ИСТИНА

This work is related to the theory of nonlocal stabilization by starting control for the equations of normal type . We consider the simplest parabolic equation of normal type connected with the Burgers equation with periodic boundary condition and study the problem of stabilization to zero of its solution with arbitrary initial condition y0 by starting control in the form u(x) = λu0(x), where λ is some constant depending on y0, and u0 is a universal function, depending only on a given arbitrary subinterval (a, b) ⊂ [0, 2π), which contains the support of control u0. This problem can be reduced to proving inequality $int_{-\pi}^{\pi}S^3(t, x; u0)dx ≥ βe−6t ∀t > 0$, where S(t, x; u0) is the solution of the heat equation with initial condition u0, and β > 0 is some constant. The proof of this estimate given previously in [1] is too complicated to be generalized for a NPE connected with three-dimensional Helmholz equation which is our future goal. In this work we provide a simplified proof. [1] A.V.Fursikov, Stabilization of the simplest normal parabolic equation.-Communications on pure and applied analysis v.13, N5, September 2014, p.1815-1854