Problem of determining of minor coefficient and right-hand side function in
semilinear ultraparabolic equation

N. P. Protsakh

doi:10.15330/ms.50.1.60-74

The problem of determining of the right-hand side function and the time depended minor coefficient in semilinear ultraparabolic equation from the initial, boundary and overdetermination conditions, is considered in this paper. The sufficient conditions of the existence and the uniqueness of solution on some interval [0; T]; where T depends on the coefficients of the equation, for the problem are obtained.

1. N.P. Protsakh, B.Yo. Ptashnyk, Nonlinear ultraparabolic equations and variational inequalities, “Naukova dumka”, Kyiv, 2017, 278 p. (in Ukrainian)

2. S.D. Eidelman, S.D. Ivasyshen, A.N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Birkhauser Verlag, 2004, 390 p.

3. A.N. Kolmogorov, Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung), Ann. Math., 35 (1934), 116–117.

4. E. Lanconelli, A. Pascucci, S. Polidoro, Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance, Nonlinear problems in mathematical physics and related topics II. In honour of Prof. O.A. Ladyzhenskaya. New York, NY: Kluwer Academic Publishers. Int. Math. Ser., N.Y., 2002, 243–265.

5. S.D. Ivasishen, I.P. Medynsky, The Fokker-Planck-Kolmogorov equations for some degenerate diffusion processes, Theory Stoch. Process., 16(32) (2010), №1, 57-66.

6. G. Avalishvili, M. Avalishvili, Nonclassical problem for ultraparabolic equation in abstract spaces, Journal of Function Spaces, 2016 (2016), 1-15.

7. A.S. Tersenov, On the global solvability of the Cauchy problem for a quasilinear ultraparabolic equation, Asymptotic Analysis, 82 (2013), №3-4, 295-314.

8. S. Lavrenyuk, N. Protsakh, Boundary value problem for nonlinear ultraparabolic equation in unbounded with respect to time variable domain, Tatra Mt. Math. Publ., 38 (2007), 131-146.

9. S.P. Lavrenyuk, N.P. Protsakh, Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia, Ukrainian Mathematical Journal, 58 (2006), №9, 1347-1368.

10. N. Protsakh, Inverse problem for a weakly nonlinear ultraparabolic equation with three unknown functions of different arguments on the right-hand side, Journal of Mathematical Sciences, 217 (2016), №4, 476-514.

11. N. Protsakh, Determining of right-hand side of higher order ultraparabolic equation, Open Math., 15 (2017), 1048-1062.

12. N. Protsakh, Inverse problem for an ultraparabolic equation with unknown function of the space variable on the right-hand side, Journal of Mathematical Sciences, 203 (2014), №1, 16-39.

13. N. Protsakh, Inverse problem for semilinear ultraparabolic equation of higher order, Mathematica Bohemica, 140 (2015), №4, 335-404.

14. Yu.A. Kosheleva, On the solvability of the nonlinear inverse problems for ultraparabolic equations, Sib. Elektron. Mat. Izv., 13 (2016), 1187-1206.

15. M. Dehghan, Determination of an unknown parameter in a semi-linear parabolic equation, Mathematical problems in engineering, 8 (2002), №2, 111-122.

16. J.R. Cannon, Y. Lin, Determination of a parameter p(t) in some quasi-linear parabolic differential equations, Inverse Probl., 4 (1988), 35-45.

17. F. Parzilvand, A.M. Shahrezaee, An inverse problem of identifying the coefficient of semilinear parabolic equation, Mathematical Researches (Sci. Kharazmi University), 2 (2016), №1, 79-88.

18. V.T. Borukhov, G.M. Zayats, Identification of a time-dependent source term in nonlinear hyperbolic or parabolic heat equation, International Journal of Heat and Mass Transfer, 91 (2015), 1106-1113.

19. P.N. Vabishchevich, V.I. Vasilev, Numerical solving the identifcation problem for the lower coencient of parabolic equation, arXiv:1304.5923v1 [cs.NA] 22 Apr 2013, 12p.

20. A.I. Prilepko, V.L. Kamynin, A.B. Kostin, Inverse source problem for parabolic equation with the condition of integral observation in time, Computational Mathematics and Mathematical Physics, 57 (2017), №6, 956-966.

21. H.P. Lopushanska, Determination of a minor coefficient in a time fractional diffusion equation, Mat. Stud., 45 (2016), №1, 57-66.

22. E. Ozbilge, A. Demir, Identification of the unknown diffusion coefficient in a linear parabolic equation via semigroup approach, Advances in Difference Equations, 2014 (2014), №47, 1-8.

23. M.I. Ivanchov, Inverse problems for equations of parabolic type, 238 p., Mathematical Studies, Monograph Series, VNTL Publishers., V.10, 2003.

24. A.I. Kozhanov, An inverse problem with an unknown coefficient and right-hand side for a parabolic equation II, J. of Inverse and ill posed problems,11 (2003), №5, 505-522.

25. S.G. Pyatkov, E.I. Safonov, Determination of the source function in mathematical models of convection diffusion, Mat. Zametki Severo-Vost. Fed. Univ., 21 (2014), №2, 117-130.

26. H. Gajewski, K. Groger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Berlin, Akademie-Verlag, 1974.

27. O.A. Ladyzhenskaya, Boundary-value problems of mathematical physics, Nauka, Moscow, 1973, 407 p. (in Russian)

	Problem of determining of minor coefficient and right-hand side function in semilinear ultraparabolic equation
Author	N. P. Protsakh protsakh@ukr.net National Forestry Engineering University of Ukraine, Lviv, Ukraine
Abstract	The problem of determining of the right-hand side function and the time depended minor coefficient in semilinear ultraparabolic equation from the initial, boundary and overdetermination conditions, is considered in this paper. The sufficient conditions of the existence and the uniqueness of solution on some interval [0; T]; where T depends on the coefficients of the equation, for the problem are obtained.
Keywords	inverse problem; ultraparabolic equation; boundary-value problem; unique solvability
DOI	doi:10.15330/ms.50.1.60-74
Reference	1. N.P. Protsakh, B.Yo. Ptashnyk, Nonlinear ultraparabolic equations and variational inequalities, “Naukova dumka”, Kyiv, 2017, 278 p. (in Ukrainian) 2. S.D. Eidelman, S.D. Ivasyshen, A.N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Birkhauser Verlag, 2004, 390 p. 3. A.N. Kolmogorov, Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung), Ann. Math., 35 (1934), 116–117. 4. E. Lanconelli, A. Pascucci, S. Polidoro, Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance, Nonlinear problems in mathematical physics and related topics II. In honour of Prof. O.A. Ladyzhenskaya. New York, NY: Kluwer Academic Publishers. Int. Math. Ser., N.Y., 2002, 243–265. 5. S.D. Ivasishen, I.P. Medynsky, The Fokker-Planck-Kolmogorov equations for some degenerate diffusion processes, Theory Stoch. Process., 16(32) (2010), №1, 57-66. 6. G. Avalishvili, M. Avalishvili, Nonclassical problem for ultraparabolic equation in abstract spaces, Journal of Function Spaces, 2016 (2016), 1-15. 7. A.S. Tersenov, On the global solvability of the Cauchy problem for a quasilinear ultraparabolic equation, Asymptotic Analysis, 82 (2013), №3-4, 295-314. 8. S. Lavrenyuk, N. Protsakh, Boundary value problem for nonlinear ultraparabolic equation in unbounded with respect to time variable domain, Tatra Mt. Math. Publ., 38 (2007), 131-146. 9. S.P. Lavrenyuk, N.P. Protsakh, Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia, Ukrainian Mathematical Journal, 58 (2006), №9, 1347-1368. 10. N. Protsakh, Inverse problem for a weakly nonlinear ultraparabolic equation with three unknown functions of different arguments on the right-hand side, Journal of Mathematical Sciences, 217 (2016), №4, 476-514. 11. N. Protsakh, Determining of right-hand side of higher order ultraparabolic equation, Open Math., 15 (2017), 1048-1062. 12. N. Protsakh, Inverse problem for an ultraparabolic equation with unknown function of the space variable on the right-hand side, Journal of Mathematical Sciences, 203 (2014), №1, 16-39. 13. N. Protsakh, Inverse problem for semilinear ultraparabolic equation of higher order, Mathematica Bohemica, 140 (2015), №4, 335-404. 14. Yu.A. Kosheleva, On the solvability of the nonlinear inverse problems for ultraparabolic equations, Sib. Elektron. Mat. Izv., 13 (2016), 1187-1206. 15. M. Dehghan, Determination of an unknown parameter in a semi-linear parabolic equation, Mathematical problems in engineering, 8 (2002), №2, 111-122. 16. J.R. Cannon, Y. Lin, Determination of a parameter p(t) in some quasi-linear parabolic differential equations, Inverse Probl., 4 (1988), 35-45. 17. F. Parzilvand, A.M. Shahrezaee, An inverse problem of identifying the coefficient of semilinear parabolic equation, Mathematical Researches (Sci. Kharazmi University), 2 (2016), №1, 79-88. 18. V.T. Borukhov, G.M. Zayats, Identification of a time-dependent source term in nonlinear hyperbolic or parabolic heat equation, International Journal of Heat and Mass Transfer, 91 (2015), 1106-1113. 19. P.N. Vabishchevich, V.I. Vasilev, Numerical solving the identifcation problem for the lower coencient of parabolic equation, arXiv:1304.5923v1 [cs.NA] 22 Apr 2013, 12p. 20. A.I. Prilepko, V.L. Kamynin, A.B. Kostin, Inverse source problem for parabolic equation with the condition of integral observation in time, Computational Mathematics and Mathematical Physics, 57 (2017), №6, 956-966. 21. H.P. Lopushanska, Determination of a minor coefficient in a time fractional diffusion equation, Mat. Stud., 45 (2016), №1, 57-66. 22. E. Ozbilge, A. Demir, Identification of the unknown diffusion coefficient in a linear parabolic equation via semigroup approach, Advances in Difference Equations, 2014 (2014), №47, 1-8. 23. M.I. Ivanchov, Inverse problems for equations of parabolic type, 238 p., Mathematical Studies, Monograph Series, VNTL Publishers., V.10, 2003. 24. A.I. Kozhanov, An inverse problem with an unknown coefficient and right-hand side for a parabolic equation II, J. of Inverse and ill posed problems,11 (2003), №5, 505-522. 25. S.G. Pyatkov, E.I. Safonov, Determination of the source function in mathematical models of convection diffusion, Mat. Zametki Severo-Vost. Fed. Univ., 21 (2014), №2, 117-130. 26. H. Gajewski, K. Groger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Berlin, Akademie-Verlag, 1974. 27. O.A. Ladyzhenskaya, Boundary-value problems of mathematical physics, Nauka, Moscow, 1973, 407 p. (in Russian)
Pages	60-74
Volume	50
Issue	1
Year	2018
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Problem of determining of minor coefficient and right-hand side function in semilinear ultraparabolic equation