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Boundary-Value Problems for Loaded Third-Order Parabolic-Hyperbolic Equations in Infinite Three-Dimensional Domains

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Abstract

In this paper, we study an analogue of the Gellerstedt problem for a loaded parabolic-hyperbolic equation of the third order in an infinite three-dimensional domain. The main method to study this Gellerstedt problem is the Fourier transform. Based on the Fourier transform, we reduce the considering problem to a planar analogue of the Gellerstedt spectral problem with a spectral parameter. The uniqueness of the solution of this problem is proved by the new extreme principle for loaded third-order equations of the mixed type. The existence of a regular solution of the Gellerstedt spectral problem is proved by the method of integral equations. In addition, the asymptotic behavior of the solution of the Gellerstedt spectral problem is studied for large values of the spectral parameter. Sufficient conditions are found such that all differentiation operations are legal in this work.

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REFERENCES

  1. I. M. Gelfand, ‘‘Some questions of analysis and differential equations,’’ Usp. Mat. Nauk 14 (3), 3–19 (1959).

    MathSciNet  Google Scholar 

  2. G. M. Struchina, ‘‘The problem of conjugation of two equations,’’ Inzh.-Fiz. Zh. 4 (11), 99–104 (1961).

    Google Scholar 

  3. Ya. S. Uflyand, ‘‘On the question of constructing the propagation of oscillations in composite electric lines,’’ Inzh.-Fiz. Zh. 7 (1), 89–92 (1964).

    Google Scholar 

  4. L. A. Zolina, ‘‘On a boundary value problem for a model equation of hyperbolo-parabolic type,’’ USSR Comput. Math. Math. Phys. 6 (6), 63–78 (1966).

    Article  MathSciNet  Google Scholar 

  5. M. S. Salakhitdinov, Equation of Mixed-Composite Type (Fan, Tashkent, 1974) [in Russian].

  6. T. D. Dzhuraev, Boundary-Value Problems for Equations of Mixed and Mixed-Composite Type (Fan, Tashkent, 1979) [in Russian].

  7. T. Sh. Kal’menov and M. A. Sadybekov, ‘‘On a Frankl-type problem for a mixed parabolic-hyperbolic equation,’’ Sib. Math. J. 58, 227–231 (2017).

    Article  MathSciNet  Google Scholar 

  8. K. B. Sabitov, ‘‘Dirichlet problem for a third-order equation of mixed type in a rectangular domain,’’ Diff. Equat. 47, 706–714 (2011).

    Article  MathSciNet  Google Scholar 

  9. T. K. Yuldashev, ‘‘A mixed differential equation of Boussinesq type,’’ Vestn. Volgogr. Univ., Ser. 1: Mat. Fiz., No. 2 (33), 13–26 (2016).

  10. T. K. Yuldashev, ‘‘Nonlocal problem for a mixed type differential equation in rectangular domain,’’ Tr. Yerevan. Univ., Fiz. Mat. Nauki, No. 3, 70–78 (2016).

    Google Scholar 

  11. T. K. Yuldashev, ‘‘On a mixed fourth-order differential equation,’’ Izv. IMI UdGU 47 (1), 119–128 (2016).

    MATH  Google Scholar 

  12. T. K. Yuldashev, ‘‘On an integro-differential equation of pseudoparabolic-pseudohyperbolic type with degenerate kernels,’’ Tr. Yerevan. Univ., Fiz. Mat. Nauki 52, 19–26 (2018).

    MATH  Google Scholar 

  13. A. M. Nakhushev, Equations of Mathematical Biology (Vysshaya Shkola, Moscow, 1995) [in Russian].

    MATH  Google Scholar 

  14. A. B. Nakhushev, ‘‘Nonlocal problem and the Goursat problem for loaded hyperbolic equation and their application in prediction of ground moisture,’’ Dokl. Akad. Nauk SSSR 19, 1243–1247 (1978).

    MATH  Google Scholar 

  15. A. M. Nakhushev and V. N. Borisov, ‘‘Boundary value problems for loaded parabolic equations and their applications to predicting groundwater levels,’’ Differ. Uravn. 13, 105–110 (1977).

    Google Scholar 

  16. M. Kh. Shkhanukov, ‘‘On some boundary value problems for a third-order equation that arise when modeling fluid filtration in porous media,’’ Differ. Uravn. 18, 689–699 (1982).

    MathSciNet  MATH  Google Scholar 

  17. A. I. Kozhanov, ‘‘Nonlinear loaded equations and inverse problems,’’ Comput. Math. Math. Phys. 44, 657–675 (2004).

    MathSciNet  MATH  Google Scholar 

  18. M. T. Dzhenaliev and M. I. Ramazanov, Loaded Equations as Perturbations of Differential Equations (FYLYM, Almata, 2010) [in Russian].

  19. A. M. Nakhushev, ‘‘On the Darboux problem for one degenerate loaded second-order integro-differential equation,’’ Differ. Uravn. 12, 103–108 (1976).

    Google Scholar 

  20. A. M. Nakhushev, ‘‘Loaded equations and their applications,’’ Differ. Uravn. 19, 86–94 (1983).

    MathSciNet  MATH  Google Scholar 

  21. A. B. Borodin, ‘‘On an estimate for elliptic equations and its application to loaded equations,’’ Differ. Uravn. 13, 17–22 (1977).

    MathSciNet  Google Scholar 

  22. S. Z. Dzhamalov and R. R. Ashurov, ‘‘On a nonlocal boundary-value problem for second kind second-order mixed type loaded equation in a rectangle,’’ Uzb. Mat. Zh., No. 3, 63–72 (2018).

  23. V. A. Eleev, ‘‘On some boundary value problems for mixed loaded second and third order equations,’’ Diff. Equat. 30, 210–217 (1994).

    MathSciNet  MATH  Google Scholar 

  24. B. Islomov and D. M. Kuryazov, ‘‘Boundary value problems for a mixed loaded third-order equation of parabolic-hyperbolic type,’’ Uzb. Mat. Zh., No. 2, 29–35 (2000).

  25. O. S. Zikirov, ‘‘A nonlocal boundary value problem for third order linear partial differential equation of composite type,’’ Math. Model. Anal. 14, 407–421 (2009).

    Article  MathSciNet  Google Scholar 

  26. B. S. Kishin and O. Kh. Abdullaev, ‘‘About a problem for loaded parabolic-hyperbolic type equation with fractional derivatives,’’ Int. J. Differ. Equat. 6, 9815796 (2016).

    MathSciNet  MATH  Google Scholar 

  27. B. Islomov and U. I. Baltaeva, ‘‘Boundary value problems for a third-order loaded parabolic-hyperbolic equation with variable coefficients,’’ Electron. J. Differ. Equat. 2015 (221), 1–10 (2015).

    Article  MathSciNet  Google Scholar 

  28. B. Islomov and U. I. Baltaeva, ‘‘Boundary value problems for the classical and mixed integro-differential equations with Riemann-Liuovil operators,’’ Int. J. Part. Diff. Equat. ID 157947, 11–17 (2013).

    MATH  Google Scholar 

  29. A. V. Bitsadze, ‘‘On equations of mixed type in three-dimensional domains,’’ Dokl. Akad. Nauk SSSR 143, 1017–1019 (1962).

    MathSciNet  Google Scholar 

  30. A. M. Nakhushev, ‘‘On a three-dimensional analogue of the Gellerstedt problem,’’ Differ. Uravn. 4, 52–62 (1968).

    Google Scholar 

  31. T. D. Dzhuraev, A. Sopuev, and M. Mamajonov, Boundary Value Problems for Equations of Parabolic-Hyperbolic Type (FAN, Tashkent, 1986) [in Russian].

    Google Scholar 

  32. M. S. Salakhitdinov and A. K. Urinov, Boundary Value Problems for Equations of Mixed Type with a Spectral Parameter (Fan, Tashkent, 1997) [in Russian].

  33. M. S. Salakhitdinov and B. Islomov, ‘‘On a three-dimensional analogue of the Tricomi problem for an equation of mixed type,’’ Dokl. Akad. Nauk UzSSR 311, 797–801 (1990).

    MATH  Google Scholar 

  34. E. K. Alikulov, ‘‘Uniqueness of the solution of a three-dimensional analogue of the Gellerstedt problem for a loaded equation of mixed type,’’ Uzb. Mat. Zh., No. 2, 29–39 (2011).

  35. V. A. Il’in and E. I. Moiseev, ‘‘An upper bound taken on the diagonal for the spectral function of the multidimensional Schrodinger operator with a potential satisfying the Kato condition,’’ Diff. Equat. 34, 358–368 (1998).

    MathSciNet  MATH  Google Scholar 

  36. N. Yu. Kapustin and E. I. Moiseev, ‘‘A spectral problem for the Laplace operator in the square with a spectral parameter in the boundary condition,’’ Differ. Equat. 34, 663–668 (1998).

    MathSciNet  MATH  Google Scholar 

  37. N. Yu. Kapustin and E. I. Moiseev, ‘‘Convergence of spectral expansions for functions of the Holder class for two problems with a spectral parameter in the boundary condition,’’ Diff. Equat. 36, 1182–1188 (2000).

    Article  Google Scholar 

  38. T. K. Yuldashev, ‘‘On inverse boundary value problem for a Fredholm integro-differential equation with degenerate kernel and spectral parameter,’’ Lobachevskii J. Math. 40, 230–239 (2019).

    Article  MathSciNet  Google Scholar 

  39. T. K. Yuldashev, ‘‘Spectral features of the solving of a Fredholm homogeneous integro-differential equation with integral conditions and reflecting deviation,’’ Lobachevskii J. Math. 40 (12), 2116–2123 (2019).

    Article  Google Scholar 

  40. T. K. Yuldashev, ‘‘Nonlocal mixed-value problem for a Boussinesq-type integrodifferential equation with degenerate kernel,’’ Ukr. Math. J. 68, 1278–1296 (2016).

    Article  Google Scholar 

  41. T. K. Yuldashev, ‘‘Mixed problem for pseudoparabolic integrodifferential equation with degenerate kernel,’’ Differ. Equat. 53, 99–108 (2017).

    Article  Google Scholar 

  42. A. B. Muravnik, ‘‘On qualitive properties of solutions to quasilinear parabolic equations admitting degenerations at infinity,’’ Ufa Math. J. 10 (4), 77–84 (2018).

    Article  MathSciNet  Google Scholar 

  43. G. Bateman and A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.

    MATH  Google Scholar 

  44. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge Univ. Press, Cambridge, 1966).

    MATH  Google Scholar 

  45. S. Agmon, L. Nirenberg, and M. H. Protter, ‘‘A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type,’’ Comm. Pure Appl. Math. 6, 455–470 (1953).

    Article  MathSciNet  Google Scholar 

  46. M. M. Smirnov, Equations of Mixed Type (Nauka, Moscow, 1970) [in Russian].

    Google Scholar 

  47. A. M. Ezhov and S. P. Pulkin, ‘‘Estimation of the solution of the Tricomi problem for one class of equations of mixed type,’’ Dokl. Akad. Nauk SSSR 193, 978–980 (1970).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. National University of Uzbekistan, Uzbek-Israel Joint Faculty of High Technology and Engineering Mathematics, 100174, Tashkent, Uzbekistan

    T. K. Yuldashev

  2. National University of Uzbekistan, Department of Differential Equations and Mathematical Physics, 100174, Tashkent, Uzbekistan

    B. I. Islomov

  3. Tashkent University of Information Technologies, Department of Algorithmization and Mathematical Modeling, 100200, Tashkent, Uzbekistan

    E. K. Alikulov

Authors
  1. T. K. Yuldashev
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  2. B. I. Islomov
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  3. E. K. Alikulov
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Correspondence to T. K. Yuldashev, B. I. Islomov or E. K. Alikulov.

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Yuldashev, T.K., Islomov, B.I. & Alikulov, E.K. Boundary-Value Problems for Loaded Third-Order Parabolic-Hyperbolic Equations in Infinite Three-Dimensional Domains. Lobachevskii J Math 41, 926–944 (2020). https://doi.org/10.1134/S1995080220050145

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