Abstract
In the domain that is the product of a segment and a p-dimensional torus, we investigate the well-posedness of a problem with nonlocal boundary conditions for a partial differential equation unsolved with respect to the leading derivative with respect to a selected variable. We establish conditions for the the classical well-posedness of the problem and prove metric theorems on the lower bounds of small denominators appearing in the course of its solution.
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REFERENCES
A. A. Dezin, ”Simplest solvable extensions for ultrahyperbolic and pseudoparabolic operators,” Dokl. Akad. Nauk SSSR, 148, No. 5, 1013-1016 (1963).
A. A. Dezin, General Problems of the Theory of Boundary-Value Problems [in Russian], Nauka, Moscow (1980).
V. K. Romanko, “On systems of operator differential equations,” Differents. Uravn., 23, No. 9, 1574-1585 (1987).
P. I. Kalenyuk and Z. M. Nytrebych, “Construction of solutions of certain boundary-value problems for linear partial differential equations admitting separation of variables,” in: Boundary-Value Problems with Various Degenerations and Singularities [in Ukrainian], Ruta, Chernivtsi (1990), pp. 62-71.
T. I. Kiguradze, “On solutions of quasilinear hyperbolic systems bounded and periodic in a strip,” Differents. Uravn., 30, No. 10, 1760-1773 (1994).
Yu. A. Mitropol'skii and L. B. Urmancheva, “On a two-point problem for a system of hyperbolic equations,” Ukr. Mat. Zh., 42, No. 2, 1657-1663 (1990).
Yu. A. Mitropol'skii, M. Kh. Shkhanukov, and A. A. Berezovskii, “On one nonlocal problem for a parabolic equation,” Ukr. Mat. Zh., 47, No. 6, 790-800 (1995).
V. K. Romanko, “Nonlocal boundary-value problems for certain systems of equations,” Mat. Zametki, 37, No. 5, 727-733 (1985).
V. K. Romanko, “On general boundary-value problems for nonlinear systems of equations with constant coefficients,” Mat. Met. Fiz.-Mekh. Polya, Issue 23, 3-7 (1986).
L. V. Fardigola, “Well-posed problems in a layer with differential operators in a boundary condition,” Ukr. Mat. Zh., 44, No. 8, 1083-1090 (1992).
L. V. Fardigola, “A nonlocal boundary-value problem in a layer for an evolution equation of the second order with respect to time variable,” Differents. Uravn., 31, No. 4, 662-671 (1995).
L. V. Fardigola, “Nonlocal two-point boundary-value problems in a layer with differential operators in the boundary condition,” Ukr. Mat. Zh., 47, No. 8, 1122-1128 (1995).
M. Kh. Shkhanukov, “On some boundary-value problems for a third-order equation appearing in the simulation of filtration of a fluid in porous media,” Differents. Uravn., 18, No. 4, 689-699 (1982).
T. P. Hoi, “A problem with nonlocal conditions for a partial differential equation perturbed by a nonlinear integro-differential term,” Mat. Stud., 8, No. 1, 71-78 (1997).
N. M. Zadorozhna and B. I. Ptashnyk, “A nonlocal boundary-value problem for parabolic equations with varying coefficients,” Ukr. Mat. Zh., 47, No. 7, 915-921 (1995).
V. S. Il'kiv, V. N. Polishchuk, and B. I. Ptashnik, “A nonlocal boundary-value problem for a system of pseudodifferential equations,” in: Methods for Investigation of Differential and Integral Operators [in Russian], Naukova Dumka, Kiev (1989), pp. 75-79.
L. I. Komarnyts'ka, “A nonlocal boundary-value problem for an equation with varying coefficients unsolved with respect to the leading derivative,” Visn. L'viv Univ., Ser. Mekh.-Mat., Issue 40, 17-23 (1994).
L. I. Komarnyts'ka and B. I. Ptashnyk, “A problem with nonlocal conditions for a partial differential equation unsolved with respect to the leading time derivative,” in: Boundary-Value Problems with Various Degenerations and Singularities [in Ukrainian], Ruta, Chernivtsi (1990), pp. 86-95.
L. I. Komarnyts'ka, Boundary-Value Problems for Partial Differential Equations and Systems Unsolved with Respect to the Leading Time Derivative[in Ukrainian], Authors's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Lviv (1995).
V. N. Polishchuk, “A problem with nonlocal boundary conditions for hyperbolic systems of differential equations with constant coefficients,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 3, 171-175 (1979).
B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).
B. I. Ptashnyk, M. M. Symotyuk, and N. M. Zadorozhna, “A problem with nonlocal conditions for quasilinear hyperbolic equations,” in: Nonlinear Boundary-Value Problems [in Ukrainian], Issue 11 (2001), pp. 161-167.
M. M. Symotyuk, “A nonlocal problem for nonisotropic partial differential equations,” in: Proceedings of the Open Scientific Conference of Young Researchers and Specialists of the Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences [in Ukrainian], Lviv (2002), pp. 161-164.
G. E. Shilov, Mathematical Analysis. Second Special Course [in Russian], Nauka, Moscow (1965).
D. K. Faddeev and I. S. Sominskii, Higher Algebra Problem Book [in Russian], Nauka, Moscow (1972).
V. I. Bernik, B. I. Ptashnik, and B. O. Salyga, “An analog of a multipoint problem for a hyperbolic equation with constant coefficients,” Differents. Uravn., 13, No. 4, 637-645 (1977).
V. G. Sprindzhuk, Metric Theory of Diophantine Approximations [in Russian], Nauka, Moscow (1977)
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Vlasii, O.D., Ptashnyk, B.I. A Problem with Nonlocal Conditions for Partial Differential Equations Unsolved with Respect to the Leading Derivative. Ukrainian Mathematical Journal 55, 1238–1253 (2003). https://doi.org/10.1023/B:UKMA.0000010756.46401.2f
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DOI: https://doi.org/10.1023/B:UKMA.0000010756.46401.2f