# To the Question of the Solvability of the Ionkin Problem for Partial Differential Equations

## Abstract

**:**

## 1. Introduction

## 2. Statement of the Problems

## 3. Main Results

**Problem**

**1.**

**Problem**

**2.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Problem**

**3.**

**Problem**

**4.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

## 4. Examples

**Example**

**1.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**5.**

**Example**

**2.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**6.**

**Example**

**3.**

**Theorem**

**7.**

**Example**

**4.**

**Theorem**

**8.**

**Example**

**5.**

**Proposition**

**3.**

**Proof.**

**Theorem**

**9.**

## 5. Comments and Supplements

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Ionkin, N.I. Solution of a Boundary-Value Problem in Heat Conduction with a Nonclassical Boundary Condition. Differ. Equ.
**1977**, 13, 204–211. [Google Scholar] - Nakhushev, A.M. Problems with Shift for Partial Differential Equation; Nauka: Moscow, Russia, 2006. (In Russian) [Google Scholar]
- Sadybekov, M.A. Initial-Boundary Value Problem for a Heat Equation with not Strongly Regular Boundary Conditions. In Functional Analysis in Interdisciplinary Applications; Springer Proceedings in Mathematics & Statistics; Springer: Cham, Switzerland, 2017; pp. 330–348. [Google Scholar]
- Orazov, I.; Sadybekov, M.A. On a Class of Problems of Determining the Temperature and Density of Heat Sources Given Initial and Final Temperature. Sib. Math. J.
**2012**, 53, 146–151. [Google Scholar] [CrossRef] - Nakhusheva, Z.A. The Samarskii Problem for the Fractal Diffusion Equation. Math. Notes
**2014**, 95, 815–819. [Google Scholar] [CrossRef] - Kozhanov, A.I. Nonlocal Problems with Generalized Samarskii–Ionkin Condition for some Classes for Nonstationary Differential Equations. Dokl. Math.
**2023**, 107, 40–43. [Google Scholar] [CrossRef] - Kozhanov, A.I.; Dyuzheva, A.V. Well–Posedness of the Generalized Samarskii–Ionkin Problem for Elliptic Equations in a Cylindrical Domain. Differ. Equ.
**2023**, 59, 230–242. [Google Scholar] [CrossRef] - Yurchuk, N.I. Mixed Problem with an Integral Condition for Certain Parabolic Equation. Differ. Equ.
**1986**, 22, 1457–1463. [Google Scholar] - Ionkin, N.I. The Stability of a Problem in the Theory of Heat Equations with Nonclassical Boundary Conditions. Diff. Uravn.
**1979**, 15, 1279–1283. [Google Scholar] - Berdyshev, A.S.; Cabada, A.; Kadirkulov, B.J. The Samarskii–Ionkin Type Problem for the Fourth Order Parabolic Equation with Fractional Differential Operator. Comput. Math. Appl.
**2011**, 62, 3884–3893. [Google Scholar] [CrossRef] - Lions, J.-L. Quelques Methods de Resolution des Problems aux Limites Nonlineaires; Dunod Gauthier: Villars, Paris, France, 1969. [Google Scholar]
- Edwards, R.E. Functional Analysis; Holt, Rinehart and Winston: New York, NY, USA, 1965. [Google Scholar]
- Trenogin, V.A. Functional Analysis; Nauka: Moscow, Russia, 1980. (In Russian) [Google Scholar]
- Beilin, S.A. Existence of Solutions for One–Dimentional Wave Equations with Nonlocal Conditions. Election J. Differ. Equ.
**2001**, 76, 1–8. [Google Scholar] - Vragov, V.N. On the Theory of Boundary Value Problems for Equations of Mixed Type. Diff. Uravn.
**1977**, 13, 1098–1105. (In Russian) [Google Scholar] - Egorov, I.E.; Fedorov, V.E. Higher–Order Nonclassical Equations of Mathematical Physics; Computer Center Press: Novosibirsk, Russia, 1995. (In Russian) [Google Scholar]
- Kozhanov, A.I.; Pinigina, N.R. Boundary-value problems for some higher-order nonclassical differential equations. Math. Notes
**2017**, 101, 467–474. [Google Scholar] [CrossRef] - Kozhanov, A.I.; Tarasova, G.I. The Samarsky–Ionkin Problem with Integral Perturbation for a Pseudoparabolic Equation. The Bulletin of Irkutsk State University. Ser. Math.
**2022**, 42, 59–74. (In Russian) [Google Scholar] - Larkin, N.A. Existence Theorems for Quasilinear Pseudohyperbolic Equations. Rep. Ussr Acad. Sci.
**1982**, 6, 1316–1319. (In Russian) [Google Scholar] - Khudaverdiev, K.I.; Veliyev, A.A. Study of a One-Dimensional Mixed Problem for One Class of Third-Order Pseudohyperbolic Equations with a Nonlinear Operator Right-Hand Side; Chashyoglu: Baku, Azerbaijan, 2010; p. 168. [Google Scholar]
- Showalter, R.E.; Ting, T.W. Pseudoparabolic Partial Differential Equations. SIAM J. Math. Anal.
**1970**, 1, 1–26. [Google Scholar] [CrossRef] - Sveshnikov, A.G.; Alshin, A.B.; Korpusov, M.O.; Pletner, Y.D. Linear and Nonlinear Equations of Sobolev Type; Fizmatlit: Moscow, Russian, 2007. [Google Scholar]
- Whitham, G.B. Linear and Nonlinear Waves; John Wiley and Sons: New York, NY, USA, 1974. [Google Scholar]
- Lonngren, K.; Scott, A. (Eds.) Solitons in Action; Academic: New York, NY, USA, 1978. [Google Scholar]
- Demidenko, G.V.; Uspenskii, S.V. Partial Differential Equations and Systems not Solvable with Respect to Highest Order Derivatives; Marcel Dekker: New York, NY, USA; Basel, Switzerland, 2003. [Google Scholar]
- Zhegalov, V.I.; Mironov, A.N.; Utkina, E.A. Equations with Dominant Partial Derivative; Kazan Federal University: Kazan, Russia, 2014. [Google Scholar]
- Moiseev, E.I. Solvability of a Nonlocal Boundary Value Problem. Differ. Equ.
**2001**, 37, 1643–1646. [Google Scholar] [CrossRef] - Moiseev, E.I. On the Solution of a Nonlocal Boundary Value Problem by the Spectral Method. Differ. Equ.
**1999**, 35, 1105–1112. [Google Scholar] - Bondar, L.N.; Demidenko, G.V. On the Cauchy Problem for Pseudohyperbolic Equations with Lower Order Terms. Mathematics
**2023**, 11, 3943. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kozhanov, A.I.
To the Question of the Solvability of the Ionkin Problem for Partial Differential Equations. *Mathematics* **2024**, *12*, 487.
https://doi.org/10.3390/math12030487

**AMA Style**

Kozhanov AI.
To the Question of the Solvability of the Ionkin Problem for Partial Differential Equations. *Mathematics*. 2024; 12(3):487.
https://doi.org/10.3390/math12030487

**Chicago/Turabian Style**

Kozhanov, Aleksandr I.
2024. "To the Question of the Solvability of the Ionkin Problem for Partial Differential Equations" *Mathematics* 12, no. 3: 487.
https://doi.org/10.3390/math12030487