Sobolev type equations in spaces of differential forms on Riemannian manifolds without boundary

The article contains a review of the results obtained by the author both independently and in collaboration with other members of the Chelyabinsk scientific school founded by G.A. Sviridyuk and devoted to Sobolev-type equations in specific spaces, namely the spaces of differential forms defined on some Riemannian manifold without boundary. Sobolev type equations are nonclassical equations of mathematical physics and are characterized by an irreversible operator at the highest derivative. In our spaces, we need to use special generalizations of operators to the space of differential forms, in particular, the Laplace operator is replaced by its generalization, the Laplace–Beltrami operator. We consider specific interpretations of equations with the relatively bounded operators: linear Barenblatt–Zheltov–Kochina, linear and semilinear Hoff, linear Oskolkov ones. For these equations, we investigate the solvability of the Cauchy, Showalter–Sidorov and initial-final value problems in different cases. Depending on the choice of the type of equation (linear or semi-linear), we use the corresponding modification of the phase space method. In the spaces of differential forms, in order to use this method based on domain splitting and the actions of the corresponding operators, the basis is the Hodge–Kodaira theorem on the splitting of the domain of the Laplace-Beltrami operator.