Abstract
We consider initial-boundary-value problems for three classes of inhomogeneous degenerate equations of mixed parabolic-hyperbolic type: mixed-type equations with degenerate hyperbolic part, mixed-type equations with degenerate parabolic part, and mixed-type equations with power degeneration. In each case, we state a criterion of uniqueness of a solution to the problem. We construct solutions as series with respect to the system of eigenfunctions of the corresponding one-dimensional spectral problem. We prove that the uniqueness of the solution and the convergence of the series depend on the ratio of sides of the rectangular from the hyperbolic part of the mixed domain. In the proof of the existence of solutions to the problem, small denominators appear that impair the convergence of series constructed. In this connection, we obtain estimates of small denominators separated from zero and the corresponding asymptotics, which allows us, under certain conditions, to prove that the solution constructed belongs to the class of regular solutions.
Similar content being viewed by others
References
V. I. Arnol’d, “Small denominators,” Izv. Ross. Akad. Nauk. Ser. Mat., 25, 21–86 (1961).
V. I. Arnol’d, “Small denominators and the problem on the stability of motion in classical and celestial mechanics,” Usp. Mat. Nauk, 18, No. 6 (114), 91–192 (1963).
H. Bateman and A. Erdélyi, Higher Transcendental Functions, McGraw Hills, New York–Toronto–London (1953).
I. M. Gel’fand, “Some problems of analysis and differential equations,” Usp. Mat. Nauk, 14, No. 3, 3–19 (1959).
T. D. Dzhuraev, Boundary-Value Problems for Equations of Mixed and Mixed-Composite Types [in Russian], Fan, Tashkent (1979).
T. D. Dzhuraev, A. Sopuev, and M. Mamazhanov, Boundary-Value Problems for Equations of Parabolic-Hyperbolic Type [in Russian], Fan, Tashkent (1986).
A. Zygmund, Trigonometric Series, Vols. 1, 2, Cambridge Univ. Press (1959).
N. Yu. Kapustin, “Tricomi problem for a parabolic-hyperbolic equation with degenerate hyperbolic part, I,” Differ. Uravn., 23, No. 1, 72–78 (1987).
N. Yu. Kapustin, “Tricomi problem for a parabolic-hyperbolic equation with degenerate hyperbolic part, II,” Differ. Uravn., 24, No. 8, 1379–1386 (1988).
N. Yu. Kapustin, Problems for parabolic-hyperbolic equations and the corresponding spectral problems with parameters at boundary points [in Russian], thesis, Moscow State Univ., Moscow (2012).
N. Yu. Kapustin and E. I. Moiseev, “On a spectral problem from the theory of parabolic-hyperbolic heat equations,” Dokl. Ross. Akad. Nauk, 352, No. 4, 451 (1997).
A. Ya. Khinchin, Continued Fractions, Univ. of Chicago Press, Chicago–London (1964).
O. A. Ladyzhenskaya and L. Stupyalis, “On equations of mixed type,” Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron., 19, No. 4, 38–46 (1965).
I. S. Lomov, “Small denominators in the analytical theory of degenerate differential equations,” Differ. Uravn., 29, No. 12, 2079–2089 (1993).
K. B. Sabitov, Functional, Differential, and Integral Equations [in Russian], Vysshaya Shkola, Moscow (2005).
K. B. Sabitov, “Tricomi problem for an equation of mixed parabolic-hyperbolic type in a rectangular domain,” Mat. Zametki, 86, No. 2, 273–279 (2009).
K. B. Sabitov, “Initial-boundary-value problem for a parabolic-hyperbolic equation with power degeneration on the transition line,” Differ. Uravn., 47, No. 1, 1–8 (2011).
K. B. Sabitov, Equations of Mathematical Physicss [in Russian], Fizmatlit, Moscow (2013).
K. B. Sabitov, Direct and Inverse Problems for Equations of Mixed Parabolic-Hyperbolic type [in Russian], Gilem, Ufa (2015).
K. B. Sabitov and L. Kh. Rakhmanova, “Initial-boundary-value problem for an equation of mixed parabolic-hyperbolic type in a rectangular domain,” Differ. Uravn., 44, No. 9, 1175–1181 (2008).
K. B. Sabitov and E. M. Safin, “Inverse problem for an equation of mixed parabolic-hyperbolic type in a rectangular domain,” Izv. Vyssh. Ucheb. Zaved. Ser. Mat., 56, No. 4, 55–62 (2010).
K. B. Sabitov and E. M. Safin, “Inverse problem for an equation of mixed parabolic-hyperbolic type,” Mat. Zametki, 87, No. 6, 907–918 (2010).
K. B. Sabitov and S. N. Sidorov, “On a nonlocal problem for a degenerate parabolic-hyperbolic equation,” Differ. Uravn., 50, No. 3, 356–365 (2014).
K. B. Sabitov and S. N. Sidorov, “Inverse problem for a degenerate parabolic-hyperbolic equation with nonlocal boundary condition,” Izv. Vyssh. Ucheb. Zaved. Ser. Mat., 1, 46–59 (2015).
A. B. Shidlovsky, Diophantine Approximations and Transcendental Numbers [in Russian], Moscow State Univ., Moscow (1982).
L. Stupyalis, “Initial-boundary-value problems for equations of mixed type,” Tr. Mat. Inst. Steklova, 27, 115–145 (1975).
Ya. S. Ufland, “On the problem on the propagation of oscillations in composite transmission lines,” Inzh.-Fiz. Zh., 7, No. 1, 89–92 (1964).
Ya. S. Ufland and I. T. Lozanovskaya, “On one class of problems of mathematical physics with mixed spectrum,” Dokl. Akad. Nauk SSSR, 164, No. 5, 1005–1007 (1965).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 137, Mathematical Physics, 2017.
Rights and permissions
About this article
Cite this article
Sabitov, K.B., Sidorov, S.N. Initial-Boundary-Value Problem for Inhomogeneous Degenerate Equations of Mixed Parabolic-Hyperbolic Type. J Math Sci 236, 603–640 (2019). https://doi.org/10.1007/s10958-018-4136-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-018-4136-y
Keywords and phrases
- mixed parabolic-hyperbolic equation
- initial-boundary-value problem
- spectral method
- uniqueness
- existence
- series
- small denominators
- uniform convergence