01.01.02 (Differential equations, dynamical systems, and optimal control)
E-mail:
Keywords:
non-local, non-linear, singular and degenerative differential and functional-differential equations; non-classical problems of mathematical physics; qualitative properties of solutions; integral transformations.
Subject:
Singular and quasi-linear parabolic and elliptic equations with singularities of Bessel operator type and non-linearities of $g(u)|\nabla u|^2$ kind are investigated. Necessary and sufficient conditions of stabilization of their solutions are found. Cauchy problem for differential-difference parabolic equations is investigated. Its classical unique solvability is proved, integral representation of its solution is obtained, asymptotic behaviour of that solution is investigated. Non-classical Cauchy problem for singular functional-differential parabolic equations, containing Bessel operators, translation operators and (integral) generalized translation operators, is investigated. Its classical unique solvability is proved, integral representation of its solution is obtained. Fourier–Bessel transforms of measures were studied. Estimates of $\Vert r^\mu\cdot\Vert_\infty\le C\Vert r^{\mu-1}\cdot\Vert_1}$ type are obtained for their weighted spherical means. Specified estimates are applied to the investigation of qualitative properties of solutions of singular differential and operator equations (including non-existence theorems).
Biography
Graduated from Faculty of Applied Mathematics and Mechanics of Voronezh State University (VSU) in 1983 (department of differentila equations). Ph.D. thesis was defended in 1986. A list of my works contains about 50 titles.
Main publications:
Muravnik A. B. On weighted norm estimates for mixed Fourier–Bessel transforms of non-negative functions // Pitman Res. Notes Math. Ser., 1997, 374, 119–123.
Muravnik A. B. Properties of stabilization functional for parabolic Cauchy problem // Progr. in Nonlin. Diff. Equations and Their Applications, 2000, 42, 217–221.
Muravnik A. B. Fourier–Bessel transformation of measures with several special variables and properties of singular differential equations // J. Korean Math. Soc., 2000, 37(6), 1043–1057.
Muravnik A. B. Fourier–Bessel transformation of compactly supported non-negative functions and estimates of solutions of singular differential equations // Functional Differential Equations, 2001, 8(3–4), 353–363.
Muravnik A. B. Fourier–Bessel transformation of measures and singular differential equations // North-Holland Math. Studies, 2001, 189, 335–345.
A. B. Muravnik, “On Hyperbolic Equations with Arbitrarily Directed Translations of Potentials”, Math. Notes, 115:5 (2024), 772–778
2023
2.
Viktoriia V. Liiko, Andrey B. Muravnik, “Elliptic equations with arbitrarily directed translations in half-spaces”, Bulletin of Irkutsk State University. Series Mathematics, 43 (2023), 64–77
N. V. Zaitseva, A. B. Muravnik, “Classical solutions of hyperbolic differential-difference equation with shift by an arbitrary vector”, Izv. Vyssh. Uchebn. Zaved. Mat., 2023, no. 5, 34–40
4.
A. B. Muravnik, “On hyperbolic equations with arbitrarily directed translations of potentials”, Applied Mathematics & Physics, 55:4 (2023), 299–304
5.
A. B. Muravnik, “Elliptic differential-difference problems in half-spaces: case of summable functions”, Ufimsk. Mat. Zh., 15:3 (2023), 100–108; Ufa Math. J., 15:3 (2023), 97–105
2022
6.
A. B. Muravnik, “Elliptic Equations with Translations of General Form in a Half-Space”, Mat. Zametki, 111:4 (2022), 571–580; Math. Notes, 111:4 (2022), 587–594
A. B. Muravnik, “Elliptic Differential-Difference Equations of General Form in the Half-Space”, Mat. Zametki, 110:1 (2021), 90–98; Math. Notes, 110:1 (2021), 92–99
A. B. Muravnik, “Elliptic Differential-Difference Equations in the Half-Space”, Mat. Zametki, 108:5 (2020), 764–770; Math. Notes, 108:5 (2020), 727–732
A. B. Muravnik, “On qualitative properties of sign-constant solutions of some quasilinear parabolic problems”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 160 (2019), 85–94
13.
A. B. Muravnik, “Elliptic Problems with Nonlocal Potential Arising in Models of Nonlinear Optics”, Mat. Zametki, 105:5 (2019), 747–762; Math. Notes, 105:5 (2019), 734–746
A. B. Muravnik, “Asymptotic Properties of Solutions of the Dirichlet Problem in the Half-Plane for Differential-Difference Elliptic Equations”, Mat. Zametki, 100:4 (2016), 566–576; Math. Notes, 100:4 (2016), 579–588
A. B. Muravnik, “Functional differential parabolic equations: integral transformations and qualitative properties of solutions of the Cauchy problem”, CMFD, 52 (2014), 3–141; Journal of Mathematical Sciences, 216:3 (2016), 345–496
A. B. Muravnik, “On blow-up of solutions of some systems of quasilinear parabolic inequalities”, CMFD, 48 (2013), 84–92; Journal of Mathematical Sciences, 202:6 (2014), 859–868
A. B. Muravnik, “On stabilization of solutions of singular elliptic equations”, Fundam. Prikl. Mat., 12:4 (2006), 169–186; J. Math. Sci., 150:5 (2008), 2408–2421
A. B. Muravnik, “On asymptotics of solutions of parabolic equations with nonlocal high-order terms”, Tr. Semim. im. I. G. Petrovskogo, 25 (2006), 143–183; J. Math. Sci. (N. Y.), 135:1 (2006), 2695–2720
A. B. Muravnik, “On the Asymptotics of the Solution of the Cauchy Problem for Some Differential-Difference Parabolic Equations”, Differ. Uravn., 41:4 (2005), 538–548; Differ. Equ., 41:4 (2005), 570–581
A. B. Muravnik, “Uniqueness of the solution of the Cauchy problem for some differential-difference parabolic equations”, Differ. Uravn., 40:10 (2004), 1385–1389; Differ. Equ., 40:10 (2004), 1461–1466
A. B. Muravnik, “On the Unique Solvability of the Cauchy Problem for Some Difference-Differential Parabolic Equations”, Differ. Uravn., 40:5 (2004), 692–701; Differ. Equ., 40:5 (2004), 742–752
A. B. Muravnik, “Stabilization of Solutions of Certain Singular Quasilinear Parabolic Equations”, Mat. Zametki, 74:6 (2003), 858–865; Math. Notes, 74:6 (2003), 812–818
A. B. Muravnik, “The Cauchy Problem for Certain Inhomogeneous Difference-Differential Parabolic Equations”, Mat. Zametki, 74:4 (2003), 538–548; Math. Notes, 74:4 (2003), 510–519
V. N. Denisov, A. B. Muravnik, “On Stabilization of the Solution of the Cauchy Problem for Quasilinear Parabolic Equations”, Differ. Uravn., 38:3 (2002), 351–355; Differ. Equ., 38:3 (2002), 369–374
A. Ya. Belov, V. M. Buchstaber, V. B. Vasilyev, Yu. P. Virchenko, S. K. Kuizheva, A. G. Kusraev, D. K. Mamyi, A. B. Muravnik, A. V. Pskhu, S. A. Rozanova, Z. A. Saidov, A. L. Semenov, S. M. Sitnik, A. L. Skubachevskii, A. P. Soldatov, E. L. Shishkina, A. G. Yagola, “Sultan Nazhmudinovich Askhabov (on the 70-th anniversary of his birth)”, Chebyshevskii Sb., 25:2 (2024), 5–19
2020
29.
O. V. Anashkin, E. M. Varfolomeev, V. I. Voytitsky, V. I. Donskoy, D. A. Zakora, A. B. Muravnik, M. A. Muratov, Yu. S. Pashkova, V. È. Petrov, E. V. Plokhaya, A. L. Skubachevskii, P. A. Starkov, T. A. Suslina, D. O. Tsvetkov, V. N. Chekhov, A. A. Shkalikov, A. I. Yakovlev, “Nikolay Dmitrievich Kopachevsky. March 25, 1940 — May 18, 2020”, CMFD, 66:2 (2020), 157–159
Гиперболические уравнения с положительными коэффициентами при нелокальных потенциалах A. B. Muravnik VI International Conference "Function Spaces. Differential Operators. Problems of Mathematical Education", dedicated to the centennial anniversary of the corresponding member of Russian Academy of Sciences, academician of European Academy of Sciences L.D. Kudryavtsev November 17, 2023 18:20
6.
Blow-Up phenomena for coercive KPZ-type inequalities A. B. Muravnik International Scientific Conference "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis - 2023 (OTHA-2023)" August 21, 2023 12:10
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