Speciality:
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).
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.
