Фундаментальні розв’язки для вироджених параболічних рівнянь: існування, властивості та деякі їх застосування

I. P. Medynsky

Анотація


Зроблено огляд результатів побудови, дослідження і застосувань фундаментальних розв’язків задачі Коші для декількох класів вироджених параболічних рівнянь.

 

Зразок для цитування: І. П. Мединський, “Фундаментальні розв’язки для вироджених параболічних рівнянь: існування, властивості та деякі їх застосування,” Мат. методи та фіз.-мех. поля, 64, No. 2, 5–30 (2021), https://doi.org/10.15407/mmpmf2021.64.2.5-30

Translation: I. P. Medynsky, “Fundamental solutions for degenerate parabolic equations: Existence, properties, and some applications,” J. Math. Sci., 277, No. 1, 1–32 (2023), https://doi.org/10.1007/s10958-023-06810-6


Ключові слова


параболічні рівняння, виродження на початковій гіперплощині, вироджені рівняння типу Колмогорова, фундаментальний розв’язок задачі Коші, об’ємний потенціал, інтегральні зображення розв’язків

Посилання


M. S. Agranovich, M. I. Vishik, “Elliptic problems with a parameter and parabolic problems of general type,” Uspekhi Mat. Nauk, 19, No. 3(117), 53–161 (1964); English translation: Russ. Math. Surv., 19, No. 3, 53–157 (1964), https://doi.org/10.1070/RM1964v019n03ABEH001149

V. I. Bogachev, N. V. Krylov, M. Röckner, S. V. Shaposhnikov, Fokker–Planck-Kolmogorov Equations [in Russian], Inst. Komp. Issled., Moscow (2013); [in English] Am. Math. Soc. (2015).

O. H. Voznyak, S. D. Ivasyshen, I. P. Medyns’kyi, “On the fundamental solution of the Cauchy problem for Kolmogorov ultraparabolic equation with degeneration in the initial plane,” Bukov. Mat. Zh., 3, No. 3-4, 41–51 (2015) (in Ukrainian).

O. H. Voznyak, S. D. Ivasyshen, I. P. Medyns’kyi, “On the fundamental solution of the Cauchy problem for the Kolmogorov ultraparabolic equation with two groups of space variables and degeneration in the initial hyperplane,” Visn. Nats. Univ. “L’vivs’ka Politekhnika”, Ser. Fiz.-Mat. Nauky, No. 871, 46–64 (2017) (in Ukrainian).

O. H. Voznyak, S. D. Ivasyshen, I. P. Medyns’kyi, “The fundamental solution of the Cauchy problem for the ultraparabolic equation of Kolmogorov type with two groups of space variables and degeneration in the initial hyperplane,” Visn. Nats. Univ. “L’vivs’ka Politekhnika”, Ser. Fiz.-Mat. Nauky, No. 898, 13–21 (2018) (in Ukrainian), https://doi.org/10.30970/vmm.2019.88.107-127

O. Voznyak, S. Ivasyshen, I. Medyns’kyi, “The fundamental solution of the Cauchy problem for the ultraparabolic equation of Kolmogorov type with three groups of space variables and degeneration in the initial hyperplane,” Visn. L’viv Univ., Ser. Mekh.-Mat., No. 88, 107–127 (2019) (in Ukrainian), https://dx.doi.org/10.30570/vmm.2019.88.107-127

N. V. Zhitarashu, S. D. Eidelman, Parabolic Boundary Value Problems [in Russian], Chişinău, "Ştiinţa" (1992); [in English] Birkhäuser, Basel (1998).

S. D. Ivasishen, “Integral representation and initial values of the solutions of 2b-parabolic systems,” Ukr. Mat. Zh., 42, No. 4, 500-506 (1990); English translation: Ukr. Math. J., 42, 443–448 (1990), https://doi.org/10.1007/BF01071332

S. D. Ivasishen, Green’s Matrices of Parabolic Boundary-Value Problems [in Russian], Vyshcha Shkola, Kyiv (1990).

A. M. Il’in, A. S. Kalashnikov, O. A. Oleinik, “Linear equations of the second order of parabolic type,” Uspekhi Mat. Nauk, 17, No. 3, 3–146 (1962); English translation: Russ. Math. Surv., 17, No. 3, 1–143 (1962)..

S. D. Ivasyshen, I. P. Medyns’kyi, “Properties of integrals which have the type of derivatives of the volume potentials for 2b-parabolic systems with degenerations in the initial hyperplane,” Mat. Met. Fiz.-Mekh. Polya, 45, No. 4, 76-86 (2002) (in Ukrainian).

S. D. Ivasyshen, I. P. Medyns’kyi, “The Cauchy problem for 2b-parabolic systems with degeneration in initial hyperplane,” Mat. Met. Fiz.-Mekh. Polya, 46, No. 3, 15-24 (2003) (in Ukrainian).

S. D. Ivasyshen, I. P. Medyns’kyi, “Local solvability of Cauchy problem for quasi-linear 2b-parabolic systems with weak degeneration in initial hyperplane,” Mat. Met. Fiz.-Mekh. Polya, 47, No. 4, 110-114 (2004) (in Ukrainian).

S. D. Ivasyshen, I. P. Medyns’kyi, “A priori estimates for solutions of parabolic systems with degeneration in the initial hyperplane and their applications,” in: Nonlinear Analysis: Proc. Ukr. Math. Congress–2001, Inst. Matem. Nats. Akad. Nauk Ukr. (2005), pp. 28-41 (in Ukrainian).

S. D. Ivasyshen, “Solutions of parabolic equations from the families of Banach spaces dependent of time,” Mat. Stud., 40, No. 2, 172–181 (2013) (in Ukrainian).

S. D. Ivasyshen, I. P. Medyns’kyi, “Classical fundamental solution of degenerate Kolmogorov equation whose coefficients are independent of the variables of degeneration,” Bukov. Mat. Zh., 2, No. 2-3, 94-106 (2014) (in Ukrainian).

S. D. Ivasyshen, I. P. Medyns’kyi, “Classical fundamental solutions of the Cauchy problem for ultraparabolic Kolmogorov-type equations with two groups of spatial variables,” in: V. A. Mykhailets’ (editor), Differential Equations and Related Problems of Analysis, Collection of Works, Institute of Mathematics of National Academy of Sciences of Ukraine, Vol. 13, No. 1, Kyiv (2016), pp. 108–155 (in Ukrainian).

S. D. Ivasyshen, I. P. Medyns’kyi, “On the classical fundamental solutions of the Cauchy problem for ultraparabolic Kolmogorov-type equations with two groups of spatial variables,” Mat. Met. Fiz.-Mekh. Polya, 59, No. 2, 28–42 (2016); English translation: J. Math. Sci., 231, No. 4, 507–526 (2018), https://doi.org/10.1007/s10958-018-3830-0

S. D. Ivasyshen, I. P. Medyns’kyi, H. S. Pasichnyk, “Parabolic equations with degenerations in the initial hyperplane,” Bukov. Mat. Zh., 4, No. 3-4, 57–68 (2016) (in Ukrainian).

S. D. Ivasyshen, I. P. Medyns’kyi, H. S. Pasichnyk, “Parabolic equations with different singularities and degenerations,” in: Nonclassical Problems of the Theory of Differential Equations: Collection of works dedicated to the 80th anniversary of B. Yo. Ptashnyk [in Ukrainian], Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine (2017), pp. 68-76.

S. D. Ivasyshen, I. P. Medynsky, “Classical fundamental solution of the Cauchy problem for ultraparabolic Kolmogorov-type equations with two groups of spatial variables of degeneration. I,” Mat. Met. Fiz.-Mekh. Polya, 60, No. 3, 9–31 (2017); English translation: J. Math. Sci., 246, No. 2, 121–151 (2020), https://doi.org/10.1007/s10958-020-04726-z

S. D. Ivasyshen, I. P. Medynsky, “Classical fundamental solution of the Cauchy problem for ultraparabolic Kolmogorov-type equations with two groups of spatial variables of degeneration. II,” Mat. Met. Fiz.-Mekh. Polya, 60, No. 4, 7–24 (2017); English translation: J. Math. Sci., 247, No. 1, 1–23 (2020), https://doi.org/10.1007/s10958-020-04786-1

S. D. Ivasyshen, I. P. Medynsky, “Properties of fundamental solutions, theorems on integral representations of solutions and correct solvability of the Cauchy problem for ultraparabolic Kolmogorov-type equations with two groups of space variables of degeneration,” Mat. Met. Fiz.-Mekh. Polya, 61, No. 4, 7–16 (2018); English translation: J. Math. Sci., 256, No. 4, 363–374 (2021), https://doi.org/10.1007/s10958-021-05432-0

S. D. Ivasyshen, I. P. Medynsky, “Fundamental solution of the Cauchy problem for degenerate parabolic Kolmogorov-type equations of any order,” Mat. Met. Fiz.-Mekh. Polya, 62, No. 1, 7–24 (2019); English translation: J. Math. Sci., 258, No. 4, 369–391 (2021), https://doi.org/10.1007/s10958-021-05554-5

O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967); [in English] Transl. Math. Monogr., Vol. 23, AMS, Providence, RI (1968).

I. P. Medynsky, “Investigations by S. D. Eidel’man of nonlinear problems and their development,” Nauk. Visn. Cherniv. Nats. Univ. Ser. Matem., 1, No. 1-2, 114–128 (2011) (in Ukrainian).

I. P. Medynsky, “Correct solvability of the Cauchy problem and integral representations of the solutions of ultraparabolic equations of Kolmogorov type with two groups of spatial variables of degeneration,” Mat. Met. Fiz.-Mekh. Polya, 62, No. 4, 39–48 (2019) (in Ukrainian).

I. P. Medynsky, Fundamental Solutions of the Cauchy Problem for Degenerate Parabolic Equations, [in Ukrainian], Doctor-Degree Thesis (Physics and Mathematics), Lviv (2021).

I. G. Petrovskii, “ On the Cauchy problem for the systems of partial differential equations in the domain of non-analytic functions,” Bull. MGU, Ser. Mat. Mekh., 1, No. 7, 1–72 (1938) (in Russian).

F. O. Porper, S. D. Eidel’man, “Two-sided estimates of fundamental solutions of second-order parabolic equations and some applications,” Uspekhi Mat. Nauk, 39, No. 3, 107–156 (1984); English translation: Russ. Math. Surv., 39, No. 3, 119–178 (1984).

N. P. Protsakh, B. Yo. Ptashnyk, Nonlinear Ultraparabolic Equations and Variational Inequalities [in Ukrainian], Nauk. Dumka, Kyiv (2017).

V. A. Solonnikov, “On boundary value problems for linear parabolic systems of differential equations of general form”, Boundary Value Problems of Mathematical Physics, Part 3, Trudy Mat. Inst. Steklov., 83, 3–163 (1965) (in Russian).

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs (1964).

S. D. Eidel’man, Parabolic Systems [in Russian], Nauka, Moscow (1964); [in English] North-Holland, Amsterdam (1969).

S. D. Eidel’man, “Parabolic equations,” Itogi Nauki Tekh., Ser. Sovr.. Probl. Mat., Fundam. Naprav., Vol. 63, VINITI, Moscow (1990), pp. 201–313.

G. Citti, A. Pascucci, S. Polidoro, “On the regularity of solutions to a nonlinear ultraparabolic equations arising in mathematical finance,” Differ. Integral Equat., 14, No. 6, 701–738 (2001).

M. Di Francesco, A. Pascucci, “A continuous dependence result for ultraparabolic equations in option pricing,” J. Math. Anal. Appl., 336, No. 2, 1026–1041 (2007), https://doi.org/10.1016/j.jmaa.2007.03.031

M. Di Francesco, A. Pascucci, “On a class of degenerate parabolic equations of Kolmogorov type,” Appl. Math. Res. Express, 2005, No. 3, 77–116 (2005), https://doi.org/10.1155/AMRX.2005.77

V. S. Dron’, S. D. Ivasyshen, I. P. Medyns’kyi, “Properties of integrals which have the type of derivatives of volume potentials for one Kolmogorov-type ultraparabolic arbitrary order equations,” Karpat. Mat. Publ., 11, No. 2, 268–280 (2019), https://doi.org/10.15330/cmp.11.2.268-280

S. D. Eidelman, S. D. Ivasyshen, “On solutions of parabolic equations from families of Banach spaces dependent on time,” in: V. M. Adamyan et al. (eds), Differential Operators and Related Topics. Ser. Operator Theory: Adv. and Appl., Vol. 117, Birkhäuser, Basel (2000), pp. 111–125, https://doi.org/10.1007/978-3-0348-8403-7_10

S. D. Eidelman, S. D. Ivasyshen, A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Ser. Operator Theory: Adv. and Appl., Vol. 152, Birkhäuser, Basel (2004), https://doi.org/10.1007/978-3-0348-7844-9

P. Foschi, A. Pascucci, “Kolmogorov equations arising in finance: direct and inverse problems,” Lect. Notes of Seminario Interdisciplinare di Matematica. Universita degli Studi della Basilicata, VI, 145–156 (2007).

S. D. Ivasishen, I. P. Medynsky, “The Fokker–Planck–Kolmogorov equations for some degenerate diffusion processes,” Theory Stoch. Process, 16(32), No. 1, 57–66 (2010).

S. D. Ivasishen, I. P. Medynsky, “On applications of the Levi method in the theory of parabolic equations,” Mat. Stud., 47, No. 1, 33–46 (2017), https://doi.org/10.15330/ms.47.1.33-46

A. Kolmogoroff, “Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung),” Ann. Math., Sec. Ser., 35, No. 1, 116–117 (1934), https://doi.org/10.2307/1968123

E. Lanconelli, S. Polidoro, “On a class of hypoelliptic evolution operators,” Rend. Sem. Mat. Univ. Politec. Torino. Partial Diff. Eqs., 52, No. 1, 29–63 (1994).

I. P. Medynsky, “On properties of solutions for Fokker–Planck–Kolmogorov equations,” Math. Model. Comput., 7, No. 1, 158–168 (2020), https://doi.org/10.23939/mmc2020.01.158

A. Pascucci, “Kolmogorov equations in physics and in finance,” in: Elliptic and Parabolic Problems, Ser. Progress in Nonlinear Differential Equations and their Applications, Vol. 63, Birkhauser, Basel (2005), pp. 313–324.

S. Polidoro, “On a class of ultraparabolic operators of Kolmogorov–Fokker–Planck type,” Le Matematiche, 49, No. 1, 53–105 (1994).


Повний текст: PDF

Посилання

  • Поки немає зовнішніх посилань.


Creative Commons License
Ця робота ліцензована Creative Commons Attribution 3.0 License.