Scientific center of nonlinear problems of mathematical physics

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 partial differential equations, applications to nonlinear problems of mathematical physics.
Localized and nonlocalized boundary regimes with singular peaking for different classes of second and higher order quasilinear parabolic and hyperbolic equations: asymptotic behaviour of solutions near to the blow-up time.
Conditions of existence and non-existence of global solutions of various classes of stationary and evolutionary equations and inequalities with nonlinear terms of source type.
Conditions of existence, non-existence and uniqueness of nonnegative very singular and large solutions to stationary and evolutionary equations of diffusion-nonlinear absorption type with degenerating absorption potential.
Initial-boundary value problems for equations of wave processes in dispersive media, such as Zakharov-Kuznetsov, Kawahara equations: existence of global solutions, their regularity, large time behaviour.

Publications

Shishkov, A., “Large and very singular solutions to semilinear elliptic equations”, Calculus of Variations and Partial Differential Equation, 2022, 61(3), 102. (Q1).
Kon’kov, A.A., Shishkov, A.E., “On removable singular sets for solutions of higher order differential inequalities”, Fractional Calculus and Applied Analysis, 2023, 26(1), pp. 91–110. (Q1).
Belaud, Y., Shishkov, A., “Extinction in a finite time for solutions of a class of quasilinear parabolic equations”, Asymptotic Analysis, 2022, 127(1-2), pp. 97–119. (Q2).
Korpusov, M.O., Panin, A.A., Shishkov, A.E., “On the critical exponent instantaneous blow-up versus local solubility in the Cauchy problem for a model equation of Sobolev type”, Izvestiya Mathematics, 2021, 85(1), pp. 111–144. (Q2).
Shishkov A. E., Yevgenieva Y. A., “Localized peaking regimes for quasilinear parabolic equations”, Mathematische Nachricht., 2019, 292(6), pp. 622–635. (Q1).
Kon’kov, A.A., Shishkov, A.E., “On large time behavior of solutions of higher order evolution inequalities with fast diffusion”, Journal of Mathematical Analysis and ApplicationsЭта ссылка отключена., 2022, 506(2), 125722. (Q1).
Kon’Kov, A.A., Shishkov, A.E., “Generalization of the Keller-Osserman theorem for higher order differential inequalities”, Nonlinearity, 2019, 32(8), pp. 3012–3022. (Q1).
Marcus M., Shishkov A.E., “Propagation of strong singularities in semilinear parabolic equations with degenerate absorption”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XVI (2016), pp. 1019-1047. (Q1).
Korpusov M. O., Lukyanenko D. V., Panin A. A., “Blow-up for Joseph—Egri equation: Theoretical approach and numerical analysis”, Mathematical Methods in the Applied SciencesVolume, 2020, 43(11), pp. 6771-6800. (Q1).
Rozanova O. S., Chizhonkov E. V., “On the existence of a global solution of a hyperbolic problem”, Dokl. Math., 2020, 101(3), pp. 254–256.
Faminskii A.V., “Regular solutions to initial-boundary value problems in a half-strip for two-dimensional Zakharov—Kuznetsov equation”, Nonlinear Analysis: Real World Applications, 51 (2020), 102959. (Q1).
Pankov A., “Solitary waves on nonlocal Fermi—Pasta—Ulam lattices: Exponential localization”, Nonlinear Analysis: Real World Applications, 2019, 50, pp. 603-612 (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, 2020, 43(8), pp. 5464-5479. (Q1).
Gladkov A., Guedda M. Global existence of solutions of a semilinear heat equation with nonlinear memory condition//Applicable Analysis, 2020, 99(16), pp. 2823-2832. (Q2).
Alibaud N., Andreianov B., Ouedraogo A., “Nonlocal dissipation measure and L^1 kinetic theory for fractional conservation laws”, Communications in Partial Differential Equations, 2020, 45(9), pp.1213-1251. (Q1).
Andreianov B., Maliki M., “On classes of well-posedness for quasilinear diffusion equations in the whole space”, Discrete and Continuous Dynamical Systems Series S, 2021, 14(2), pp. 505-531. (Q2).
Kon’kov A.A., Shishkov A.E., “On Removable Singularities of Solutions of Higher-Order Differential Inequalities”, Advanced Nonlinear Studies, 2020, 20(2), pp. 385–397. (Q1).

Main scientific directions

Existence, uniqueness and asymptotic behavior of solutions of boundary value problems for different classes of stationary and evolution nonlinear partial differential equations.
Finite-time blow-up of solutions to initial–boundary value problems for nonlinear evolution equations of mathematical physics. The structure of the singularities of solutions to nonlinear stationary and evolution PDE.
Very singular and large solutions of semi-linear and quasilinear parabolic and elliptic equations of the structure of stationary and nonstationary diffusion-nonlinear absorption type.

Achievements All achievements

Existence, uniqueness and asymptotic behavior of solutions of boundary value problems for different classes of stationary and evolution nonlinear partial differential 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.

The existence of global solutions (their internal regularity, and their behavior at large times) to initial-boundary value problems for equations describing nonlinear wave processes in dispersive media (namely, non-linear evolution equations of odd order in space variables, such as the Zakharov-Kuznetsov equation, which is a multidimensional generalization of the Korteweg-de Vries equation, and a high-order nonlinear Schrödinger equation) are proved.

Qualitative behavior of solutions (existence, destruction, evolution of singularity) of the Euler-Poisson and magnetohydrodynamic equations in the cold plasma approximation for relativistic and nonrelativistic productions is described.

Finite-time blow-up of solutions to initial–boundary value problems for nonlinear evolution equations of mathematical physics. The structure of the singularities of solutions to nonlinear stationary and evolution PDE

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, was investigated. 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.

There was studied nonlinear equations of modern mathematical physics, such as the Kadomtsev-Petviashvili equation, equations of Benjamin-Bona-Mahoney-Burgers,Rosenau- Burgers. The conditions for local solvability and blow-up of the solution in finite time, as well as critical exponents that determine the boundaries between local solvability, absence of local solutions, absence of global solutions, was obtained.

Boundary regimes with finite-time singular peaking of boundary data for various classes of evolution equations (in particular, linear and quasi-linear, parabolic, pseudo-parabolic, hyperbolic, both second and high orders) was investigated: exact estimates of the profile of solutions near the blowing time, description of localized and non-localized blow-up regimes, estimates of the dimensions and geometry of the localization region of the singularity of solutions in the case of localized regimes, asymptotic behavior of the singularity wave in the case of non-localized regimes was obtained.

Very singular and large solutions of semi-linear and quasilinear parabolic and elliptic equations of the structure of stationary and nonstationary diffusion-nonlinear absorption type

There was investigated very singular and so-called large (that is, turning to infinity on the entire boundary of the domain of the consideration) solutions of quasilinear parabolic and elliptic equations of the structure of stationary and nonstationary diffusion — nonlinear degenerate and non-degenerate absorption: exact conditions for the existence and non-existence of very singular solutions and new exact conditions for the uniqueness of large solutions are obtained. 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.