Mierczynski / Shen Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications

Erscheinungsjahr 2008
ISBN: 978-1-58488-896-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 336 Seiten

Reihe: Monographs and Surveys in Pure and Applied Mathematics

Providing a basic tool for studying nonlinear problems, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications focuses on the principal spectral theory for general time-dependent and random parabolic equations and systems. The text contains many new results and considers existing results from a fresh perspective. Taking a clear, unified, and self-contained approach, the authors first develop the abstract general theory in the framework of weak solutions, before turning to cases of random and nonautonomous equations. They prove that time dependence and randomness do not reduce the principal spectrum and Lyapunov exponents of nonautonomous and random parabolic equations. The book also addresses classical Faber–Krahn inequalities for elliptic and time-periodic problems and extends the linear theory for scalar nonautonomous and random parabolic equations to cooperative systems. The final chapter presents applications to Kolmogorov systems of parabolic equations. By thoroughly explaining the spectral theory for nonautonomous and random linear parabolic equations, this resource reveals the importance of the theory in examining nonlinear problems.

Zielgruppe

Graduate students and researchers in differential equations and dynamical systems.

Autoren/Hrsg.

Mierczynski, Janusz

Shen, Wenxian

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

Introduction
Outline of the Monograph
General Notations and Concepts
Standing Assumptions
Fundamental Properties in the General Setting
Assumptions and Weak Solutions
Basic Properties of Weak Solutions
The Adjoint Problem
Perturbation of Coefficients
The Smooth Case
Remarks on Equations in Nondivergence Form
Spectral Theory in the General Setting
Principal Spectrum and Principal Lyapunov Exponent: Definitions and Properties
Exponential Separation: Definitions and Basic Properties
Existence of Exponential Separation and Entire Positive Solutions
Multiplicative Ergodic Theorems
The Smooth Case
Remarks on the General Nondivergence Case
Appendix: The Case of One-Dimensional Spatial Domain
Spectral Theory in Nonautonomous and Random Cases
Principal Spectrum and Principal Lyapunov Exponents in Random and Nonautonomous Cases
Monotonicity with Respect to the Zero Order Terms
Continuity with Respect to the Zero Order Coefficients
General Continuity with Respect to the Coefficients
Historical Remarks
Influence of Spatial-Temporal Variations and the Shape of Domain
Preliminaries
Influence of Temporal Variation on Principal Lyapunov Exponents and Principal Spectrum
Influence of Spatial Variation on Principal Lyapunov Exponents and Principal Spectrum
Faber–Krahn Inequalities
Historical Remarks
Cooperative Systems of Parabolic Equations
Existence and Basic Properties of Mild Solutions in the General Setting
Principal Spectrum and Principal Lyapunov Exponents and Exponential Separation in the General Setting
Principal Spectrum and Principal Lyapunov Exponents in Nonautonomous and Random Cases
Remarks
Applications to Kolmogorov Systems of Parabolic Equations
Semilinear Equations of Kolmogorov Type: General Theory
Semilinear Equations of Kolmogorov Type: Examples
Competitive Kolmogorov Systems of Semilinear Equations: General Theory
Competitive Kolmogorov Systems of Semilinear Equations: Examples
Remarks
References
Index