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 qualitative properties of sign-constant solutions of some quasilinear parabolic problems”, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 160 (2019), 85–94

2.

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

2018

3.

A. B. Muravnik, “On qualitative properties of solutions to quasilinear parabolic
equations admitting degenerations at infinity”, Ufimsk. Mat. Zh., 10:4 (2018), 77–84; Ufa Math. J., 10:4 (2018), 77–84

2017

4.

A. B. Muravnik, “Asymptotic properties of solutions of two-dimensional differential-difference elliptic problems”, CMFD, 63:4 (2017), 678–688

2016

5.

A. B. Muravnik, “On the Dirichlet problem for differential-difference elliptic equations in a half-plane”, CMFD, 60 (2016), 102–113

6.

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

2014

7.

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

2013

8.

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

2006

9.

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

10.

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

2005

11.

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

2004

12.

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

13.

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

2003

14.

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

15.

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

2002

16.

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