Nonclassical Equations of Mathematical Physics. Linear Sobolev Type Equations of Higher Order

Abstract

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.

On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient nonlinearity

“…For example, initial-boundary value problems for equations of Sobolev type have been considered in a general form and as examples in the papers of Sviridyuk, 1 Zagrebina, 2 and Zamyshlyaeva. 3 It should be noted that for the first time, the theory of potential for nonclassical equations of Sobolev type was considered by Kapitonov. 4 Later, the potential theory has been developed in Gabov 5 and Gabov and Sveshnikov 6 and their successors (see, for example, the work of Pletner 7 ).…”

Existence of Solutions in Quasi-Banach Spaces for Evolutionary Sobolev Type Equations in Relatively Radial Case

“…Suppose as above that U and F are quasi-Banach spaces and take two operators L ∈ L(U; F) and M ∈ Cl(U; F) so that M is strongly (L, p)-radial with p ∈ {0} ∪ N. Consider the weak (in the sense of Krein) Showalter-Sidorov problem lim t→0+ P (u(t) − u 0 ) = 0 (9) for the linear inhomogeneous evolutionary Sobolev-type equation…”

О Разрушении Решений Задач Коши Для Одного Класса Нелинейных Уравнений Теории Ферритов

“…Так, в работах Г. А. Свиридюка, С. А. Загребиной, А. А. Замышляевой (см. [6,15,18]) были рассмотрены в общем виде и в виде примеров начально-краевые задачи для большого многообразия классов линейных и нелинейных уравнений соболевского типа.…”