The Solvability of Nonstationary Problem of Filtering Theory

We discuss one problem for class unclassical equations mathematical wave theory. A distinctive feature of this problem is the time dependence of the functional coefficients of an elliptic operator on the right side of the equation. The method of investigation this theory is reduction to problem Cauchy for nonstationary equation of Sobolev type. The Sobolev type equations with time-dependent operator in this formulation are considered for the first time.
We introduced definition of relatively spectrally bounded operator-functions. The conditions that guarantee the fulfillment of this task properties allow to allocate the subspace of initial values for which there is only one solution to the Cauchy problem. This subspace we are named the generalized phase space solutions for the nonstationary equations of Sobolev type. The solution of this problem for a Sobolev type equations, as well as in the original formulation, is obtained by recursive formula.