Abstract
We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation {fx1870-01} in an unbounded domain with respect to the variables x.
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A. N. Kolmogorov, “Zufällige Bewegungen (Zur Theorie der Bownschen Bewegung),” Ann. Math., 35, 116–117 (1934).
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin (1975).
E. Landconelli, A. Pascucci, and S. Polidoro, “Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance,” in: Nonlinear Problems in Mathematical Physics and Related Topics. II. In Honour of Prof. O. A. Ladyzhenskaya, Kluwer, New York (2002), pp. 243–265.
S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser, Berlin (2004).
V. S. Dron’ and S. D. Ivasyshen, “On correct solvability of the Cauchy problem for degenerate parabolic equations of Kolmogorov type,” Ukr. Mat. Visn., No. 1, 61–68 (2004).
O. H. Voznyak and S. D. Ivasyshen, “Fundamental solutions of the Cauchy problem for one class of degenerate parabolic equations and their application,” Dopov. Nats. Akad. Nauk Ukr., No. 10, 11–16 (1996).
S. D. Éidel’man and A. P. Malitskaya, “On fundamental solutions and stabilization of a solution of the Cauchy problem for one class of degenerate parabolic equations,” Differents. Uravn., 11, No. 7, 1316–1331 (1975).
S. Polidoro, “On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance,” Nonlin. Anal., 47, 491–502 (2001).
S. P. Lavrenyuk and N. P. Protsakh, “Mixed problem for an ultraparabolic equation in an unbounded domain,” Ukr. Mat. Zh., 51, No. 8, 1053–1066 (2000).
H. M. Barabash, S. P. Lavrenyuk, and N. P. Protsakh, “Mixed problem for a semilinear ultraparabolic equation,” Mat. Met. Fiz.-Mekh. Polya, 45, No. 4, 27–34 (2002).
F. Lascialfari and D. Morbidelli, “A boundary-value problem for a class of quasilinear ultraparabolic equations,” Commun. Part. Different. Equat., 23, No. 5, 6, 847–868 (1998).
N. I. Huzil’ and S. P. Lavrenyuk, “Mixed problem for a semilinear ultraparabolic system in an unbounded domain,” Dopov. Nats. Akad. Nauk Ukr., No. 5, 11–16 (2005).
N. Huzil’, “Problem without initial conditions for a system of ultraparabolic equations,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 63, 59–76 (2004).
O. A. Oleinik and E. V. Radkevich, “Method of introduction of a parameter for the investigation of evolution equations,” Usp. Mat. Nauk, 33, Issue 5, 7–72 (1978).
H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen [Russian translation], Mir, Moscow (1978).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1661–1673, December, 2007.
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Lavrenyuk, S.P., Oliskevych, M.O. Mixed problem for a semilinear ultraparabolic equation in an unbounded domain. Ukr Math J 59, 1870–1884 (2007). https://doi.org/10.1007/s11253-008-0031-x
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DOI: https://doi.org/10.1007/s11253-008-0031-x