Il'yasov Yavdat Shavkatovich

Position: Principal Research Scientist, Professor

Degree: Doctor of Physical and Mathematical Sciences

E-mail: ilyasov02@gmail.com

Research interests

My research lies at the intersection of nonlinear analysis, spectral theory, variational methods, inverse problems, and computational mathematics. A unifying objective of my work is to develop direct methods for identifying and quantifying critical regimes in parametrised nonlinear equations, including bifurcations, loss of solvability, changes in stability, resonances, and spectral transitions.

In nonlinear mathematical models, a singular point is often more than an exceptional parameter value at which regularity fails. It may mark the boundary between qualitatively different regimes: stability and collapse, existence and disappearance of solutions, or gradual evolution and an abrupt transition to a new state. Critical parameters therefore describe both the structure of the solution set and the operational limits of the underlying physical or technological system. Problems of this kind arise in differential equations, plasma physics, quantum mechanics, chemical kinetics, biological models, and power-system stability.

Despite substantial progress in local and global bifurcation theory, there is still no general framework for determining singular parameter values directly. Continuation methods typically start from a known point on a solution branch and trace the branch until regularity is lost. The situation is more delicate for non-variational and non-selfadjoint problems, where standard energy functionals are unavailable and the spectrum of the linearised operator does not necessarily yield a direct characterisation of the critical parameter. This motivates the search for spectral and extremal principles capable of detecting singularities without first constructing the entire solution branch.

A central direction of my work is the extension of classical spectral principles — Rayleigh quotients, Courant–Weyl minimax formulas, and the Collatz–Wielandt principle — to nonlinear, non-variational, and non-selfadjoint problems. Within this programme, I develop nonlinear and extended Rayleigh quotient methods and associated minimax functionals. Their purpose is to replace the indirect continuation of solution branches by direct extremal characterisations of critical values.

The long-term objective is to develop a minimax theory that, for a broad class of nonlinear equations, determines the first or maximal singular point along a solution branch, identifies its bifurcation type, and provides robust a posteriori bounds and numerical approximations for the critical parameter. Such a theory should combine, within a single framework, the existence of a singular solution, its spectral characterisation, the location of the critical parameter, and its effective computation.

These methods have been applied to elliptic and quasilinear equations, systems involving the \(p\)-Laplacian, problems with non-Lipschitz nonlinearities, non-variational models, and the stability of ground states. They are particularly relevant to threshold problems such as pull-in instability, voltage collapse, and tipping phenomena, where one needs not only to detect the loss of stability or solvability, but also to estimate how close a given state is to the critical threshold.

A complementary research direction, developed jointly with N. Valeev, concerns inverse optimal spectral problems. Reconstructing an unknown coefficient, such as a potential or density, from finitely many observed eigenvalues is generally ill-posed and may admit infinitely many solutions. Our approach incorporates available a priori information through a reference coefficient and selects, among all coefficients consistent with the spectral data, the one closest to this reference. This provides a natural optimisation framework for regularising inverse spectral problems and a basis for studying existence, uniqueness, stability, and numerical reconstruction.

My current research develops these two complementary directions: minimax bifurcation theory and inverse optimal spectral problems. The first seeks direct extremal characterisations of critical parameter values in nonlinear equations; the second develops optimisation principles for reconstructing unknown coefficients from incomplete spectral data. Together, they encompass minimax formulas, cone-based and Galerkin approximations, a posteriori and perturbation estimates, and reliable analytical and numerical methods for nonlinear direct and inverse problems. The broader aim is to develop a nonlinear spectral theory that retains the most useful features of classical linear theory while extending them to a substantially wider class of problems.

Education and academic career

  • 1976–1984: Undergraduate and postgraduate studies at Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Department of Differential Equations.
  • 1985–1997: Assistant Professor and, subsequently, Associate Professor (Docent), Bashkir State University, Ufa, Russia.
  • 1989: Candidate of Physical and Mathematical Sciences (PhD equivalent). Thesis: Asymptotic behaviour of solutions to nonlinear parabolic equations with a small parameter under white-noise perturbations. Research supervisor: Professor A. I. Komech.
  • 1997–1999: Research Fellow, Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia.
  • 2000: Doctor of Physical and Mathematical Sciences. Thesis: Nonlocal analysis of bifurcations of solutions to nonlinear elliptic equations. Scientific consultant: Professor S. I. Pohozaev.
  • 2000–2007: Professor of Mathematics, Bashkir State University, Ufa, Russia.
  • 2002–2003: Visiting Professor, University of La Rochelle, La Rochelle, France. A one-year academic appointment combining research and teaching.
  • Since 2008: Principal Research Scientist, Institute of Mathematics, Ufa Federal Research Centre of the Russian Academy of Sciences.
  • 2018–2022: Visiting Professor, Institute of Mathematics and Statistics, Universidade Federal de Goiás, Goiânia, Brazil. A four-year academic appointment combining research, teaching, and international collaboration.
  • Supervision of PhD research in nonlinear equations, stability of solutions, critical parameter sets, and numerical methods for computing turning points of nonlinear equations.

Selected international research visits

In addition to my long-term academic appointments in Brazil and France, I have conducted research and delivered lectures at universities and research centres in Europe, Asia, and the United States.

  • 2025: Visiting Professor, School of Mathematics and Statistics, Shandong University of Technology, Zibo, China.
  • 2024, 2018, 2015, and 2014: Visiting Professor, Complutense University of Madrid, Faculty of Mathematical Sciences, Madrid, Spain.
  • 2019, 2018, and 2017: Visiting Professor, Department of Mathematics and NTIS, University of West Bohemia, Plzeň, Czech Republic.
  • Earlier research visits were hosted by Friedrich Schiller University Jena, University Paul Sabatier Toulouse 3, the University of Helsinki, the University of Rostock, the University of Catania, Los Alamos National Laboratory, and Rensselaer Polytechnic Institute.

Grants and research projects

  • Research grant from the Russian Science Foundation, 2021–2024.
  • Research grants from the Russian Foundation for Basic Research, 1995–2014.
  • INTAS grants, 1998–2007; leader of the Russian research team in several projects.
  • Deutsche Forschungsgemeinschaft research grants, 1997–2006.
  • Applied research projects on power-system stability, inverse problems for acoustic waves, and optimisation of catalytic processes.

Selected invited talks and conference participation

  • 23–27 June 2025: Conference “Nonlinear Partial Differential Equations”, invited speaker, Sirius Mathematics Center, Sirius, Russia.
  • 13–15 October 2025: All-Russian Conference “Partial Differential Equations and Their Applications”, invited speaker, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia.
  • 17–20 February 2022: XIV Summer Workshop in Mathematics MAT/UnB, invited speaker, University of Brasília, Brasília, Brazil.
  • 13–20 August 2017: The 8th International Conference on Differential and Functional Differential Equations, invited speaker, Moscow, Russia.
  • 7–11 July 2014: The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 44, Madrid, Spain.
  • 23–24 August 2013: International Conference “Nonlinear Analysis Plzeň 2013”, invited speaker, Plzeň, Czech Republic.
  • 25–28 May 2010: The 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications, invited speaker, Dresden, Germany.
  • 17–22 October 2010: Workshop on Variational Methods in Nonlinear Differential Equations, invited speaker, Oaxaca, Mexico.
  • 4–9 June 2005: Second International Conference “Abstract and Applied Analysis 2005”, invited speaker, Quy Nhon, Vietnam.
  • 28 June–4 July 2001: International Conference “Function Spaces, Differential Operators, and Nonlinear Analysis” (FSDONA-01), held in honour of Hans Triebel on the occasion of his 65th birthday, invited speaker, Teistungen, Germany.

Selected publications

Complete publication list available through ORCID and Google Scholar. h-index: 17; 1,316 citations according to Google Scholar, February 2026.

  • Díaz, J. I., Hernández, J., and Il'yasov, Y. (2026). Non-negative and positive solutions for some indefinite sublinear elliptic problems. Boundary Value Problems 2026, Article 22. DOI: 10.1186/s13661-025-02200-w .
  • Il'yasov, Y. and Valeev, N. (2025). On degenerate \((p,q)\)-Laplace equations corresponding to an inverse spectral problem. Bulletin of the London Mathematical Society 57(1), 218–235.
  • Il'yasov, Y. (2024). A finding of the maximal saddle-node bifurcation for systems of differential equations. Journal of Differential Equations 378, 610–625.
  • Il'yasov, Y. and Valeev, N. (2024). An extension of the Perron–Frobenius theory to arbitrary matrices and cones. Electronic Journal of Linear Algebra 40, 788–802.
  • Carles, R. and Il'yasov, Y. (2023). On ground states for the 2D Schrödinger equation with combined nonlinearities and harmonic potential. Studies in Applied Mathematics 150(1), 92–118.
  • Carvalho, M. L., Il'yasov, Y., and Santos, C. A. (2022). Existence of S-shaped type bifurcation curve with dual cusp catastrophe via variational methods. Journal of Differential Equations 334, 256–279. DOI: 10.1016/j.jde.2022.06.021 .
  • Il'yasov, Y. and Valeev, N. (2021). Recovery of the nearest potential field from the \(m\) observed eigenvalues. Physica D: Nonlinear Phenomena 426, 132985.
  • Il'yasov, Y. and Silva, K. (2018). On branches of positive solutions for \(p\)-Laplacian problems at the extreme value of the Nehari manifold method. Proceedings of the American Mathematical Society 146(7), 2925–2935.
  • Il'yasov, Y. (2017). On extreme values of Nehari manifold method via nonlinear Rayleigh's quotient. Topological Methods in Nonlinear Analysis 49(2), 683–714.
  • Il'yasov, Y. and Egorov, Y. (2010). Hopf boundary maximum principle violation for semilinear elliptic equations. Nonlinear Analysis: Theory, Methods & Applications 72(7–8), 3346–3355.
  • Cherfils, L. and Il'yasov, Y. (2005). On the stationary solutions of generalized reaction–diffusion equations with \(p\)- and \(q\)-Laplacian. Communications on Pure and Applied Analysis 4(1), 9–22. DOI: 10.3934/cpaa.2005.4.9 . More than 500 citations according to AIMS/CPAA.