Linear Sobolev Type Equations and Degenerate Semigroups of Operators

Georgy A. Sviridyuk, South Ural State University, Chelyabinsk, Russia; Vladimir E. Fedorov, Chelyabinsk State University, Russia.

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