Plenary Lecture
Methods of Studying Qualitative Properties of Nonlinear Differential Equations
Professor Irina V. Astashova
Faculty of Mechanics and Mathematics
Lomonosov Moscow State University
&
Faculty of Mathematical Economics, Statistics and Informatics
Plekhanov Russian University of Economics
Russia
E-mail: ast@diffiety.ac.ru
Abstract: Different methods for studying qualitative properties of solutions to nonlinear equations and several author's results obtained for higher-order equations with regular and singular power-law nonlinearity are presented. Positive and oscillatory solutions to equations are studied. We consider application of dynamical systems of the (n-1)-th order to study the n-th order equations. We will discuss the existence and asymptotic behavior of singular ("blow-up", "black hole", and "white hole") and regular (Kneser, periodic) solutions. In particular, for slightly superlinear equations we prove Kiguradze’s hypothesis about power-law behavior of all “blow –up” solutions. We will use different variants of the Hopf bifurcation theorem to prove the existence of such solutions with non-power-law behavior, it means that in general case this hypothesis is not true. The influence of spectral properties of a linear operator on the asymptotic behavior of solutions to nonlinear differential equations will describe. A method of factorization of linear differential operators will be applied to obtain uniform estimates of solutions with common domain to nonlinear higher-order equations, and to obtain an oscillation criterion and some results on asymptotic equivalence of equations. For the second-, third-, and forth-order Emden – Fowler type equations, an asymptotic classification of their solutions will be presented. Some applications of the results obtained will be proposed and open problems will be formulated.
Brief Biography of the Speaker: Irina Astashova (ast@diffiety.ac.ru), Full Professor of Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, dept. of Differential Equations, Professor of Plekhanov Russian University of Economics, Faculty MESI. Education: Lomonosov Moscow State University, faculty of Mechanics and Mathematics (1978), Post-graduate in 01.01.02 (“Differential Equations”), Lomonosov Moscow State University (1984) Ph.D. in 01.01.02 (“Differential Equations”); thesis: “Asymptotic Properties of Solutions to Some Nonlinear Differential Equations”( 1989.) Doctor of Physics and Mathematics, thesis: «Qualitative properties of solutions to quasilinear ordinary differential equations» (2008) Teaching Experience: Assistant Professor, Moscow Technical University of Communications and Informatics (MTUCI) (1977-1992); Associated Professor, Moscow Aviation Technology University (MATI) (1992-2002); Associated Professor, after 2005 professor, Moscow State University of Economics, Statistics, and Informatics (MESI) (1992-2015); Professor, Lomonosov Moscow State University (2008 up to now); Professor, Plekhanov Russian University of Economics, faculty MESI (2015 up to now). Courses given: Differential Equations, Partial Differential Equations, Functional Analysis, Complex Analysis, Geometry and Topology, Qualitative Theory of Differential Equations for students and post-graduates. Seminars: Head of a scientific Seminar on qualitative theory of differential equations, Moscow Aviation Technology University (MATI) (1994-2002), of Seminar on qualitative theory of differential equations, Lomonosov Moscow State University (2009 up to now), of Interuniversity Seminar on qualitative theory of differential equations and related topics (Lomonosov MSU - Bauman MGTU - Plekhanov REU (MESI)) ( 2002 up to now). Publications: more than 120 articles, 7 monographs. Societies: Member of Moscow Mathematical Society since 1989, Member of European Mathematical Society since 2015. Award: Keldysh medal of the Russian Cosmonautics Federation.