# Modified Cauchy Problem with Impulse Action for Parabolic Shilov Equations

Authors: Galina Unguryan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

In the mediumℝn, we consider a certain process whose evolution ut;x during the time 0;T is described by the partial differential equation:(1)∂tut;x=Pt;i∂xut;x,t;x∈Π0;T≔0;T×ℝn,in which(2)Pt;i∂x=∑k≤paktik∂xk,differential expression of order p>1 with continuous coefficients ak· on the set 0;T. Here, i is an imaginary unit, ∂xk 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,x1/2.We consider that equation (1) on the set Π0;T is a uniformly parabolic Shilov equation with the index of parabolicity h, 0<h≤p, i.e., such that(3)∃δ>0∃δ0≥0∀t∈0;T∀ξ∈ℝn:RePt;ξ≤−δξh+δ0.A simpler example (1) is the classical thermal conductivity equation(4)∂tut;x=a2Δut;x,t;x∈Π0;T,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 statefx, x∈ℝn, is known, this process at time t=0, i.e., the initial condition is set,(5)ut;·|t=0=f,then to find its evolution ut;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 theut;x process described by equation (1), based only on the known information about its final (or intermediate) state in time:(6)ut;·|t=T=f,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)ut;·|t=0+νut;·|t=T=f,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 fx, 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)ut;·|t=t∗=f,for t∗≤T. In this regard, the corresponding condition (7) takes the form(9)ut;·|t=0+νut;·|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. 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 ti, ti≠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)uti+;·−uti−;·=χ,where uti±;·≔limt⟶ti+0ut;· and limt⟶ti−0ut;·. In this case, if ti<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. Section2 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

Letℂ∞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 ℤ+n. And let l=l1+⋯+ln, zl=z1l1,…,znln, if z=z1;…;zn∈ℝn, l=l1;…;ln∈ℤ+n.Forα>0 and β>0,(11)Sα=φ∈S|∃A>0∀k∈ℤ+n∃ck>0∀q∈ℤ+n∀x∈ℝn:xq∂xkφx≤ckAqqαq;Sβ=φ∈S|∃B>0∀q∈ℤ+n∃cq>0∀k∈ℤ+n∀x∈ℝn:xq∂xkφx≤cqBkkβk;Sαβ=φ∈S|∃A>0∃B>0∃c>0∀k,q⊂ℤ+n∀x∈ℝn:xq∂xkφx≤cAqBkqαqkβk.The setsSα, Sβ, and Sαβ 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 isSαβ=Sα∩Sβ [41].The spaceSαβ is nontrivial for α+β≥1 and consists only of those functions φ∈ℂ∞ℝn that satisfy the inequality(12)∂xkφx≤cBkkβke−δx1/α,k∈ℤ+n,x∈ℝn,with positive constants c, B, and δ, dependent on the function φ only [40].In spaces of the typeS, continuous addition, multiplication, convolution, and the operator F of the Fourier transform are defined, and the following topological equations are satisfied [40]: FSα=Sα, FSβ=Sβ, and FSαβ=Sβα.We shall denote byΦ′ the space topologically conjugate to the space Φ∈S1/h;Sβ1/h,β≥p−1/h, where h>0 and p>1.Forα1≤α2 and β1≤β2, the following continuous inclusions are satisfied [40]:(13)Sα1β1⊂Sα2β2⊂Sβ1⊂Sβ2Sα1⊂Sα2⊂S⊂S′⊂Sβ2′⊂Sβ1′Sα2′⊂Sα1′⊂Sα2β2′⊂Sα1β1′.The Fourier transform of the generalized functionf∈Φ′ and the convolution f∗g of the elements f,g⊂Φ′ are determined by the relations [40]:(14)Ff,Fφ=2πnf,φ;f∗g,φ=f,g∗φ,φ∈Φ,where the angle brackets , indicate the effect of the generalized function on the basic one.The convolution operation of the generalized functionf∈Φ′ with the main function φ∈Φ is determined by the equality(15)f∗φ·=fx,φx+·.The convolutionf∗φ· is a classic function from the class ℂ∞ℝn [40].It is obvious that the convolution operationf∗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·≔<gx,φx+·>∈Φ,∀φ∈Φ(2)
The convolution operationg with the elements φ∈Φ is continuous in the space ΦFrom here we come to the equalityFf∗g=FfFg, from which it becomes clear that the element g∈Φ′ is a convoluter in Φ only when its Fourier transform is Fg—a multiplier in the corresponding space FΦ.The following criterion of the multiplier [6, 7] is correct: let θτt·=exp∫τtPς;·dς, 0≤τ<t≤T, then to be a multiplier in the space FΦ, it is necessary and important for the function μ·∈ℂ∞ℝn that for each fixed t, 0<t≪1, the product μθ0t· belongs to the space FΦ.In [6], the properties of the function θτt· are studied, in particular, the belonging of θτt· to the space S1/hp−1/h for each fixed t>τ, and the following estimates are obtained:(16)∂ξkθτtξ≤ceδt−τAkkp−1/hkt−τn+γke−δ0t−τξh,k∈ℤ+n,ξ∈ℝn,0≤τ<t≤T,with positive constants c, δ, δ0, and A. Here, γk=1−pk/h, if 0<t−τ<1 and γk=1+hk/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 Π0;T is the function u, which on Π0;T satisfies equation (1) in the usual sense and the initial condition (5)—in the sense of convergence in the space Φ′:(17)ut;x,φx⟶t→+0f,φx,∀φ∈Φ.
The fundamental solution of the Cauchy problem for equation (1) is a function(18)Gt,τ;·=F−1θτtξt,τ;·,0≤τ<t≤T.
Obviously, the solutionGt,0;·∈Sp−1/h1/h for each fixed t∈0;T.
The next statement is correct.Theorem 1 (see [6]).
Letf be a real functional from the space Φ′, then the corresponding Cauchy problem (1) and (5) on the set Π0;T is correctly solvable and its solution ut;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∂tut;·=∂tFut;·,t∈0;T(2)
ut;x=f∗Gt,0;x,t;x∈Π0;TThese 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 Theorem1 for a better understanding of our further considerations. Since Gt,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)ut;·=fξ,Gt,0;ξ+·,with(20)∂tut;x=fξ,∂tGt,0;ξ+x,∂xkut;x=fξ,∂xkGt,0;ξ+x,while the equality(21)F∂tut;·=∂tFut;·,is considered in the space Φ′.


## 3. The Modified Cauchy Problem

We arbitrarily fix the real-valued functionalf 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)ut;·⟶t→+0Φ′f−νut∗;·.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)∂tvt;ξ=Pt;ξvt;ξ,t;ξ∈Π0;T,(24)vt;·⟶t→+0FΦ′f˜−νvt∗;·,where v=Fu≡u˜.According to the classical Cauchy theorem, all solutions of equation (23) are described by the formula vt;·=c·θ0t· with an arbitrary function c·. Hence, for 1+νθ0t∗ξ≠0, ξ∈ℝn, we arrive at the equivalence of the condition (24) to the following initial condition:(25)vt;·⟶t→+0FΦ′f˜1+νθ0t∗·.Thus, for1+νθ0t∗ξ≠0, 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)vt;ξ=f˜1+νθ0t∗ξθ0tξ,t;ξ∈Π0;T.Then, taking into account the statement of Theorem1, 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−11+νθ0t∗ξ−1∗f to the space Φ′ and its real-valuedness. For this, it would obviously be enough to show that the function μν·=1+νθ0t∗·−1 is a multiplier in the space FΦ.Definition 2.
We assume that for problems (1) and (9), the following condition holds:(27)infξ∈ℝn1+νθ0t∗ξ≠0.
In particular, condition (27) for problem (1) and (9) will be satisfied if ν∈ℝn is such that(28)ν<e−δ0t∗,where δ0 is the corresponding constant from the condition of parabolicity (3).
Indeed, directly from (3), we come to the estimate(29)θ0t∗ξ≤eδ0t∗,ξ∈ℝn.
Then, according to (28), we have(30)1+νθ0t∗ξ≥1−νθ0t∗ξ≥1−νeδ0t∗>0,ξ∈ℝn.
If the symbolPt;· of equation (1) for each t∈0;T acquires only valid values on ℝn, then condition (27) will already be satisfied for all ν>−e−δ0t∗.Lemma 1.
Suppose that condition (27) is satisfied for problem (1) and (9), then the corresponding function μν· is a multiplier in the space FΦ.Proof.
To simplify the calculations, we present a scheme of proof for the casen=1.
According to the well-known Faa di Bruno formula of the differentiation of a composite function(31)∂xkfφx=∑rkk!q!j!,…,m!drfφdφrdφx1!dxqd2φx2!dxj,…,dlφxl!dxm,k∈ℤ+,x∈ℝ,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)∂ξkμνξ≤∑rkk!r−1!q!j!,…,m!νinfξ∈ℝn1+νθ0t∗ξrdθ0t∗ξ1!dxqd2θ0t∗ξ2!dxj,…,dlθ0t∗ξl!dxm,k∈ℤ+,ξ∈ℝ.
Hence, according to Stirling’s formula(33)n!=2πnnen1+O1n,and the estimates (16),(34)r!q!j!,…,m!≤2k,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)∂ξkμνξ≤cAkkp−1/hk,which in combination with the estimate (16) ensures that the product μνθ0t· belongs to the space S1/hp−1/h for each fixed t∈0;1.
Therefore,μν· is a multiplier in the space S1/hp−1/h.
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 Π0;T is correctly solvable. Its solution u is determined by the following formula:(36)ut;x=f∗F−1μν∗Gt,0;x,t;x∈Π0;T.In this case,ut;x is a classical function on Π0;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 ti∈0;T. In this case, equation (1) will be considered on the set Π0;T\ti with an additional impulse condition (10), in which χ∈ℝ\0. We assume that the solution u at the point ti is continuous on the left. Then we understand condition (10) in the following limiting limiting sense:(37)ut;·⟶t→ti+0Φ′χ+uti−;·.Taking into account the specificity of condition (9), for t∗≠T there are two possible cases: t∗<ti(impulse after-effect) and ti<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” evolutionut;· of the considered process, i.e., when t∗<ti. In this case, on the set Π0;T\ti, we have problem (1), (9), and (10) for t∗<ti.We shall solve this problem thinking as follows.If we take into account the continuity on the left of the functionut;· at the point ti, then on the time interval 0;ti the initial problem (1), (9), and (10) obviously reduces to problem (1) and (9) with T=ti. Then, according to Theorem 2, under the appropriate conditions, the desired solution can be written as(38)ut;x=f∗F−1μν∗Gt,0;x,t;x∈Π0;ti,with this solution being unique on the set Π0;ti.Further, the impulse condition (10) takes the form(39)ut;·|t=ti+0=ϱ,where ϱ=χ+f∗F−1μν∗Gti,0;·—regular functionality from the space Φ′. Therefore, finding the solution u of problem (1), (9), and (10) on the interval ti;T is reduced to solving the Cauchy problem (1) and (39) on the set Πti;T. Using the statement of Theorem 1, we find that(40)ut;x=ϱ∗Gt,ti;x,t;x∈Πti;T,and this solution is also unique on Πti;T.The use of the Heaviside function(41)Ht=1,0<t,0,t<0,enables to write the found solution on the set Π0;T\ti in the form(42)ut;x=f∗F−1μν∗Gt,0;xHti−t+χ+f∗F−1μν∗Gti,0;x∗Gt,ti;xHt−ti.Hence, according to the convolution formula(43)Gt,τ;·=Gt,ς;·∗Gς,τ;·,0≤τ<ς<t≤T,we get(44)ut;x=f∗F−1μν∗Gt,0;xHti−t+χ∗Gt,ti;x+f∗F−1μν∗Gt,0;xHt−ti.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∗<ti the corresponding problem (1), (9), and (10) on the set Π0;T\ti is correctly solved. Its solution ut;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\ti is correct.


## 5. The Problem with Impulse Preeffect

We consider now the case when the impulse occurs before the “measurement” evolutionut;·: ti<t∗. Then on the set Π0;T\ti we have problem (1), (9), and (10) for ti<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 ti<t∗ on the set Π0;T\ti is correctly solvable and its solution ut;x is determined by the formula(45)ut;x=f−νχ∗Gt∗,ti;x∗F−1μν∗Gt,0;xHti−t+f∗F−1μν∗Gt,0;x+χ∗F−1μν∗Gt,ti;xHt−ti.In this case,ut;x is a classical function on Π0;T\ti, 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\ti.Proof.
The Fourier transform operatorF reduces problem (1), (9), and (10) to problem(46)∂tvt;ξ=Pt;ξvt;ξ,t;ξ∈Π0;T\ti,(47)vt;·⟶t→+0FΦ′f˜−νvt∗;·,(48)vt;·⟶t→ti+0FΦ′χ˜+vti−;·,where v=u˜.
According to the classical Cauchy theorem, all solutions of equation (46) on the set τ;T are described by the formula vt;·=c·θτt· with a random function c·. Then the general solution of equation (46) on the set 0;T\ti is the set(49)vt;·=c1·θ0t·Hti−t+c2·θtit·Ht−ti,with random functions cj·.
From here and from conditions (47) and (48), we get the system(50)c1·+νc2·θtit∗·=f˜,c2·−c1·θ0ti·=χ˜,to find the quantities cj·∈FΦ′ for which the function v from equality (49) is the solution to problem (46)–(48).
Provided that(51)1+νθ0t∗·≠0,system (50) has a single solution(52)c1·=f˜−νχ˜θtit∗·1+νθ0t∗·,c2·=f˜θ0ti·+χ˜1+νθ0t∗·.
Therefore, the function(53)vt;ξ=f˜−νχ˜θtit∗ξ1+νθ0t∗ξθ0tξHti−t+f˜θ0tiξ+χ˜1+νθ0t∗ξθtitξHt−ti,t;ξ∈Π0;T\ti,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)ut;·=F−1μνξf˜−νχ˜θtit∗ξθ0tξHti−t+f˜θ0tiξ+χ˜θtitξHt−tit;·,which is also the only one. This solution allows images(55)ut;x=f−νχ∗Gt∗,ti;x∗F−1μνξ∗Gt,0;xHti−t+f∗Gti,0;x+χ∗F−1μνξ∗Gt,ti;xHt−ti.
This is easy to see, if we take into account equation (18), the properties of the function θτt·, statement of Lemma 1, and the known formula(56)f∗g=F−1FfFg.
Further, ift∈0;ti, then(57)ut;x=f−νχ∗Gt∗,ti;x∗F−1μνξ∗Gt,0;x,x∈ℝn,and, if t∈ti;T, then(58)ut;x=f∗Gti,0;x+χ∗F−1μνξ∗Gt,ti;x,x∈ℝn.
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 Π0;T\ti, besides this ut;·∈ℂ∞ℝn, 0;T\ti, and the equality is satisfied(59)F∂tut;·=∂tFut;·,t∈0;T\ti.
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)Gt,τ;·=2aπt−τ−ne−·2/4a2t−τ,0≤τ<t≤T,∀T>0,with(61)θτt·=e−a2t−τ·2,0≤τ<t≤T.
SincePt;ξ=−a2ξ2∈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 Gt,0;·∈S1/21/2, t>0.
Further, in the spaceS1/21/2′, we have(62)FF−1μν∗Gt,0;·=μνFGt,0;·=e−a2t·21+νe−a2t∗·2≡ϕ,t>0.
The functionalϕ is a regular generalized function with S1/21/2′ generated by the element ϕ·∈S1/21/2. Then, F−1μνGt,0;·∈S1/21/2, in this case, the following equality is satisfied(63)F−1μν∗Gt,0;·=F−1e−a2tξ21+νe−a2t∗ξ2t;·,t>0.
If we now choose a really significant generalized functionf, the δ-Dirac function from the space S1/21/2′, for the initial one, is then(64)f∗F−1μν∗Gt,0;·=F−1e−a2tξ21+νe−a2t∗ξ2t;·,t>0.
Hence, according to formula (36), we obtain the following classical solution of the corresponding modified Cauchy problem (4) and (9):(65)ut;x=F−1e−a2tξ21+νe−a2t∗ξ2t;x,t;x∈Π0;T.
The solutionut;· for each t>0 is an element of the space S1/21/2.
Forχ from ℝ, we have(66)χ∗Gt,ti;x=χ,Gt,ti;ξ+x=χ∫ℝnGt,ti;ξ+xdξ=χ.
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 Π0;+∞\ti:(67)ut;x=Hti−tF−1e−a2tξ21+νe−a2t∗ξ2t;x+Ht−tiχ+F−1e−a2tξ21+νe−a2t∗ξ2t;x.
Obviously,ut;·∈S1/21/2, t∈0;ti, and ut;·∈ℂ∞ℝn, t>ti.
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 Π0;+∞\ti:(68)ut;x=F−1e−a2tξ21+νe−a2t∗ξ2t;x−νχ1+νHti−t+F−1e−a2tξ21+νe−a2t∗ξ2t;x+χ1+νHt−ti.
This solution is an element of the classℂ∞ℝn for each t∈0;+∞\ti.


## 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.



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