A problem with nonlocal conditions for a one-dimensional parabolic equation

Cover Page

Cite item

Full Text

Abstract

In present paper, we consider a problem with nonlocal conditions for parabolic equation and show that there exists a unique weak solution in Sobolev space. The main tool to prove the existence of a unique weak solution to the problem is a priori estimates derived by authors. We also note a connection between Steklov nonlocal conditions and first kind integral conditions. This connection enables interpret the problem under consideration as a problem with perturbed Steklov nonlocal conditions. Obtained results may be useful for certain class of problems including inverse problems.

About the authors

Alexander B. Beylin

Samara State Technical University

Email: abeilin@mail.ru
ORCID iD: 0000-0002-4042-2860
SPIN-code: 8390-6910
http://www.mathnet.ru/person100342

Cand. of Techn. Sci., Associate Professor, Dept. Mechanical Engineering, Machine Tools and Tools

Russian Federation, 244, Molodogvardeyskaya st., Samara, 443100

Andrey V. Bogatov

Samara National Research University

Email: andrebogato@mail.ru
ORCID iD: 0000-0001-5797-1930
http://www.mathnet.ru/person152395

Postgraduate Student

Russian Federation, 34, Moskovskoye shosse, Samara, 443086

Ludmila S. Pulkina

Samara National Research University

Author for correspondence.
Email: louise@samdiff.ru
ORCID iD: 0000-0001-7947-6121
SPIN-code: 9768-0196
Scopus Author ID: 6506395220
ResearcherId: C-1180-2017
http://www.mathnet.ru/person17853

Dr. Phys. & Math. Sci., Professor, Dept. of Differential Equations and Control Theory

Russian Federation, 34, Moskovskoye shosse, Samara, 443086

References

  1. Bažant, Zdeněk P., Jirásek M. Nonlocal integral formulation of plasticity and damage: Survey of progress, J. Eng. Mech., 2002, vol. 128, no. 11, pp. 1119–1149. DOI: https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1119).
  2. Cannon J. R. The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 1963, vol. 21, no. 2, pp. 155–160. DOI: https://doi.org/10.1090/qam/160437.
  3. Kamynin L. I. On certain boundary problem of heat conduction with nonclassical boundary conditions, Comput. Math. Math. Phys., 1964, vol. 4, no. 6, pp. 33–59. EDN: XNNPFM. DOI: https://doi.org/10.1016/0041-5553(64)90080-1.
  4. Ionkin N. I. The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Differ. Uravn., 1977, vol. 13, no. 2, pp. 294–304 (In Russian).
  5. Kartynnik A. V. A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic equations, Differ. Equ., 1990, vol. 26, no. 9, pp. 1160–1166.
  6. Bouziani A. On the solvability of parabolic and hyperbolic problems with a boundary integral condition, Int. J. Math. Math. Sci., 2002, vol. 31, no. 4, pp. 201–213. DOI: https://doi.org/10.1155/S0161171202005860.
  7. Kerefov A. A., Shkhanukov–Lafishev M. Kh., Kuliev R. S. Boundary-value problems for a loaded heat equation with nonlocal Steklov-type conditions, In: Neklassicheskie uravneniia matematicheskoi fiziki [Nonclassical Equations of Mathematical Physics]. Novosibirsk, Inst. Math. SB RAS, 2005, pp. 152–159 (In Russian).
  8. Kozhanov A. I. On a nonlocal boundary value problem with variable coefficients for the heat equation and the Aller equation, Differ. Equ., 2004, vol. 40, no. 6, pp. 815–826. EDN: PJGDYR. DOI: https://doi.org/10.1023/B:DIEQ.0000046860.84156.f0.
  9. Ivanchov N. I. Boundary value problems for a parabolic equation with integral conditions, Differ. Equ., 2004, vol. 40, no. 4, pp. 591–609. EDN: XLSRJL. DOI: https://doi.org/10.1023/B:DIEQ.0000035796.56467.44.
  10. Orazov I., Sadybekov M. A. One nonlocal problem of determination of the temperature and density of heat sources, Russian Math. (Iz. VUZ), 2012, vol. 56, no. 2, pp. 60–64. EDN: XMXTBB. DOI: https://doi.org/10.3103/S1066369X12020089.
  11. Cannon J. R., van der Hoek J. The classical solution of the one-dimensional two-phase Stefan problem with energy specification, Ann. Mat. Pura Appl., IV. Ser., 1982, vol. 130, no. 1, pp. 385–398. DOI: https://doi.org/10.1007/BF01761503.
  12. Cannon J. R., Lin Y. Determination of a parameter p(t) in some quasi-linear parabolic differential equations, Inverse Problems, 1988, vol. 4, no. 1, pp. 35–45. DOI: https://doi.org/10.1088/0266-5611/4/1/006.
  13. Kamynin V. L. The inverse problem of determining the lower-order coefficient in parabolic equations with integral observation, Math. Notes, 2013, vol. 94, no. 2, pp. 205–213. EDN: RFQRGX. DOI: https://doi.org/10.1134/S0001434613070201.
  14. Steklov V. A. The problem of cooling of inhomogeneous solid, Commun. Kharkov Math. Soc., 1896, vol. 5, no. 3–4, pp. 136–181 (In Russian).
  15. Pul’kina L. S. Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind, Russian Math. (Iz. VUZ), 2012, vol. 56, no. 4, pp. 62–69. EDN: PDSZMV. DOI: https://doi.org/10.3103/S1066369X12040081.
  16. Pulkina L. S. Nonlocal problems for hyperbolic equation from the view point of strongly regular boundary conditions, Electron. J. Differ. Equ., 2020, vol. 2020, no. 28, pp. 1–20. https://ejde.math.txstate.edu/Volumes/2020/28/abstr.html.
  17. Kozhanov A. I., Dyuzheva A. V. Non-local problems with integral displacement for high-order parabolic equations, Bulletin of Irkutsk State University, Ser. Mathematics, 2021, vol. 36, pp. 14–28 (In Russian). DOI: https://doi.org/10.26516/1997-7670.2021.36.14.
  18. Danyliuk I. M., Danyliuk A. O. Neumann problem with the integro-differential operator in the boundary condition, Math. Notes, 2016, vol. 100, no. 5, pp. 687–694. EDN: XNSUDH. DOI: https://doi.org/10.1134/S0001434616110055.
  19. Kozhanov A. I. On the solvability of some nonlocal and associated with them inverse problems for parabolic equations, Math. Notes of Yakutsk State Univ., 2011, vol. 18, no. 2, pp. 64–78 (In Russian).
  20. Kozhanov A. I., Dyuzheva A. V. The second initial-boundary value problem with integral displacement for second-order hyperbolic and parabolic equations, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2021, vol. 25, no. 3, pp. 423–434 (In Russian). DOI: https://doi.org/10.14498/vsgtu1859.
  21. Evans L. C. Uravneniia s chastnymi proizvodnymi [Partial Differential Equations]. Novosibirsk, Tamara Rozhkovskaya, 2003, 560 pp. (In Russian)
  22. Ladyzhenskaya O. A. Kraevye zadachi matematicheskoi fiziki [Boundary Problems of Mathematical Physics]. Moscow, Nauka, 1973, 407 pp. (In Russian)

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2022 Authors; Samara State Technical University (Compilation, Design, and Layout)

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).