Dynamics Reported: Expositions in Dynamical Systems New Series: Volume 3

ISBN-13: 9783642782367 / Angielski / Miękka / 2011 / 162 str.

G. Fournier; I. Lasiecka; D. Lupo

Opis

This book contains three contributions with topics in dynamical systems: Limit relative category and critical point theory, coexistence of infinitely many stable solutions to reaction diffusion systems, and second-order hyperbolic mixed problems. All the authors give a careful and readable presentation of recent research results, which are of interest to mathematicians, mathematical biologists, chemists and physicists. The book is written for graduate students and researchers in these fields and it is also suitable as a text for graduate level seminars in dynamical systems.

Spis treści

Limit Relative Category and Critical Point Theory.- 1. Introduction.- 2. Relative Category.- 3. Relative Cuplength.- 4. Limit Relative Category.- 5. The Deformation Lemma.- 6. Critical Point Theorems.- 7. Some Applications.- 8. A Perturbation Theorem.- References.- Coexistence of Infinitely Many Stable Solutions to Reaction Diffusion Systems in the Singular Limit.- 1. Introduction and singular limit slow dynamics.- 2. Intuitive Approach to the Stability of Multi-layered Solutions.- 2.1. Slow Manifold for Mono-layered Solution.- 2.2. Local Slow Manifold for Double-layered Solution.- 2.3. Formal linearization.- 3. The SLEP Method for the Stability of Normal N-layered Solutions.- 3.1. Normal N-layered Solution.- 3.2. Asymptotic Behaviors of Critical Eigenvalues and Eigenfunctions of Le,a.- 3.3. Derivation of the SLEP System.- 3.4. Eigenvalue problem for the SLEP matrix.- 4. Recovery Process of Stability.- 4.1. Instability of Multi-layered Solutions for the Shadow System.- 4.2. Recovery of Stability (From the shadow system to the full system).- 5. Concluding Remarks.- Appendix A.- Appendix B.- Appendix C.- References.- Recent advances in regularity of second-order hyperbolic mixed problems, and applications.- 1. Introduction.- I: Regularity Theory.- 2. Regularity under Dirichlet boundary conditions.- 3. Regularity under Neumann Boundary Conditions.- 4. Cosine/Sine (Semigroup) Representation Formulae of the Solutions.- 4.1. Dirichlet Case.- 4.2. Neumann Case.- II: Applications.- 5. Well-posedness of Semi-linear Wave Equations with Neumann Boundary Conditions.- 5.1. Statement of Local Well-posedness.- 5.2. Proof of Theorem 5.1.- 5.3. Proof of Theorem 5.2.- 6. Local Exponential Stability of Damped Wave Equations with Semi-linear Boundary Conditions.- 6.1. Orientation.- 6.2. Dirichlet Boundary Conditions: Theorem 6.1.- 6.3. Dirichlet Boundary Conditions: Proof of Theorem 6.1.- 6.4. Neumann Boundary Conditions. Theorem 6.2.- 6.5. Neumann Boundary Conditions. Proof of Theorem 6.2.- 7. Exact Controllability of Semi-linear Hyperbolic Problems.- 7.1. Semi-linear Wave Equation of Neumann Type. Exact Controllability.- 7.2. Semi-linear Wave Equation of Dirichlet Type. Exact Controllability.- 7.3. An Exact Controllability Result for Abstract Semi-linear Systems. Theorem 7.4.- 7.4. Application of Theorem 7.4 to the Semi-linear Wave Problem (7.12) with Dirichlet Control.- 8. Riccati Operator Equations and Hyperbolic Mixed Problems.- 8.1. Introduction; the Case where B and 1Z are Bounded.- 8.2. The Unbounded Case: Main statements for T < oo.- 8.2.1. The Unbounded Case for T < oo. The Differential Riccati Equation. Theorems 8.1, 8.2.- 8.3. Application of Theorems 8.1 and 8.2: The DRE for Boundary Control and Boundary Observation for Hyperbolic Mixed Problems of Neumann Type.- 8.4. The Unbounded Case for T = oo. The Algebraic Riccati Equation. Theorem 8.4.- 8.5. Application of Theorem 8.4: The ARE for Boundary Control for Hyperbolic Mixed Problems of Dirichlet Type.- References.