The Higher-Order Sobolev-Type Models

This paper surveys the author's results concerning mathematical models based on Sobolev-type equations of higher order. The theory is built using the available facts on the solvability of initial (initial-final) problems for first-order Sobolev-type equations. The main idea is a generalization of the theory of degenerate (semi)groups of operators to the case of higher-order equations: decomposition of spaces and actions of the operators, construction of propagators and the phase space for the homogeneous equation, as well as the set of valid initial values for the inhomogeneous equation. We use the phase space method, which is quite useful for solving Sobolev-type equations and consists in a reduction of a singular equation to a regular one defined on a certain subspace of the original space. We reduce mathematical models to initial (initial-final) problems for abstract Sobolev-type equations of higher order. The results may find further applications in the study of optimal control problems and nonlinear mathematical models, and to the construction of the theory of Sobolev-type equations of higher order in quasi-Banach spaces.

Keywords

Sobolev-type equations; phase space; propagators; initial-final problem; relative spectrum.

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