Recent progress on reaction-diffusion systems and viscosity solutions / editors, Yihong Du, Hitoshi Ishii, Wei-Yueh Lin.

Includes bibliographical references.

This book consists of survey and research articles expanding on the theme of the 'International Conference on Reaction-Diffusion Systems and Viscosity Solutions', held at Providence University, Taiwan, during January 3-6, 2007. It is a carefully selected collection of articles representing the recent progress of some important areas of nonlinear partial differential equations. The book is aimed for researchers and postgraduate students who want to learn about or follow some of the current research topics in nonlinear partial differential equations. The contributors consist of international exp.

Cover -- CONTENTS -- Preface -- Long-Time Dynamics in Semilinear Parabolic Problems with Autocatalysis N. Ackermann -- 1. Introduction -- 2. Generalities -- 2.1. Existence of the Parabolic Flow -- 2.2. Invariant Sets -- 2.3. Invariant Order Relations -- 2.4. Blow- Up in Finite Time and A Priori Bounds -- 3. Existence of Equilibria and Connecting Orbits -- 3.1. Invariant Manifolds and the Comparison Principle -- 3.2. The Definite Homogeneous Case -- 3.3. Invariant Manifolds and the Zero Number -- 3.4. Invariant Manifolds and Linking Theorems -- 3.5. Order Intervals, Sub- and Supersolutions -- Acknowledgment -- Bibliography -- A Note on Reaction-Diffusion Systems with Skew-Gradient Structure C.-N. Chen, T.-L. Horng, D. Lee and G.-H. Tsai -- 1. Introduction -- 2. Stability Criteria -- 3. Applications of Theorem 1 and Theorem 2 -- 4. Numerical Results -- 4.1. -- 4.2. -- Acknowledgments -- References -- Change of Environment in Model Ecosystems: Effect of a Protection Zone in Diffusive Population Models Y. Du -- 1. Introduction -- 2. The competition model -- 2.1. Preliminaries -- 2.2. Main results -- 3. The Holling type II predator-prey model -- 3.1. Protection zone above critical size -- 3.2. Protection zone below critical size -- 4. The Leslie predator-prey model -- References -- The State of the Art for a Conjecture of de Giorgi and Related Problems A. Farina and E. Valdinoci -- 1. De Giorgi conjecture -- 1.1. (Crude) physical motivation -- 1.2. Possible motivation for the conjecture -- 2. Available results -- 2.1. The case n = 2 -- 2.2. The case n = 3 -- 2.3. The case 4 n 8 -- 2.4. The case of the quasiminima -- 2.5. The case in which the level sets are global graphs -- 2.6. The fully nonlinear case -- 2.7. The fractional Laplacian case -- 2.8. The Heisenberg group case -- Acknowledgments -- References -- Two Remarks on Periodic Solutions of Hamilton-Jacobi Equations H. Ishii and H. Mitake -- 1. Introduction -- 2. Existence of periodic solutions -- 3. Constancy on Aubry sets -- Acknowledgments -- References -- Asymptotic Expansion Method for Local Volatility Models N. Ishimura and K. Nishida -- 1. Introduction -- 2. Proof of Theorem -- 3. Discussions -- Acknowledgments -- References -- Recent Developments on Maximum Principle for LP-Viscosity Solutions of Fully Nonlinear Elliptic/Parabolic PDEs s. Koike -- 1. Introduction -- 2. Preliminaries -- 3. Elliptic PDEs -- 3.1. Linear growth (i.e. (1) and (2)) -- 3.2. Superlinear growth (i.e. (3) and (4)) -- 3.3. Linear and superlinear growth -- 4. Parabolic PDEs -- 4.1. Bounded coefficients (i.e. (1)) -- 4.2. Linear growth (i.e. (2) and (3)) -- 4.3. Superlinear growth (i.e. (4) and (5)) -- 4.4. Linear and superlinear growth -- References -- Multiscale Modeling of Electrical Activities of the Heart C.-P. Lo, H.-C. Tien, D. Lee and C.-H. Chang -- 1. Introduction -- 2. Cellular or subcellular level modeling (microscopic): Ordinary differential equations -- 3. Tissue level modeling (mesoscopic) -- 4. Whole heart modeling (macroscopic) -- 4.1. Buildup of geometric model of heart -- 4.2. Numerical methods -- 4.3. EGG computing -- 5. Applications in hear arrhythmia -- 6. Conclusion -- References -- Symmetry Properties of Positive Solutions of Parabolic Equations: A Survey P. Polacik -- T.

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