Singular solutions of quasilinear elliptic and parabolic equations
Singular solutions of quasilinear elliptic and parabolic equations

Very singular and “large” solutions of semi-linear and quasilinear equations of stationary and non-stationary nonlinear diffusion-absorption type.

Localized and non-localized boundary regimes with infinite peaking for different classes of parabolic and evolution equations.

The blow-up problems for solutions of the Cauchy problem for nonlinear equations of modern mathematical physics such as Khokhlov-Zabolotskaya equation, ion-acoustic waves in plasma equation, Rosenau-Burgers equation, Benjamin-Bona-Mahoney-Burgers equation.

The peaked solitary wave solutions (peakons) for different dispersive equations that are model in shallow water theory.

The problem of averaging of solutions of nonlinear elliptic and parabolic equations.

  • 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).
  • 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). 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).
  • M.O. Korpusov, E.A. Ovsyannikov. Blow-up instability in non-linear wave models with distributed parameters. Izvestiya: Mathematics (Q2).
  • 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).
Project goals
  • Limit behavior of solutions of quasilinear parabolic equations near the time of singular peaking of the boundary data is studied.
  • Conditions for existence and nonexistence (blow-up) of global solutions of different classes of stationary and evolution equations with nonlinear source term are considered.
  • Conditions for existence and nonexistence of very singular and large solutions of diffusion-nonlinear absorption equations with degenerate absorption potential are investigated.
  • Homogenization of families of boundary value problems for nonlinear elliptic equations and variational inequalities are studied.
Project leader All participants
Andrey Shishkov

Andrey Shishkov

Doctor of Physics and Mathematics, professor
Project results
There are investigated initial-boundary value problems for doubly quasilinear parabolic equations of Newtonian-non-Newtonian diffusion type with boundary regimes with singular peaking in some finite time moment. There are described so called localized regimes with peaking and for corresponding solutions there are obtained sharp upper estimates of profile of solution near to the blow-up time. Using these estimates there are considered large solutions( solutions with infinite boundary data) for equations of nonlinear diffusion-absorption type with absorption potential degenerating in some finite moment. Sharp upper estimates of profile of large solution in the moment of degeneration of absorption potential are obtained.
Nonlinear nonlocal equations of mathematical physics were considered, for which sufficient blow up conditions were found. In addition, for a nonlinear Benjamin-Bohn-Mahony-burgers equation, results on time-local solvability for a new class of initial functions were proved. Finally, for one nonlinear equation of Sobolev type with gradient nonlinearity, critical parameters are found. The considered equations describe quasi-stationary processes in semiconductors, ferromagnets and plasma. The obtained sufficient conditions for the occurrence of blow up in terms of the occurrence of breakdown in semiconductors and the development of instability in plasma. In particular, a model nonlinear equation of ion-sound waves is considered, for which the results on the occurrence of collapse are obtainedsorption potential are obtained.
There are obtained conditions guaranteeing that every entire solution of higher order differential inequalities with nonlinearity of the Emden-Fowler type is trivial (the so called blow-up conditions). These results are exact and can be applied to a wide class of differential inequalities. We generalized the Keller-Osserman theorem for higher order differential inequalities. Our results are exact and can be applied to inequalities with a nonlinearity of the general form. In the case of higher order differential inequalities, such results were earlier known for nonlinearity of the Emden-Fowler type only.
Application area
  • The qualitative theory of nonlinear partial differential equations, applications to nonlinear problems of mathematical physics.