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

In the problem of describing the asymptotic properties of generalized solutions of quasilinear parabolic equations in a neighborhood of the time of the singular exacerbation of the boundary regime (i.e. boundary data), at the present time, it were found limiting restrictions on the intensity of the exacerbation leading to solutions with a non zero but finite measure of the blow- up, i.e. the so-called S-modes are described. For more intensive boundary modes (so-called HS-modes) exact estimates for the propagation of a singularity wave were found. Within the framework of the project it is proposed to study arbitrary less singular than S-regimes (so-called LS-regimes) and to establish exact upper bounds of the limiting profile of the solution in the vicinity of the exacerbation time, depending on the asymptotics of the corresponding LS-mode. On the basis of this analysis, it is planned to study the properties of large solutions of various classes of equations of the type of nonstationary diffusion / nonlinear absorption with a degenerate absorption potential at a finite time. We suupose to give an accurate description 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. In addition, for various classes of elliptic and parabolic equations of the type of stationary and nonstationary diffusion / nonlinear absorption with an absorption potential that degenerates on manifolds extending to the boundary of the corresponding domain or the initial plane, it is proposed to establish exact necessary and sufficient conditions (and in some cases also a criterion) on the character of the degeneracy of the potential which guarantee the existence or non-existence of very singular solutions with a point singularity at the point from the intersection of the above manifold and the boundary of the corresponding domain or the initial plane. We suppose to consider a number of problems on the existence and non-existence of global solutions of various classes of stationary and evolution equations with a nonlinear source in the spirit of Fujita's theorems. Thus, it is planned to establish the blow-up conditions, conditions for removability of singularities, and stabilization at infinity conditions for solutions of broad classes of elliptic, parabolic, and higher order equations without type. We shall continue the study of the blow-up effect for solutions of the Cauchy problem and the quaternary space problems for model nonlinear equations of the modern mathematical physics such as generalized Khokhlov-Zabolotskaya equations, ion-acoustic wave equations in plasma, Rosenau-Burgers equations, Benjamin-Bon-Mahoney-Burgers, as well as various versions of the equations of acoustic and electromagnetic waves in the continuous media. We'll study the destruction and instantaneous destruction of solutions of nonlinear evolution equations with non coercive nonlinearities of the form of the derivative of a quadratic nonlinearity.

It is also planned to study the influence of the behavior of the coefficients for large time values on the global solvability of initial-boundary value problems for nonlinear parabolic equations with nonlocal terms in the equation and the boundary condition. In so doing, it is proposed to investigate problems with nonlocal terms both in the spatial variables and in time.

We'll also study the problem of averaging of solutions nonlinear elliptic equations by methods closely related to the theory of singular solutions. Thus, for families of elliptic operators with nonstandard growth, in the presence of the so-called Lavrentiev effect, a limit homogenized problem will be constructed on the basis of a suitable generalization of the concept of G-convergence. To describe the growth conditions for the averaged operators, we use the T-convergence of anisotropic integral functionals.

We plan to study the asymptotic behavior of solutions of variational problems with implicit constraints and of variational inequalities with bilateral obstacles in variable domains. Moreover, the summability of entropy solutions will be investigated for nonlinear elliptic equations with right-hand side in weak logarithmic classes.

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
Application area
  • The qualitative theory of nonlinear partial differential equations, applications to nonlinear problems of mathematical physics.