Attractors, Shadowing, and Approximation of Abstract Semilinear Differential Equations

Contents
Preface
Introduction
Preliminaries
1. Semilinear Equations
	1.1 Differential Equations in Banach Spaces
	1.2 Some Interesting Properties
	1.3 Theorem ABC
	1.4 Convergence of Semidiscretizations for Periodic Problem
2. Convergence of Attractors
	2.1 Discrete Convergence of Sets
	2.2 Equiattraction and Continuous Convergence
	2.3 Abstract Parabolic Problems
3. Convergence of Attractors in the Hyperbolic Case
	3.1 Attractors for Abstract Parabolic Problems
		3.1.1 Equilibrium points
		3.1.2 Unstable manifolds
	3.2 Approximation of Operators on the General Discretization Scheme
		3.2.1 Approximation of spectra of linear operators
		3.2.2 Regions of convergence
	3.3 Approximation of Attractors
		3.3.1 Upper semicontinuity of attractors
		3.3.2 Approximation of the set of equilibria
	3.4 Approximation of Unstable Manifolds
	3.5 Auxiliary Results
		3.5.1 Convergence of linear semigroups and uniform estimates
4. Shadowing for Abstract Parabolic Equations
	4.1 Preliminaries
	4.2 Discretization of Operators and Semigroups
		4.2.1 Estimates for the linear case
		4.2.2 Estimates for the nonlinear case
	4.3 Shadowing in Space
		4.3.1 Shadowing in space and the condition Δcc ≠ ∅
	4.4 Shadowing in Time
		4.4.1 Shadowing in time and the condition Δcc≠ ∅
	4.5 Checking of the Condition Δcc ≠ ∅
	4.6 Dichotomy for Semigroups
		4.6.1 Preliminary results on dichotomy of semigroups
		4.6.2 Discretization of dichotomy semigroups
	4.7 General Dichotomy for Semidiscretization
		4.7.1 Dichotomy for condensing operators in semidiscretization
	4.8 General Dichotomy for Discretization in Time Variable
		4.8.1 Discretization in the time variable and the condensing property
	4.9 Semidiscrete Dichotomy for Condensing Operators
		4.9.1 Shadowing for discretization in space and the condensing property
		4.9.2 Shadowing for discretizing in time variable and condensing property
	4.10 Verification of Condensing Conditions
5. Semilinear Fractional Equations
	5.1 Setting of the Problem
	5.2 Semidiscrete Approximation for a Special Case
	5.3 The Case of a Classical Right-Hand Side
	5.4 Semidiscrete Approximation for the Classical Case
	5.5 Maximal Regularity in the Space Cα0 ([0, T];E)
		5.5.1 Setting of the problem
		5.5.2 Difference schemes for fractional equations
6. Stable and Unstable Manifolds for Fractional Semilinear Equations
	6.1 Setting of the Fractional Semilinear Problems
	6.2 Existence of Stable Manifolds
	6.3 Approximation of Stable Manifolds
	6.4 Existence of Unstable Manifolds
	6.5 Approximation of Unstable Manifolds
Bibliography
Index