Initial-boundary problem for parabolic-hyperbolic equation with loaded summands

In this paper we find necessary and sufficient conditions for uniqueness of solution to the initial-boundary problem for a loaded equation of mixed parabolic-hyperbolic type. The solution is constructed as a sum of a series in eigenfunctions of the corresponding one-dimensional problem on eigenvalues. In the proof of convergence of the series the problem of small denominators arises. Under certain conditions on these problems we obtain an estimate for a small denominator to be separated from zero that allows to prove the existence theorem in the class of regular solutions.

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