Nonclassical equations of mathematical physics. Linear Sobolev type equations of higher order

Аннотация

The article presents the review of authors’ results in the field of non-classical equations of mathematical physics. The theory of Sobolev-type equations of higher order is introduced. The idea is based on generalization of degenerate operator semigroups theory in case of the following equations: decomposition of spaces, splitting of operators' actions, the construction of propagators and phase spaces for a homogeneous equation, as well as the set of valid initial values for the inhomogeneous equation. The author uses a proven phase space technology for solving Sobolev type equations consisting of reduction of a singular equation to a regular one defined on some subspace of initial space. However, unlike the first order equations, there is an extra condition that guarantees the existence of the phase space. There are some examples where the initial conditions should match together if the extra condition can’t be fulfilled to solve the Cauchy problem. The reduction of nonclassical equations of mathematical physics to the initial problems for abstract Sobolev type equations of high order is conducted and justified.