Fourier transform method for partial differential equations. Part 2. Existence and uniqueness of solutions to the Cauchy problem for linear equations

The article proposes a method for analyzing the Cauchy problem wide class of evolutionary linear partial differential equations with variable coefficients. By applying (inverse) Fourier transform, original equation is reduced to an integro-differential equation, which can be considered as ordinary in corresponding Banach space. selection this space carried out such way that principle contraction mappings used. To carry estimates operators generated by transformed we impose conditions finiteness inverse transform coefficients, and spaces coefficients are determined from Paley-Wiener theorems. In case, apparatus theory Bochner integral pseudo-normed spaces, countably-normed Sobolev Classes functions distinguished existence uniqueness solutions proved. For separated variables, exact obtained form finite sums operator exponentials.