1975-1980

Student of the Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University.

1980-1983

Postgraduate student at the Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University.

1984

Defended his Postgraduate diploma. PhD thesis: “The Cauchy problem and the mixed problem in a half-strip for equations of the Korteweg-de Vries type”.

1983-1988

Assistant of the Mathematics Department of the Moscow Institute of Radio Engineering, Electronics and Automation (MIREA).

1988-1990

Assistant professor at the Mathematics Department of MIREA.

1990-2002

Associate Professor of the Department of Differential Equations and Functional Analysis of RUDN University.

2001

Defended his Doctoral thesis. Dr.Sci. thesis: “Boundary value problems for the Korteweg-de Vries equation and its generalizations”.

2002-2018

Professor at the Department of Differential Equations and Functional Analysis (Department of Nonlinear Analysis and Optimization after 2005), RUDN University.

2012

Received the academic title of Professor.

Since 2018

Professor of the S. Nikolsky Mathematical Institute.

2014

Winner of the RUDN prize in the field of science and innovations.

Teaching

  1. Made training courses, the most significant are the following:
    •  “Evolutionary type functional spaces” - Moscow: RUDN University, 2011, 2016 - 2nd edition.
    • "Selected chapters of the theory of evolutionary equations" - Moscow: RUDN University, 2014.
  2. Conducts the following lecture courses for bachelor and master’s degree students in RUDN University:
    • "Complex analysis" ("Mathematics" direction, bachelor degree),
    • "Partial differential equations" ("Mathematics" direction, bachelor degree),
    • "Nonlinear evolution equations" ("Mathematics" direction, magistracy)

Science

  • Results on the global solvability and correctness of the Cauchy problem and initial boundary value problems for nonlinear evolution equations of odd order were obtained. These equations include the Korteweg-de Vries, Kawahary, Kadomtsev-Petviashvili, Zakharov-Kuznetsov equations. These equations describe nonlinear wave processes in dispersion media. These results have been established both for wide classes of equations of the type under consideration and for individual equations. For the considered boundary problems, the problems of internal regularity of solutions, the behavior of solutions at large times and controllability are also studied.
  • The studies were started in the 1980s together with Professor Stanislav N. Kruzhkov for solving the Cauchy problem for the Korteweg-de Vries equation - the most famous representative of this class of equations. In particular, the so-called local smoothing effect was discovered (at the same time with the famous American mathematician T. Kato), which allowed us to establish the results of the existence of global time solutions with irregular initial data. In addition, the property was found to increase the internal smoothness of solutions depending on the decay rate at infinity of the initial data. The research method essentially used the properties of the corresponding linearized equation, namely, the Airy equation. In particular, the idea of ​​inversion of the linear part of the Korteweg-de Vries equation was applied in order to establish the results on the uniqueness of the considered solutions.
  • Further study of the properties of the Airy equation allowed us to construct special solutions of this equation of the type of boundary potentials. The use of these boundary potentials made it possible to obtain results on the global correctness of initial boundary value problems for the Korteweg-de Vries equation with natural or close to natural conditions for the smoothness of boundary data.
  • In the following, these methods, developed for the Korteweg-de Vries equation, were applied to a wider class of equations of odd order. Results were obtained on the existence and uniqueness of global solutions. Assumptions on the class of the equations considered allowed us to use such analogs of the properties of the Korteweg-de Vries equation, such as the existence of conservation laws and the effect of local smoothing. Special solutions of the type of boundary potentials of the corresponding linearized equations were also constructed, which were used in the study of initial-boundary value problems.
  • In the process of studying the generalizations of the Korteweg-de Vries equation, special attention was paid to those that were model for describing the propagation of nonlinear waves. In particular, for the Kadomtsev-Petviashvili equation in the case of two spatial variables, the first results on the existence of global solutions to the Cauchy problem were obtained.
  • Another example of a multidimensional generalization of the Korteweg-de Vries equation is the Zakharov-Kuznetsov equation. For this equation, global correctness classes were constructed, both in the case of the Cauchy problem and in the initial-boundary value problems. In this case, cases of both two-dimensional and three-dimensional equations are considered. The use of boundary potentials made it possible to obtain these results under the natural conditions of smoothness of the boundary data. Results on the internal regularity of solutions were also established.
  • Another important equation that generalizes the Korteweg-de Vries equation is the Kawahara equation, which models the propagation of waves in media with a higher order of dispersion. Here, results are also obtained on the global correctness of initial-boundary problems under the natural conditions of smoothness of boundary data and on the internal regularity of solutions.
  • In recent years, issues of the behavior of solutions at large times, as well as some control problems, have also been studied for the equations above.

Scientific interests

  • solvability and correctness of boundary problems for nonlinear evolution equations of odd order
  • internal regularity of solutions of boundary value problems for nonlinear evolution equations of odd order
  • behavior at large times of solutions of boundary value problems for nonlinear evolution equations of odd order
  • controllability of boundary value problems for nonlinear evolution equations of odd order
An initial boundary-value problem in a half-strip with one boundary condition for the Korteweg-de Vries equation is considered and results on global well--posedness of this problem are established in Sobolev spaces of various orders, including fractional. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation. An essential part of the study is the investigation of special solutions of a "boundary potential" type for this linearized KdV equation.
In the present paper we establish results concerning the decay of the energy related to the damped Korteweg–de Vries equation posed on infinite domains. We prove the exponential decay rates of the energy when an initial value problem and a localized dissipative mechanism are in place. If this mechanism is effective in the whole line, we get a similar result in H k -level, k∈ℕ. In addition, the decay of the energy regarding an initial boundary value problem posed on the right half-line, is obtained considering convenient a smallness condition on the initial data but a more general dissipative effect.
An initial-boundary value problem in a strip with homogeneous Dirichlet boundary conditions for two-dimensional generalized Zakharov--Kuznetsov equation is considered. In particular, dissipative and absorbing degenerate terms can be supplemented to the original Zakharov-Kuznetsov equation. Results on global existence, uniqueness and long-time decay of weak solutions are established.
An initial–boundary value problem in a strip with homogeneous Dirichlet boundary conditions for two-dimensional Zakharov–Kuznetsov–Burgers equation is considered. Results on global well-posedness and large-time decay of solutions in the spaces Hs for s∈[0,2] are established.
An initial–boundary value problem with homogeneous Dirichlet boundary conditions for three-dimensional Zakharov–Kuznetsov equation is considered. Results on global existence, uniqueness and large-time decay of weak solutions in certain weighted spaces are established.
Initial-boundary value problems in a bounded rectangle with different types of boundary conditions for two-dimensional Zakharov-Kuznetsov equation are considered. Results on global well-posedness in the classes of weak and regular solutions are established. As applications of the developed technique results on boundary controllability and long-time decay of weak solutions are also obtained.