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.
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