ИСТИНА

This talk is related to the theory of nonlocal stabilization by starting control for the equations of normal type. We consider the semilinear parabolic equation of normal type connected with the 3D Helmholtz equation with periodic boundary condition. Normal parabolic equation is derived from 3D Helmholtz equation by replacing the nonlinear term by its orthogonal projection B(v) on the vector v of velocity. We will discuss the problem of stabilization to zero of the solution for NPE with arbitrary initial condition y0 by starting control in the form v(x) = λu(x), where λ is some constant depending on y0, and u is a universal function, depending only on a given arbitrary subset Ω ⊂ T^3, which contains the support of control u. This problem is reduced to three inequalities connected with starting control. Semilinear normal parabolic equations have been derived in order to get better understanding of the structure of the solutions of the three-dimensional Helmholtz and Navier-Stokes system and other similar equations. In future we hope to use the obtained results to prove stabilization for Navier-Stokes and Helmholtz equations.