Evgeny Galakhov
Doctor of Physical and Mathematical Sciences
Professor,
s.M. Nikol’skii Mathematical Institute
1972
Was born in Moscow.
1989-1995
Student of the Department of Differential Equations and Mathematical Physics (DDEMP), Faculty of Applied Mathematics and Physics, Moscow State Aviation Institute (Technical University) (MAI).
1998
Defended his Postgraduate diploma. PhD thesis : "Some classes of non-local elliptic problems and Feller semigroups", Faculty of Computational Mathematics of M.V. Lomonosov Moscow State University.
1998-2000
Assistant at DDEMP, MAI.
1999
Intern at the Department of Mathematics, University of Trieste (Italy).
2000-2006
Assistant at the Department of Functional Analysis, University of Rostock, Germany.
2006
Received Dr. rer. nat. habil. degree, thesis theme: “Positive Solutions of Some Partial Differential Inequalities and Systems” at the University of Rostock, Germany.
2006-2009
Assistant professor at DDEMP, MAI. Doctoral student of the department of theory of functions at the V.A. Steklov Mathematical Institute, Russian Academy of Sciences (RAS).
2010
Defended his Doctoral thesis. Dr.Sci. thesis : "On the blow-up of solutions of nonlinear singular partial differential equations", V.A. Steklov Mathematical Institute, Russian Academy of Sciences (RAS).
2009-2018
Associate Professor of the Department of Mathematical Analysis and Theory of Functions of the RUDN University.
2017
Deputy Director of the S. Nikolsky Mathematical Institute.
Since 2018
Associate professor at the S. M. Nikolskii Mathematical Institute.
Teaching
- Developer of educational courses, the most significant are the following:
- “Destruction of solutions of nonlinear inequalities” (Scientific and Educational Center at the V.A. Steklov Mathematical Institute, 2009)
- "Modern problems of mathematics" (RUDN University, 2017)
- “Destruction of solutions of nonlinear differential inequalities” (RUDN University, 2018)
- In 2006 conducted the “Functional Analysis” lecture course in German (“Mathematics” direction, bachelor degree) at the University of Rostock (Germany)
- In 2014 conducted the “Theory of Functional Spaces” course of lectures (“Mathematics” direction, magistracy) at RUDN University
- Teaches the following disciplines at RUDN University:
- "Mathematical analysis" ("Fundamental informatics and information technology", "Mathematics and computer science" directions, bachelor degree, lectures),
- "Functional analysis" ("Mathematics" direction, bachelor degree, practical classes in Russian and English),
- "Modern problems of mathematics" (“Mathematics” direction, master's, lectures and practical classes),
- “Destruction of solutions of nonlinear differential inequalities” (“Mathematics” direction, Master's program, lectures and practical classes)
Science
- As a student, under the leadership of Alexander L. Skubachevsky, investigated differential operators with non-local conditions and obtained the geometric characteristics of their spectrum, which can be used to calculate distributed loads in aircraft engineering, machinery, etc.
- Studied Feller semigroups generated by elliptic operators with nonlocal conditions, and obtained sufficient conditions for the existence of these semigroups. The theory of Feller semigroups is used in the study of multidimensional diffusion processes in biological cells.
- Since 1999 (together with the scientific groups of Enzo Mitidieri and Stanislav I. Pokhozhaev), and since 2011 (together with Olga A. Salieva and other co-authors) has investigated the solvability of nonlinear differential and functional differential equations and inequalities. Sufficient conditions for the absence of solutions (catastrophes) for equations and inequalities containing coefficients with singularities on unbounded domains, variable nonlinearity parameters, fractional powers of the Laplace operator, etc. were obtained. For inequalities with power singularities on unbounded domains, the optimality of the conditions obtained was proved. The developed analytical and numerical methods have been used in modeling the sintering of aluminum oxide in metallurgy, the phenomena of chemotaxis and haptotaxis during the propagation of microorganisms and the growth of malignant tumors, the emergence of financial bubbles, etc.
- Since 2009 has investigated the monotonicity of bounded positive solutions of the Dirichlet quasilinear problem in half-space. Together with Olga A. Salieva he obtained sufficient conditions for the monotony of such solutions in terms of nonlinearity indicators. It is assumed that the obtained results are used to predict the occurrence of phase transitions in an activated plasma.
Scientific interests
- Lack of solutions of nonlinear differential and functional differential equations and inequalities.
- Monotonicity of solutions of nonlinear boundary value problems.
- Spectral theory of differential operators with non-local conditions.
- Operator Feller semigroups.
We obtain sufficient conditions for the existence of a Feller semigroup generated by an elliptic operator with integro-differential boundary conditions. In this paper we study both transversal and non-transversal cases under very general assumptions on nonlocal terms.
It is shown that various quasilinear elliptic and parabolic differential inequalities and systems of such inequalities defined on bounded domains, and which have point singularities on the boundary do not have solutions. The method of nonlinear capacity is used in the proof. Examples show that the conditions obtained by this method cannot be improved in the class of problems under consideration.
By the nonlinear capacity method, conditions of solvability are obtained for some classes of stationary and evolutional differential inequalities with coefficients that have singularities on unbounded sets.
We consider a nonlinear PDEs system of two equations of Parabolic–Elliptic type with chemotactic terms. The system models the movement of a biological population towards a higher concentration of a chemical agent in a bounded and regular domain. We study the range of parameters and constrains for which the solution exists globally in time.
We prove the monotonicity of nonnegative bounded solutions of the Dirichlet problem for the quasilinear elliptic equation −Δ pu = f(u), p ≥ 3, in a half-space. This assertion implies new results on the nonexistence of solutions for the case in which f(u) = uq with appropriate values of q.
Galakhov E., Salieva O. Blow-up for nonlinear inequalities with singularities on unbounded sets // Current Trends in Analysis and its Applications: Proceedings of the IXth ISAAC Congress, Birkhäuser, Basel, 2015. pp. 299-305.
Sufficient conditions for the absence of solutions for some nonlinear elliptic inequalities with coefficients having singularities on unbounded sets are obtained.
Review of some research in the field of modeling of heterogeneous complex systems.
An estimate of the failure time of the solution for some problems for the nonlinear heat equation is obtained.
Sufficient conditions for the destruction of the solution for some problems for the nonlinear heat equation are obtained.
Sufficient conditions for the absence of solutions for some nonlinear inequalities with coefficients having singularities on unbounded sets and with terms depending on the gradient of the desired function are obtained.
Sufficient conditions for the absence of solutions for some nonlinear elliptic and parabolic inequalities containing the p-Laplace operator with a variable exponent are obtained.
Sufficient conditions for the absence of solutions for some nonlinear inequalities with coefficients having singularities on unbounded sets and with terms depending on the gradient of the desired function are obtained.
Estimates of the failure time of the solution for the nonlinear heat transfer equation with the source are obtained.
Sufficient conditions for the absence of solutions for some nonlinear inequalities containing the p-Laplace operator with a variable exponent are obtained.
Sufficient conditions for the absence of non-negative monotone solutions for some quasilinear elliptic inequalities in the half-space are obtained.
Uvarova L., Salieva O., Devyaterikova E., Galakhov E., Devyaterikov I. Estimation of the critical parameters of the solution of the Cauchy problem for a nonlinear heat equation with a source // Fundamental physical and mathematical problems and modeling of technical and technological systems, Vol. 17. Bulletin of the Moscow State Technical University “Stankin”, 2016, pp. 304-310.
An estimate of the parameters of the Cauchy problem for a nonlinear heat equation with a source is obtained, at which the solution is destroyed.
Sufficient conditions for the destruction of the solution and an estimate of its lifetime for the model problem of nonlinear heat transfer are obtained.
A model of waves with evaporation from the phase surface is constructed on the basis of the modified Korteweg-de Vries equation.
Sufficient conditions for the absence of solutions to some nonlinear elliptic inequalities and systems with coefficients having singularities on the boundary are obtained.
Sufficient conditions for the absence of solutions to some nonlinear partial differential inequalities in unbounded domains are obtained.
Sufficient conditions are obtained for the absence of solutions to certain nonlinear inequalities containing the fractional degree of the Laplace operator and terms that depend on the gradient of the desired function.
Sufficient conditions for the absence of non-negative monotone solutions of some coercive nonlinear elliptic inequalities in the half-space are obtained.
Sufficient conditions for the absence of solutions for some nonlinear elliptic functional-differential inequalities are obtained.
Sufficient conditions for the absence of solutions to certain inequalities and systems with variable nonlinearity indices and singular coefficients on the boundary are obtained.
Sufficient conditions for the absence of non-negative solutions to some nonlinear parabolic inequalities in the half-space are obtained.
Sufficient conditions for the uniqueness of the trivial solution of some nonlinear inequalities containing the fractional degree of the Laplace operator are obtained.
Sufficient conditions for the absence of solutions for some nonlinear inhomogeneous elliptic inequalities are obtained.
Sufficient conditions for the absence of alternating solutions for some nonlinear elliptic and parabolic inequalities are obtained.
Sufficient conditions for the absence of alternating solutions for some nonlinear elliptic inequalities in bounded domains are obtained.
Sufficient conditions for the absence of global weak solutions for parabolic equations containing the fractional power of the Laplace operator are obtained.
In this paper, we make modification of the results obtained by Mitidieri and Pokhozhaev on sufficient conditions for the nonexistence of nontrivial weak solutions of nonlinear inequalities and systems with integer power of the Laplacian with the nonlinearity term of the form a(x) vertical bar Delta(m)u vertical bar(q) + b(x)vertical bar u vertical bar(s). We obtain an optimal a priori estimate by employing the nonlinear capacity method under a special choice of test functions. Finally, we prove the nonexistence of nontrivial weak solutions of the considered inequalities and systems by contradiction.
In this article, we modify the results obtained by Mitidieri and Pohozaev on sufficient conditions for the absence of nontrivial weak solutions to nonlinear inequalities and systems with integer powers of the Laplace operator and with a nonlinear term of the form a(x)|∇(Δmu)|q+b(x)|∇u|s. We obtain optimal a priori estimates by applying the nonlinear capacity method with an appropriate choice of test functions. As a result, we prove the absence of nontrivial weak solutions to nonlinear inequalities and systems by contradiction.