Andrey Shishkov Biography

Abstract and Applied Analysis. Science has developed a number of options for the method of local energy assessments in the qualitative theory of nonlinear boundary tasks that made it possible to establish a number of fundamental results about asymptotic and qualitative properties of generalized solutions of the broad classes of quasi -alien elliptical, parabolic and some components of differential equations.

In the year, he received local variants of energy and entropy a priori assessments of generalized solutions of the equations of the course of thin viscous films, specific quasilic parabolic equations of the fourth order. On the basis of these assessments, a number of works establish the conditions of the limb of the speed of distribution of carriers of the corresponding solutions, as well as a description of various scenarios of the evolution of these carriers in time.

In the year, he proposed a new energy method for studying linear parabolic boundary tasks with unlimitedly exacerbating border data at the end of time. Based on this method, the accurate classes of localized boundary regimes of the S-mode, that is, with exacerbation, generating solutions with spatially localized at the time of exacerbation of the Singularity zone.

Together with Professor L. Veron, he established accurate conditions on the nature of the degeneration of potential conditions of the Dini type, which ensure the absence of the effect of the spread of the singular solutions and, therefore, the existence of super -orgular and “large” solutions. In recent joint work with Professor M. Marcus, he established that on a wide class of potentials, the condition of Dini is a necessary condition for the absence of the spread of singularity along the variety variations, that is, the criterion for the existing classes of the singular solutions was found.

Scientific interests are a qualitative theory of solutions to the boundary problems for quasilinear and nonlinear elliptical and parabolic equations of the second and high orders; Equations of the course of thin capillary films; Caano-Hilliard equation; The theory of the singular, super-sylumular and "large" solutions of the equations of the structure of nonlinear diffusion of-aborption; Localized and unfobs boundary bounds with a singular exacerbation.

Shishkov A. Permissible growth of initial data for diffusion equations with slightly super -linear absorption, Commun. Marcus M. The spread of strong singularities in semi -linear parabolic equations with degenerating absorption, Ann. Scuola Norm. Pisa, Cl. Beli and, Shishkov A. The attenuation of the solutions of some semi -linear parabolic equations over the final time. Equat, t.

The task of a koshman is considered for a semi-linear equation of thermal conductivity with the absorption potential, degenerating on some of the subsets of the initial plane. The exact conditions are established on the nature of this degeneration, guaranteeing the complete attenuation of any solution over the final time. Galactionov, A. The principle of Saint-Wenan to study the singularity of solutions of quasi-icilinear parabolic equations of high order.

The task of Koshi is studied - a conductor for a wide class of quasilinear sabotage parabolic equations of an arbitrary even with boundary data, syngularly aggravating at the end of time. The behavior of decisions in the vicinity of the exacerbation time is being studied. The exact conditions for the boundary regime are established, guaranteeing the localization of the Sellularity zone in the vicinity of the boundary of the initial localization area.

The study is carried out on the basis of a new study of energy technology for this area, based on a priori assessment of Saint -Venan's decisions in the theory of elasticity. Jacquelly L. Distribution of the carriers of solutions of one -dimensional equation of the course of thin films with convection, Indiana Univ. The property of the limb of the distribution of the carriers of strong generalized solutions of the equations of the course of thin films in the presence of members modeling nonlinear convection is studied.

In cases of severe and weak slip, accurate assessments of the evolution of the quick and slow fronts of the solid medium at large and small times are established. Corps, A. Panin, A. Yevgenieva Ye. The equation of slow diffusion with the singular boundary data is considered. The assessment of all the weak solutions of such a problem was obtained, provided that the boundary regime is localized.

A comparative analysis of the results obtained by the method of energy assessments and the barrier method for the equation of the porous medium is presented. Shishkov, E. Based on energy methods, in a certain sense, accurate assessments of the final profile of a generalized solution in the vicinity of the exacerbation time, depending on the rate of increase in the global energy of this solution.