**Sobolev
Institute of Mathematics **

**Main fields of research (mathematical physics)**

Laboratory "Evolution Equations"The Laboratory was organized in 1996. It is headed by doctor of physical and mathematical sciences M.V. Fokin. The staff of the Laboratory is doctors of physical and mathematical sciences V.S. Belonosov, M.P. Vishnevskii, V.V. Skazka and candidates of physical and mathematical sciences M.A. Sychev, D.R. Akhmetov, and A.A. Lyashenko. Investigations in the following main directions are carried on in the Laboratory:

general theory of initial boundary value problems for linear equations and systems of elliptic, parabolic, and quasielliptic types, including the questions of existence of classical solutions, solutions estimating in uniform norms, boundary value problems in domains with a nonsmooth boundary;

theory of nonlinear evolutionary equations of parabolic and hyperbolic types, questions of solvability of basic initial boundary value problems “on the whole” with respect to time, asymptotic behavior of solutions, their stabilization and stability, attraction domains of stable stationary or periodic solutions, theorems on integral manifolds;

Lyapunov methods in the problem of stability of solutions to linearized parabolic and hyperbolic systems, localization of the spectrum of corresponding differential operators, and solvability of hereby arising operator equations;

application of the spectral theory of differential operators to solving inverse problems for evolutionary equations of mathematical physics;

general questions of calculus of variations and its applications to solution of problems of nonlinear theory of elasticity;

spectral properties of operators arising in the theory of small oscillations of rotating ideal fluid and dependence of qualitative characteristic features of solutions on the character of the spectrum of the corresponding problem.

**<Main fields of research>**** ****<Sobolev
Institute of Mathematics>**