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

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