Scientific center of nonlinear problems of mathematical physics
Scientific center of nonlinear problems of mathematical physics

Type

Center

Department

S.M. Nikol’skii Mathematical Institute

Head:

Andrey Shishkov

Doctor of Physics and Mathematics, professor

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

Scientific activities (specialization): 

  • Morrie-type spaces, hysteresis, dynamical systems, mathematical modeling, partial differential equations; 
  • spectral properties of various operators of mathematical physics, in particular the Schrodinger, Dirac operator; 
  • discrete and continuous problems with indefinite quadratic part and lack of compactness;
  • semigroups of holomorphic maps, functional Schroeder and Abel equations; - “large” and supersingular solutions for semi-linear and quasi-linear elliptic and parabolic equations, localized and non-localized singularities, propagation of singularities;
  • study of reaction-diffusion equations in biomedical applications; investigation of the Hamilton-Crane index for non-self-adjoint spectral problems; 
  • generalized indefinite strings and nonlinear wave equations.
Main scientific directions
  • 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.
Achievements
Large time asymptotic behavior of solutions of nonlinear boundary value problems.

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.

Finite-time blow-up of solutions to initial–boundary value problems for nonlinear nonstationary equations of mathematical physics.

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.
 

Homogenization of elliptic and parabolic boundary value problems for nonlinear equations.

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.