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.

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.

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.

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.