Sobolev type equations now constitute a vast area of nonclassical equations of mathematical physics. Those called nonclassical equations of mathematical physics, whose representation in the form of equations or systems of equations partial does not fit within one of the classical types (elliptic, parabolic or hyperbolic). In this paper we prove a generalized splitting theorem of spaces and actions of the operators for Sobolev type equations with respect to the relatively radial operator. The main research method is the Sviridyuk theory about relatively spectrum. Abstract results are applied to prove the unique solvability of the multipoint initial-final problem for the evolution equation of Sobolev type, as well as to explore the dichotomies of solutions for the linearized phase field equations.
Apart from the introduction and bibliography article comprises three parts. The first part provides the necessary information regarding the theory of p-radial operators, the second contains the proof of main result about generalized splitting theorem for strongly (L,p)-radial operator M. The third part contains the results of the application of the preceding paragraph for different tasks, namely to prove the unique solvability of the multipoint initial-final problem for Dzektser and to explore the dichotomies of solutions of the linearized phase field equations. References not purport to, and reflects only the authors' tastes and preferences.

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