Dynamics Reported. New

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.