is a classic function from the class ℂ^∞(ℝⁿ) [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

(1)
(g∗φ)(·)≔<g(x), φ(x + ·) > ∈Φ, (∀φ ∈ Φ)
(2)
The convolution operation g with the elements φ ∈ Φ is continuous in the space Φ

From here we come to the equality F[f∗g] = F[f]F[g], from which it becomes clear that the element g ∈ Φ^′ is a convoluter in Φ only when its Fourier transform is F[g]—a multiplier in the corresponding space F[Φ].

The following criterion of the multiplier [6, 7] is correct: let , 0 ≤ τ < t ≤ T, then to be a multiplier in the space F[Φ], it is necessary and important for the function μ(·) ∈ ℂ^∞(ℝⁿ) that for each fixed t, 0 < t ≪ 1, the product belongs to the space F[Φ].

In [6], the properties of the function

are studied, in particular, the belonging of

to the space

for each fixed t > τ, and the following estimates are obtained:

(16)

with positive constants c, δ, δ₀, and A. Here, γ(k) = (1 − p)|k|/h, if 0 < t − τ < 1 and γ(k) = (1 + h)|k|/h at 1 ≤ t − τ.

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. The 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^′:

(17)

The fundamental solution of the Cauchy problem for equation (1) is a function

(18)

Obviously, the solution for each fixed t ∈ (0; T].

The 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:

(1)
F[∂_tu(t; ·)] = ∂_tF[u(t; ·)], t ∈ (0; T]
(2)
u(t; x) = f∗G(t, 0; x), (t; x) ∈ Π_{(0; T]}

These 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 Theorem 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

(19)

with

(20)

while the equality

(21)

is considered in the space Φ^′.

3. 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^′:

(22)

We shall solve the obtained problem (1) and (9) by the Fourier transform method.

To do this, we write the corresponding dual Fourier problem:

(23)

(24)

where

According to the classical Cauchy theorem, all solutions of equation (23) are described by the formula

with an arbitrary function c(·). Hence, for

, ξ ∈ ℝⁿ, we arrive at the equivalence of the condition (24) to the following initial condition:

(25)

Thus, for

, ξ ∈ ℝⁿ, 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

(26)

Then, taking into account the statement of Theorem 1, to prove the correct solvability of the original problem (1) and (9), it will be enough to substantiate the belonging of the functional to the space Φ^′ and its real-valuedness. For this, it would obviously be enough to show that the function is a multiplier in the space F[Φ].

Definition 2. We assume that for problems (1) and (9), the following condition holds:

(27)

In particular, condition (27) for problem (1) and (9) will be satisfied if ν ∈ ℝⁿ is such that

(28)

where δ₀ is the corresponding constant from the condition of parabolicity (3).

Indeed, directly from (3), we come to the estimate

(29)

Then, according to (28), we have

(30)

If the symbol P(t; ·) of equation (1) for each t ∈ [0; T] acquires only valid values on ℝⁿ, then condition (27) will already be satisfied for all .

Lemma 1. Suppose that condition (27) is satisfied for problem (1) and (9), then the corresponding function μ_ν(·) 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

(31)

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

(32)

Hence, according to Stirling’s formula

(33)

and the estimates (16),

(34)

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:

(35)

which in combination with the estimate (16) ensures that the product

belongs to the space

for each fixed t ∈ (0; 1).

Therefore, μ_ν(·) is a multiplier in the space .

Now that we have relation (13), we obtain the statement of the original lemma.

The lemma is proved.

We summarize the previous reflections in the form of the following statement.

Theorem 2. Suppose that condition (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.

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

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. Then we understand condition (10) in the following limiting limiting sense:

(37)

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.

4. 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 , 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. Then, according to Theorem 2, under the appropriate conditions, the desired solution can be written as

(38)

with this solution being unique on the set

Further, the impulse condition (10) takes the form

(39)

where ϱ = χ + f∗F⁻¹[μ_ν]∗G(t_i, 0; ·)—regular functionality from the space Φ^′. Therefore, 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

. Using the statement of Theorem 1, we find that

(40)

and this solution is also unique on

The use of the Heaviside function

(41)

enables to write the found solution on the set

in the form

(42)

Hence, according to the convolution formula

(43)

we get

(44)

Therefore, the following statement is correct.

Theorem 3. If condition (27) is satisfied and f is a real-valued functional from the space Φ^′, then for t_∗ < t_i the corresponding problem (1), (9), and (10) on the set is correctly solved. Its solution u(t; x) is a classical function represented by formula (44). It is once differentiable with respect to the variable t and infinitely differentiable with respect to the variable x, and equality (21) on the set (0; T]\{t_i} is correct.

5. The Problem with Impulse Preeffect

We consider now the case when the impulse occurs before the “measurement” evolution u(t; ·): t_i < t_∗. Then on the set we have problem (1), (9), and (10) for t_i < t_∗.

The next statement is correct.

Theorem 4. Let condition (27) be satisfied and f be a real-valued functional from the space Φ^′. Then the corresponding problem (1), (9), and (10) for t_i < t_∗ on the set is correctly solvable and its solution u(t; x) is determined by the formula

(45)

In this case, u(t; x) is a classical function on , which is once differentiable with respect to the variable t and infinitely differentiable with respect to the variable x, for which equation (21) is correct on the set (0; T]\{t_i}.

Proof. The Fourier transform operator F reduces problem (1), (9), and (10) to problem

(46)

(47)

(48)

where

According to the classical Cauchy theorem, all solutions of equation (46) on the set [τ; T] are described by the formula with a random function c(·). Then the general solution of equation (46) on the set [0; T]\{t_i} is the set

(49)

with random functions c_j(·).

From here and from conditions (47) and (48), we get the system

(50)

to find the quantities c_j(·) ∈ F[Φ^′] for which the function v from equality (49) is the solution to problem (46)–(48).

Provided that

(51)

system (50) has a single solution

(52)

Therefore, the function

(53)

is the only solution of problem (46)–(48).

Let condition (27) be satisfied. Then acting on equality (53) by the operator F⁻¹, we obtain the solution of problem (1), (9), and (10)

(54)

which is also the only one. This solution allows images

(55)

This is easy to see, if we take into account equation (18), the properties of the function , statement of Lemma 1, and the known formula

(56)

Further, if t ∈ (0; t_i], then

(57)

and, if t ∈ (t_i; T], then

(58)

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. Therefore, according to Theorem 1, the found function u is a classical solution of equation (1) on the set , besides this u(t; ·) ∈ ℂ^∞(ℝⁿ), (0; T]\{t_i}, and the equality is satisfied

(59)

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).

The theorem is proved.

In conclusion, we demonstrate the obtained results on the example of the classical equation of thermal conductivity (4). The fundamental solution of the Cauchy problem for (4) is a function

(60)

with

(61)

Since P(t; ξ) = −a²‖ξ‖² ∈ ℝ, ξ ∈ ℝⁿ, we choose ν > −1. This choice holds condition (27) for the corresponding task (4) and (9).

Equation (4) is a Shilov parabolic one, for which p = h = 2. Therefore, its fundamental solution is , t > 0.

Further, in the space , we have

(62)

The functional ϕ is a regular generalized function with generated by the element . Then, , in this case, the following equality is satisfied

(63)

If we now choose a really significant generalized function f, the δ-Dirac function from the space , for the initial one, is then

(64)

Hence, according to formula (36), we obtain the following classical solution of the corresponding modified Cauchy problem (4) and (9):

(65)

The solution u(t; ·) for each t > 0 is an element of the space .

For χ from ℝ, we have

(66)

Considering this, directly from formula (44), we arrive at the following solution of the modified Cauchy problem (4), (9), and (10) with impulse after-effect on the set :

(67)

Obviously, , t ∈ (0; t_i), and u(t; ·) ∈ ℂ^∞(ℝⁿ), t > t_i.

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 :

(68)

This solution is an element of the class ℂ^∞(ℝⁿ) for each t ∈ (0; +∞)\{t_i}.

6. Conclusions

This research deals with a modified Cauchy problem for parabolic Shilov equations with variable coefficients, which arises at approximately solving the time-inverse parabolic problem. The 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. This 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.

The 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. They are also appropriate for 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.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Open Research

Data Availability

The 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.

References

1 Shilov G. E., On conditions of correctness of Cauchy’s problem for systems of partial differential equations with constant coefficients, Uspekhi Matematicheskikh Nauk. (1955) 10, no. 4, 89–100, in Russian.
Google Scholar
2 Gel’fand I. and Shilov G., Generalized functions, Theory of Differential Equations. (1967) Academic Press, Boston, MA, USA.
Google Scholar
3 Eidelman S. D., Ivasishen S. D., and Porper F. O., The Liouville theorems for systems parabolic by Shilov, Central European Journal of Mathematics. (1961) 169, no. 6, in Russian.
Google Scholar
4 Gorodetskii V. V., The Cauchy problem for Shilov parabolic equations in classes of generalized periodic functions, Central European Journal of Mathematics. (1988) 10, no. 5, 82–84, in Russian.
Google Scholar
5 Gorodetskii V. V., Some theorems on stabilization of solutions of the Cauchy problem for Shilov parabolic systems in classes of generalized functions, Ukrainian Mathematical Journal. (1988) 40, no. 1, 43–48, in Russianhttps://doi.org/10.1007/bf01056443, 2-s2.0-34250089108.
10.1007/BF01056443
Google Scholar
6 Litovchenko V. A., The Cauchy problem for Shilov parabolic equations, Siberian Mathematical Journal. (2004) 45, no. 4, 669–679, https://doi.org/10.1023/B:SIMJ.0000035831.63036.bb, 2-s2.0-4043111244.
10.1023/B:SIMJ.0000035831.63036.bb
Google Scholar
7 Litovchenko V. A., Cauchy problem for -parabolic equations with time-dependent coefficients, Mathematical Notes. (2005) 77, no. 3-4, 364–379, https://doi.org/10.1007/s11006-005-0036-9, 2-s2.0-20244378380.
10.1007/s11006-005-0036-9
Google Scholar
8 Litovchenko V. A., One method for the investigation of fundamental solution of the Cauchy problem for parabolic systems, Ukrainian Mathematical Journal. (2018) 70, 922–934.
10.1007/s11253-018-1542-8
Google Scholar
9 Zheng Q., Matrices of operators and regularized cosine functions, Journal of Mathematical Analysis and Applications. (2006) 315, no. 1, 68–75, https://doi.org/10.1016/j.jmaa.2005.10.007.
10.1016/j.jmaa.2005.10.007
Web of Science® Google Scholar
10 Kostic M., Shilov Parabolic Systems, Bulletin. (2012) 37, 19–39.
Google Scholar
11 Zhitomirskii Y. I., Cauchy problem for some types of systems of linear equations, which are parabolic by G.E. Shilov, with partial derivatives with continuous coefficients, Izvestiya Rossiiskoi Akademii Nauk. (1959) 23, 925–932, in Russian.
Google Scholar
12 Litovchenko V. A. and Dovzhitska I. M., The fundamental matrix of solutions of the Cauchy problem for a class of parabolic systems of the Shilov type with variable coefficients, Journal of Mathematical Sciences. (2011) 175, no. 4, 450–476, https://doi.org/10.1007/s10958-011-0356-0, 2-s2.0-79957657646.
10.1007/s10958-011-0356-0
Google Scholar
13 Litovchenko V. A. and Dovzhytska I. M., Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients, Central European Journal of Mathematics. (2012) 10, no. 3, 1084–1102, https://doi.org/10.2478/s11533-012-0025-7, 2-s2.0-84859164909.
10.2478/s11533-012-0025-7
Google Scholar
14 Litovchenko V. A. and Dovzhytska I. M., Stabilization of solutions to Shilov-type parabolic systems with nonnegative genus, Siber Mathematics Journal. (2014) 55, no. 2, 276–283, https://doi.org/10.1134/s0037446614020104, 2-s2.0-84899690421.
10.1134/S0037446614020104
Google Scholar
15 Litovchenko V. A. and Unguryan G. M., Conjugate Cauchy problem for parabolic Shilov type systems with nonnegative genus, Differential Equations. (2018) 54, no. 3, 335–351, https://doi.org/10.1134/S0012266118030060, 2-s2.0-85046625653.
10.1134/S0012266118030060
Google Scholar
16 Litovchenko V. and Unguryan G., Some properties of Green’s functions of Shilov-type parabolic systems, Miskolc Mathematical Notes. (2019) 20, no. 1, 365–379, https://doi.org/10.18514/MMN.2019.2089, 2-s2.0-85067422285.
10.18514/MMN.2019.2089
Google Scholar
17 Petrovskii I. G., On the Cauchy problem for systems of equations with partial derivatives in a domain of nonanalytic functions, Bulletin. (1938) 1, no. 7, 1–72, in Russian.
Google Scholar
18 Eidelman S. D., Parabolic Systems, 1969, North-Holland, Amsterdam, Netherland.
Google Scholar
19 Friedman A., Partial Differential Equations of Parabolic Type, 1964, Prentice-Hall, Englewood Cliffs, NJ, USA.
Google Scholar
20 Cannon J. R. and van der Hoek J., Diffusion subject to the specification of mass, Journal of Mathematical Analysis and Applications. (1986) 115, no. 2, 517–529, https://doi.org/10.1016/0022-247x(86)90012-0, 2-s2.0-0022709536.
10.1016/0022-247X(86)90012-0
Web of Science® Google Scholar
21 Nakhushev A. M., Equations of Mathematical Biology, 1995, Springer, Berlin, Germany, in Russian.
Google Scholar
22 Belavin I. A., Kapitsa S. P., and Kurdyumov S. P., Mathematical model of global demographic processes taking into account spatial distribution, Computational Mathematics and Mathematical Physics. (1998) 38, no. 6, 885–902, in Russian.
Google Scholar
23 Alekseeva S. M. and Yurchuk N. I., The quasi-inversion method for the initial condition control problem for the heat equation with integral boundary condition, Differential Equation. (1998) 34, no. 4, 495–502, in Russian.
Google Scholar
24 Tikhonov A. N. and Arsenin V. Y., Methods for Solving Incorrect Problems, 1979, Springer, Berlin, Germany, in Russian.
Google Scholar
25 Lattes R. and Lions J.-L., Methode de quasi-reversibilite et applications, 1967, Dunod, Paris, France.
Google Scholar
26 Vabishchevich P. N., Nonlocal parabolic problems and the inverse problem of thermal conductivity, Differential Equation. (1981) 17, no. 7, 1193–1199, in Russian.
Google Scholar
27 Ivanchov N. I., Boundary value problems for a parabolic equation with Integral Conditions, Differential equation. (2004) 40, 591–609.
10.1023/B:DIEQ.0000035796.56467.44
Google Scholar
28 Chabrowski J., On the non-local problems with a functional for parabolic equation, Funckcialaj Ekvacioj. (1984) 27, 101–123.
Google Scholar
29 Shelukhin V. V., Time-nonlocal problem for the equations of the dynamics of a barotropic ocean, Siber Mathematics Journal. (1995) 36, no. 3, 701–724, in Russianhttps://doi.org/10.1007/bf02109846, 2-s2.0-34249759242.
10.1007/BF02109846
Google Scholar
30 Gorodetskiy V. V., Kolisnyk R. S., and Martynyuk O. V., On a nonlocal problem for partial diferential equations of parabolic type, Bukovinian Mathematics Journal. (2020) 8, no. 2, 24–39, https://doi.org/10.31861/bmj2020.02.03.
10.31861/bmj2020.02.03
Google Scholar
31 Lakshmikantham V. and Bainov D. D., Simeonov Theory of Impulsive Differential Equations, 1989, World Scientific, Singapore.
10.1142/0906
Google Scholar
32 Samoilenko A. M. and Perestyuk N. A., Impulsive Differential Equations, 1995, World Scientific, Singapore.
10.1142/2892
Google Scholar
33 Slyusarchuk V. E., General theorems on the existence and uniqueness of solutions of differential equations with impulsive action, Ukrainian Mathematical Journal. (2000) 52, no. 7, 954–964, in Russian.
10.1023/A:1005281717641
Google Scholar
34 Radhakrishnan B. and Balachandran K., Controllability results for semilinear impulsive integrodifferential evolution systems with nonlocal conditions, Journal of Control Theory and Applications. (2012) 10, no. 1, 28–34, https://doi.org/10.1007/s11768-012-0188-6, 2-s2.0-84555194919.
10.1007/s11768-012-0188-6
Google Scholar
35 Leiva H. and Merentes N., Approximate controllability of the impulsive semilinear heat equation, Journal of Mathematics and Applications. (2015) 38, 85–104.
Google Scholar
36 Leiva H., Narvaez M., and Sivoli Z., Controllability of impulsive semilinear stochastic heat equation with delay, International Journal of Differential Equations. (2020) 2020, 10, 2515160, https://doi.org/10.1155/2020/2515160.
10.1155/2020/2515160
Google Scholar
37 Matiichuk M. I. and Luchko V. M., Cauchy problem for parabolic systems with pulse action, Ukrainian Mathematical Journal. (2006) 58, no. 11, 1525–1535, https://doi.org/10.1007/s11253-006-0165-7, 2-s2.0-34147182094.
10.1007/s11253-006-0165-7
Google Scholar
38 Luo L. P., Wang Y. Q., and Gong Z. G., New criteria for oscillation of vector parabolic equations with the influence of impulse and delay, Acta Scientiarum Naturalium Universitatis Sunyatseni. (2012) 51, no. 2, 45–48.
Google Scholar
39 Pukal’skii I. D. and Yashan B. O., The Cauchy problem for parabolic equations with degeneration, Advances in Mathematical Physics. (2020) 2020, 7, 1245143, https://doi.org/10.1155/2020/1245143.
10.1155/2020/1245143
Google Scholar
40 Gel’fand I. M. and Shilov G. E., Spaces of basic and generalized functions, 1958, Elsevier, Amsterdam, Netherlands, in Russian.
Google Scholar
41 Kashpirovsky A. I., About coincidence of spaces S_α∩S^β and Functional analysis and its application, 1980, 14, no. 2, 60–72, in Russian.
Google Scholar

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Modified Cauchy Problem with Impulse Action for Parabolic Shilov Equations

Abstract

1. Introduction

2. Preliminary Information

3. The Modified Cauchy Problem

4. The Problem with Impulse After-Effect

5. The Problem with Impulse Preeffect

6. Conclusions

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

References

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Modified Cauchy Problem with Impulse Action for Parabolic Shilov Equations

Abstract

1. Introduction

2. Preliminary Information

3. The Modified Cauchy Problem

4. The Problem with Impulse After-Effect

5. The Problem with Impulse Preeffect

6. Conclusions

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

References

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