We study the conditional well-posedness of the initial-boundary value problem for a system of inhomogeneous mixed type equations with two degeneration lines. We establish the conditional well-posedness of the problem, i.e., we prove the uniqueness and conditional stability theorems.
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L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics Dover, Mineola, NY (2016).
A. G. Kuzmin, Nonclassical Equations of Mixed Type and Their Applications to Gas Dynamics, Leningrad State Univ. Press, Leningrad (1990).
F. I. Frankl, Selected Works on Gas Dynamics [in Russian], Nauka, Moscow (1973).
V. N. Vragov, “Theory of boundary value problems for equations of mixed type in space,” Differ. Equations 13, No. 6, 759–764 (1977).
K. B. Sabitov, “On the theory of the Frankl problem for a mixed type equations,” Izv. Math. 81, No. 1, 99–136 (2017).
A. I. Kozhanov, Composite Type Equations and Inverse Problems, VSP, Utrecht (1999).
N. V. Kislov, “Nonhomogeneous boundary value problems for differential-operator equations of mixed type, and their application,” Math. USSR, Sb. 53, No. 1, 17–35 (1986).
A. A. Gimaltdinova, “The Dirichlet problem for the Lavrent’ev-Bitsadze equation with two type-change lines in a rectangular domain,” Dokl. Math. 91, No. 1, 41–46 (2015).
S. G. Krein and O. I. Prozorovskaya, “Approximate methods of solving ill-posed problems,” U.S.S.R. Comput. Math. Math. Phys. 3, No. 1, 153–167 (1963).
M. M. Lavrent‘ev and L. Y. Savel’ev, Theory of Operators and Ill-Posed Problems [in Russian], Sobolev Institute of Mathematics Press, Novosibirsk (2010).
H. A. Levine, “Logarithmic convexity, first order differential inequalities and some applications,” Trans. Am. Math. Soc. No. 152, 299–320 (1970).
A. L. Bukhgeim, “Ill-posed problems, number theory and tomography,” Sib. Math. J. 33, No. 3, 389–402 (1992).
K. S. Fayazov, “The Cauchy problem for an elliptic equation with operator coefficients,” Sib. Math. J. 36, No. 2, 404–411 (1995).
K. S. Fayazov and Z. Sh. Abdullayeva, “Conditional correctness of the internal boundary value problem of the pseudoparabolic equation with a changing time direction,” Missouri J. Math. Sci. 32, No. 1, 49–60 (2020).
K. S. Fayazov and I. O. Khajiev, “Conditional correctness of the initial-boundary value problem for a system of high-order mixed-type equations,” Russ. Math. 66, No. 2, 53–63 (2022).
I. O. Khajiev, “Conditional correctness and approximate solution of boundary value problem for the system of second order mixed-type equations,” J. Sib. Fed. Univ., Math. Phys. 11, No. 2, 231–241 (2018).
K. S. Fayazov and Y. K. Khudayberganov, “An ill-posed boundary value problem for a mixed type second-order differential equation with two degenerate lines,” Mat. Zamet. SVFU 30, No. 1, 51–62 (2023).
K. S. Fayazov and Y. K. Khudayberganov, “Ill-posed boundary-value problem for a system of partial differential equations with two degenerate lines,” J. Sib. Fed. Univ., Math. Phys. 12, No. 3, 392–401 (2019).
K. S. Fayazov and Y. K. Khudayberganov, “Ill-posed boundary value problem for mixed type system equations with two degenerate lines,” Sib. Èlectron. Mat. Izv. 17, 647–660 (2020).
K. S. Fayazov and M. M. Lavrent’ev, “Cauchy problem for partial differential equations with operator coefficients in space,” J. Inverse Ill-Posed Probl. 2, No. 4, 283–296 (1994).
A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, CRC, Boca Raton, FL (2007).
S. G. Pyatkov, “Properties of eigenfunctions of a certain spectral problem and their applications” [in Russian], In: Applications of Functional Analysis to Equations of Mathematical Physics, pp. 65–84, Sobolev Institute of Mathematics Press, Novosibirsk, (1986).
I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Am. Math. Soc., Providence, RI (1969).
S. G. Pyatkov, “Some properties of proper and attached functions of unfamiliar Sturm-Liouville problems,” In : Nonclassical Equations of Mathematical Physics, pp. 240–251, Sobolev Institute of Mathematics Press, Novosibirsk, (2005).
K. S. Fayazov and I. O. Khazhiev, “Conditional stability of a boundary value problem for a system abstract differential equations of the second order with operator coefficients” [in Russian], Uzb. Math. J. No 2, 145–155 (2017).
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Fayazov, K., Khudayberganov, Y. & Pyatkov, S. Conditional Well-Posedness of the Initial-Boundary Value Problem for a System of Inhomogeneous Mixed Type Equations with Two Degeneration Lines. J Math Sci 274, 201–214 (2023). https://doi.org/10.1007/s10958-023-06589-6
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DOI: https://doi.org/10.1007/s10958-023-06589-6