Linear Sobolev Type Equations and Degenerate Semigroups of Operators

Abstract

Auxiliary material Banach spaces and linear operators Theorems on infinitesimal generators Functional spaces and differential operators Relatively p-radial operators and degenerate strongly continuous semigroups of operators Introduction Relative resolvents Relatively p-radial operators Degenerate strongly continuous semigroups of operators Approximations of Hille-Widder-Post type Splitting of spaces Infinitesimal generators and phase spaces Generators of degenerate strongly continuous semigroups of operators Degenerate strongly continuous groups of operators Relatively p-sectorial operators and degenerate analytic semigroups of operators Introduction Relatively p-sectorial operators Degenerate analytic semigroups of operators Phase spaces for the case of degenerate analytic semigroups Space splitting Generators of degenerate analytic semigroups of operators Degenerate infinitely differentiable semigroups of operators Phase spaces for the case of degenerate infinitely continuously differentiable semigroups Kernels and images of degenerate infinitely differential semigroups of operators Relatively ?-bounded operators and degenerate analytic groups of operators Introduction Relatively ?-bounded operators Relative ?-boundedness and relative p-sectoriality Relative ?-boundedness and relatively adjoint vectors Degenerate analytical groups of operators Sufficient conditions of the relative ?-boundedness The case of a Fredholm operator Analytical semigroups of operators degenerating on the chains of relatively adjoint vectors of an arbitrary length Cauchy problem for inhomogeneous Sobolev-type equations Introduction Case of a relatively ?-bounded operator The case of a relatively p-sectorial operator Case of a relatively p-radial operator Strong solution of Cauchy problem Cauchy problem for an equation with Banach-adjoint operators Propagators Inhomogeneous Cauchy problem for high-order Sobolev-type equations Bounded solutions of Sobolev-type equations Introduction Relatively spectral theorem Bounded relaxed solutions of a homogeneous equation Bounded solutions of the inhomogeneous equation Examples Optimal control Introduction Strong solution of Cauchy problem for an equation with Hilbert-adjoint operators Problem of optimal control for an equation with relatively ?-bounded operator Problem of optimal control for quatino with a relatively p-sectorial operator Barenblatt-Zheltov-Kochina equation System of ordinary differential equations Equation of the evolution of the free filtered-fluid surface Bibliography Index