Advances in Mathematical Physics
Volume 2020 (2020), Article ID 1245143, 7 pages
https://doi.org/10.1155/2020/1245143

Research Article

The Cauchy Problem for Parabolic Equations with Degeneration

Ivan Pukal’skii and Bohdan Yashan

Department of Differential Equations, Yuriy Fedkovych Chernivtsi National University, 28 Universitetska St., Chernivtsi, Ukraine, 58012

Correspondence should be addressed to Bohdan Yashan; b.yashan@chnu.edu.ua

Received 12 February 2020; Accepted 26 May 2020; Published 16 June 2020

Academic Editor: Soheil Salahshour

Copyright © 2020 Ivan Pukal’skii and Bohdan Yashan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Annotation. For a second-order parabolic equation, the multipoint in time Cauchy problem is considered. The coefficients of the equation and the boundary condition have power singularities of arbitrary order in time and space variables on a certain set of points. Conditions for the existence and uniqueness of the solution of the problem in Hölder spaces with power weight are found.

1. Introduction

One of the important directions of development of the modern theory of partial differential equations is the study of nonlocal boundary value problems for different types of differential equations and partial differential equation systems and establishing the conditions for their correct solvability. Such problems arise when modeling different phenomena and processes in modern science, quantum mechanics, technology, economics, etc.

Multipoint problems for partial differential equations were studied in the papers by Ptashnyk and his disciples. In particular, paper [1] is dedicated to multipoint problems for partial differential equations, not resolved relative to the highest time derivative. The question of the existence and qualitative properties of solutions for equations with limited order of degeneration has been studied in papers [2–4]. The Dirichlet problem with impulse action for a parabolic equation with power singularities of arbitrary order on time and spatial variables is investigated in paper [5]. In paper [6], a multipoint one-sided boundary value problem is studied. In paper [7], presented is the result of research of optimal control of the system described by the problem with an oblique derivative and an integral condition on time variable for parabolic equations with power singularities of arbitrary order. For a second-order parabolic equation, a multipoint (in time) problem with oblique derivative is considered in paper [8]. Conditions for the existence and uniqueness of solution of the posed problem in Hölder spaces with power weight are established.

In this paper, we investigate a multipoint time-varying Cauchy problem for a parabolic second-order equation with power singularities and arbitrary order degenerations in the coefficients of spatial and time variables at some set of points. We also find conditions for the existence and uniqueness of the solution of formulated problem in Hölder spaces with power weight.

2. Statement of the Problem and Main Result

Let be fixed positive numbers, , , , , and let be some bounded domain in , .

Let us consider the problem of finding a function in the domain , which, for , satisfies the equation

and multipoint conditions for variable

The power singularities of coefficients of the differential equation (1) at the point characterize functions , :

where , , , , .

Let us denote that , , , , , , , are real numbers, , , , , , is the integer part of , , , , , , , are arbitrary points of the domain , .

Let be an arbitrary closed domain in , , .

We define the functional space in which we study problem (1) and (2).

is a set of functions of space , which are having continuous partial derivatives in of the form , and a finite value of the norm

Marked here, , , .

We assume that the initial problems (1) and (2) satisfy the following conditions:

(a)For the arbitrary vector , , the following inequality

is true, where are fixed positive constants and , , , , , , .

(b)Functions , , , ,

Let us formulate the main result of the paper.

Theorem 1. Let conditions (a) and (b) be satisfied for problem (1) and (2). Then, there exists a unique solution of problem (1) and (2) in the space , and the following estimate is correct:

To study problem (1) and (2), we construct a sequence of solutions of problems with a smooth coefficient limit value of which is the solution of problem (1) and (2).

3. Evaluation of Solutions of Problems with Smooth Coefficients

Let , , , be a sequence of domains that for , converges to .

In the domain , we consider the problem of finding the functions that satisfy the equations

and the time variable condition

Here, the coefficients , , , and functions , , into the domains coincide with , , , , , respectively, and in the domains are continuous prolongations of coefficients , , and functions , from domains into domains with preservation of their smoothness and norm ([9], p. 82).

To solve problem (7) and (8), we have a correct theorem.

Theorem 2. Let be the classical solutions of problem (7) and (8) in the domain and let conditions (a) and (b) be satisfied. Then, for , the following estimate

is true.

Proof. Let . If , then at the point , the following correlations

are true and equation (7) is satisfied. Considering (10) and equation (7) at the point , the inequality

is correct.
Let . If , then at the point , the following correlations

are true and equation (7) is satisfied. Considering correlation (12) and equation (7) at the point , we have

In the case of or from condition (8), we obtain

Considering inequality (11), (13), and (14), we obtain

The theorem is proved.

Now, we find estimates of the derivatives of solutions . In the space , we introduce a norm which is equivalent, at fixed , , to the Hölder norm, which is defined by the same way as the norm ; only, instead of functions and , we take and , respectively,

Theorem 3. Let conditions (a) and (b) be satisfied. Then, for the solution of problem (7) and (8), the estimate

is true.

Proof. Using the definition of the norm and interpolation inequalities from [9], we have

where is an arbitrary real number . Hence, it is sufficient to estimate the seminorm . As follows from the definition of seminorm, there exist in points , for which one of the inequalities

is true, where

If , and is an arbitrary real number from (0,1), then

If , then

Applying the interpolation inequalities to (21) and (22), we find

Let , and let . We assume that , and

, . In the domain , we write the problem (7) and (8) in the form

Let be a domain from , . Performing the substitution,

in problem (25) and (26), we obtain

where .

We denote

and choose a thrice differentiable function , which satisfies the conditions

We denote the function which is a solution of the Cauchy problem

On the basis of Theorem 5.3 from ([2], p. 364), for the solution of problem (31) and (32) and arbitrary points inequalities

are true, where , is the parabolic distance between the points and . With regard to the properties of function , we find estimates of the norms of expressions and :

The definition of the space implies the satisfaction of inequalities

Substituting (34) and (35) into (33) and returning to the variables , we obtain inequalities

Given the interpolation inequalities and estimates of the norm of each additive of the expressions , , we obtain the inequalities

Using inequalities (23) and (37) and choosing and sufficiently small, we obtain the estimate

Given the values of the expression and , for , we have

Because

then, given the estimate (9) and inequalities (39) and (40) for , the estimate (17) is true. The theorem is proved.

Proof of Theorem 4. The right-hand side of inequality (17) is independent of , and the sequences

are uniformly bounded and equicontinuous in . According to Arcel’s theorem, there exist subsequences which are uniformly convergent to in for , . Passing to the limit as in problem (7) and (8), we obtain that is the unique solution of problem (1) and (2) and . The theorem is proved.

4. Conclusions

The necessary and sufficient conditions for the existence of the unique solution of a multipoint problem for parabolic equations with degeneration are established. Estimates of derivatives of the solution of the problem in the Hölder spaces with power weight are found. The order of the degree weight depends on the power of the degree features of the coefficients of the equation.

Data Availability

The article contains theoretical material. This paper has links to articles and textbooks from other researchers. The article does not use any data.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this article.

References

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