Work

The project is devoted to the development of new modern algebraic and analytical methods for the study of partial differential equations. Regarding algebraic methods, the project will focus on the classification of Lie symmetries in classes of nonlinear partial differential equations that model physical and biological processes. The improved and modified approaches of group analysis of differential equations will be applied to investigate (1+1) - and (2+1) -dimensional nonlinear partial differential equations. These include, in particular, the variable coefficient reaction–diffusion and Fisher equations, Burgers and Klein–Gordon equations. We plan to obtain exhaustive classifications of Lie symmetries for these equations and then to construct their exact solutions. As to analytical tools of the study of multidimensional differential equations, the project participants will aim at the developing of the method of scales of function spaces and at the method of the interpolation of pairs of normed spaces and operators on them. We plan to introduce broad classes of the generalized Sobolev, Nikolskii–Besov, and Triebel–Lizorkin spaces that have interpolation properties with respect to their classical analogs and allow a correct definition on smooth manifolds. We expect to build a new theory of solvability of elliptic differential equations and elliptic boundary-value problems on manifolds in classes of the spaces introduced. A specific attention will be paid to elliptic problems with rough (i.e. irregular) data, specifically to problems with white noise in boundary conditions, which arise in physics and technology.