Spectral theory for random and nonautonomous parabolic equations and applications /
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| Author / Creator: | Mierczynski, Janusz. |
|---|---|
| Imprint: | Boca Raton : CRC Press, c2008. |
| Description: | xiii, 317 p. ; 25 cm. |
| Language: | English |
| Series: | Chapman & Hall/CRC monographs and surveys in pure and applied mathematics ; 139 |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/7095573 |
Table of Contents:
- Preface
- Symbol Description
- 1. Introduction
- 1.1. Outline of the Monograph
- 1.2. General Notations and Concepts
- 1.3. Standing Assumptions
- 2. Fundamental Properties in the General Setting
- 2.1. Assumptions and Weak Solutions
- 2.2. Basic Properties of Weak Solutions
- 2.3. The Adjoint Problem
- 2.4. Perturbation of Coefficients
- 2.5. The Smooth Case
- 2.6. Remarks on Equations in Nondivergence Form
- 3. Spectral Theory in the General Setting
- 3.1. Principal Spectrum and Principal Lyapunov Exponents: Definitions and Properties
- 3.2. Exponential Separation: Definitions and Basic Properties
- 3.3. Existence of Exponential Separation and Entire Positive Solutions
- 3.4. Multiplicative Ergodic Theorems
- 3.5. The Smooth Case
- 3.6. Remarks on the General Nondivergence Case
- 3.7. Appendix: The Case of One-Dimensional Spatial Domain
- 4. Spectral Theory in Nonautonomous and Random Cases
- 4.1. Principal Spectrum and Principal Lyapunov Exponents in Random and Nonautonomous Cases
- 4.1.1. The Random Case
- 4.1.2. The Nonautonomous Case
- 4.2. Monotonicity with Respect to the Zero Order Terms
- 4.2.1. The Random Case
- 4.2.2. The Nonautonomous Case
- 4.3. Continuity with Respect to the Zero Order Coefficients
- 4.3.1. The Random Case
- 4.3.2. The Nonautonomous Case
- 4.4. General Continuity with Respect to the Coefficients
- 4.4.1. The Random Case
- 4.4.2. The Nonautonomous Case
- 4.5. Historical Remarks
- 4.5.1. The Time Independent and Periodic Case
- 4.5.2. The General Time Dependent Case
- 5. Influence of Spatial-Temporal Variations and the Shape of Domain
- 5.1. Preliminaries
- 5.1.1. Notions and Basic Assumptions
- 5.1.2. Auxiliary Lemmas
- 5.2. Influence of Temporal Variation on Principal Lyapunov Exponents and Principal Spectrum
- 5.2.1. The Smooth Case
- 5.2.2. The Nonsmooth Case
- 5.3. Influence of Spatial Variation on Principal Lyapunov Exponents and Principal Spectrum
- 5.4. Faber-Krahn Inequalities
- 5.5. Historical Remarks
- 6. Cooperative Systems of Parabolic Equations
- 6.1. Existence and Basic Properties of Mild Solutions in the General Setting
- 6.1.1. The Nonsmooth Case
- 6.1.2. The Smooth Case
- 6.2. Principal Spectrum and Principal Lyapunov Exponents and Exponential Separation in the General Setting
- 6.2.1. Principal Spectrum and Principal Lyapunov Exponents
- 6.2.2. Exponential Separation: Basic Properties
- 6.2.3. Existence of Exponential Separation and Entire Positive Solutions
- 6.3. Principal Spectrum and Principal Lyapunov Exponents in Nonautonomous and Random Cases
- 6.3.1. The Random Case
- 6.3.2. The Nonautonomous Case
- 6.3.3. Influence of Time and Space Variations
- 6.4. Remarks
- 7. Applications to Kolmogorov Systems of Parabolic Equations
- 7.1. Semilinear Equations of Kolmogorov Type: General Theory
- 7.1.1. Existence, Uniqueness, and Basic Properties of Solutions
- 7.1.2. Linearization at the Trivial Solution
- 7.1.3. Global Attractor and Uniform Persistence
- 7.2. Semilinear Equations of Kolmogorov Type: Examples
- 7.2.1. The Random Case
- 7.2.2. The Nonautonomous Case
- 7.3. Competitive Kolmogorov Systems of Semilinear Equations: General Theory
- 7.3.1. Existence, Uniqueness, and Basic Properties of Solutions
- 7.3.2. Linearization at Trivial and Semitrivial Solutions
- 7.3.3. Global Attractor and Uniform Persistence
- 7.4. Competitive Kolmogorov Systems of Semilinear Equations: Examples
- 7.4.1. The Random Case
- 7.4.2. The Nonautonomous Case
- 7.5. Remarks
- References
- Index