Structural unit: S.M. Nikol’skii Mathematical Institute.

The Scientific Center for Nonlinear Problems of Mathematical Physics was created to promote scientific research in this field and to form a young generation of scientists.

**Researches of the Center: **

- The qualitative theory of nonlinear partial differential equations, applications to nonlinear problems of mathematical physics are considered.
- The limiting behavior of solutions of quasilinear parabolic equations in the vicinity of the time of singular exacerbation of the boundary regime is studied.
- The questions of the existence and non-existence of global solutions of various classes of stationary and evolutionary equations with a nonlinear source are considered.
- The conditions for the existence of supersingular and large solutions of equations of the type of diffusion-nonlinear absorption with a degenerate absorption potential are studied.
- The problems of averaging families of boundary value problems for nonlinear elliptic equations and averaging variational inequalities are studied.

### Scope of the results

The equations under consideration describe intensive processes in complex media, in particular, materials with complex properties. Therefore, the results obtained in the course of research can be applied in the theory of plasma, the theory of materials, including materials used in nanotechnology.

The scientific seminar "Seminar on nonlinear problems of PDE and mathematical physics"is organized and regularly held. Scientific seminars, master classes, work with young scientists, round tables on the research areas of the center are held.

### Publications

- Akduman, S., Pankov, A. Nonlinear Schrödinger equation with growing potential on infinite metric graphs//Nonlinear Analysis, Theory, Methods and Applications (Q1).
- Pankov A. Solitary waves on nonlocal Fermi–Pasta–Ulam lattices: Exponential localization//Nonlinear Analysis: Real World Applications (Q1).
- Konkov A. A., Shiskov A. E. Generalization ot the Keller-Osserman theorem for higher order differential inequalities // Institute of Physics Publishing, Nonlinearity (Q1).
- Korpusov M., Ovchinnikov V., Panin A. Instantaneous blow‐up versus local solvability of solutions to the Cauchy problem for the equation of a semiconductor in a magnetic field // Mathematical Methods in the Applied Sciences (Q1).
- Gladkov A., Kavitova T. Global existence of solutions of initial-boundary value problem for nonlocal parabolic equation with nonlocal boundary condition// Mathematical Methods in Applied Sciences (Q1).
- Gladkov A., Guedda M. Global existence of solutions of a semilinear heat equation with nonlinear memory condition//Applicable Analysis (Q2).
- Kon’kov, A.A., Shishkov, A.E. On blow-up conditions for solutions of higher order differential inequalities//Applicable Analysis (Q2).
- Korpusov M.O. Blow-up of Solutions of Nonclassical Nonlocal Nonlinear Model Equations//Computational Mathematics and Mathematical Physics (Q2).
- Shishkov, A.E., Yevgenieva, Y.A. Localized Blow-Up Regimes for Quasilinear Doubly Degenerate Parabolic Equations//Mathematical Notes (Q2).
- Korpusov M.O., Yablochkin D.K. Potential Theory for a Nonlinear Equation of the Benjamin–Bona–Mahoney–Burgers Type // Computational Mathematics and Mathematical Physics (Q2).
- N. Alibaud, B. Andreianov, A. Ouedraogo. Nonlocal dissipation measure and L^1 kinetic theory for fractional conservation laws. Communications in Partial Differential Equations (Q1).

https://doi.org/10.1080/03605302.2020.1768542 - B. Andreianov, M. Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete and Continuous Dynamical Systems Series S (Q2). https://doi.org/ 10.3934/dcdss.2020361
- A.A. Kon’kov, A.E. Shishkov. On Removable Singularities of Solutions of Higher-Order Differential Inequalities. Advanced Nonlinear Studies (Q1). https://doi.org/10.1515/ans-2020-2085
- M.O. Korpusov, E.A. Ovsyannikov. Blow-up instability in non-linear wave models with distributed parameters. Izvestiya: Mathematics (Q2). https://doi.org/10.1070/IM8820
- M.O. Korpusov, D.V. Lukyanenko, A.A. Panin. Blow-up for Joseph–Egri equation: Theoretical approach and numerical analysis. Mathematical Methods in the Applied Sciences (Q1). https://doi.org/10.1002/mma.6421
- hishkov, A.E., Yevgenieva, Ye.A. Localized peaking regimes for quasilinear parabolic equations // Mathematische Nachrichten. (Q1). https://doi.org/10.1002/mana.201700436

- Large time asymptotic behavior of solutions of nonlinear boundary value problems.
- Finite-time blow-up of solutions to initial–boundary value problems for nonlinear nonstationary equations of mathematical physics. The structure of the singularities of solutions of stationary and evolutionary nonlinear partial differential equations.
- Homogenization of boundary value problems for nonlinear elliptic and parabolic equations.

The conditions for the existence and nonexistence of global solutions of initial-boundary value problems for nonlinear parabolic equations with nonlocal boundary data are established. The orbital stability of one class of soliton solutions of the generalized Kawahara equation is proved. Conditions for stabilization of solutions of higher order nonlinear evolution equations are established. For high order nonlinear Emden-Fowler type inequalities conditions for the absence of nontrivial solutions are found. The Keller-Osserman theorem are generalized for the case of higher order differential inequalities.

Model nonlinear evolution equations of the third and fourth orders are considered, which describe waves and quasi-stationary processes in a plasma and in semiconductors, respectively. The existence of classical solutions of the Cauchy problem is proved and sufficient conditions for their blow- up in a finite time are obtained. Estimates from above for the time of blow- up were also obtained.

The behavior of solutions of quasilinear parabolic equations in the neighborhood of the time of singular peaking of the boundary regime is studied. Exact estimates from above of the final profile of the solution are established. A criterion for the existence of very singular non-negative solutions with a point singularity for non-stationary diffusion-nonlinear absorption equations with degenerate absorption potential is established. The

behaviour of “large” solutions for such equations is also investigated. The exact conditions of the propagation of singularities of large solutions from the boundary of the domain to the interior of the domain along the degeneracy manifold of the absorption potential are established.

There are studied maximal monotone operators in variable spaces. On this basis, a theory of averaging of boundary value problems for elliptic equations with non-standard growth conditions is constructed. The properties of integrability of entropy and weak solutions of the Dirichlet problem for nonlinear second-order elliptic equations with the right-hand side from the classes whose proximity to the space of integrable functions is characterized by compositions of a logarithmic function with small exponents are established.