Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions

✍ Scribed by Yihong Du, Hitoshi Ishii, Wei-yueh Lin

• Publisher: World Scientific Publishing Company
• Year: 2009
• Tongue: English
• Leaves: 373
• Category: Library

✦ Synopsis

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 experts and some participants of the conference, including Nils Ackermann (Mexico), Chao-Nien Chen (Taiwan), Yihong Du (Australia), Alberto Farina (France), Hitoshi Ishii (Waseda), N Ishimura (Japan), Shigeaki Koike (Japan), Chu-Pin Lo (Taiwan), Peter Polacik (Minnesota), Kunimochi Sakamoto (Hiroshima), Richard Tsai (Texas), Mingxin Wang (China), Yoshio Yamada (Waseda), Eiji Yanagida (Tohoku), and Xiao-Qiang Zhao (Canada).

✦ Table of Contents

CONTENTS......Page 8
Preface......Page 6
1. Introduction......Page 10
2. Generalities......Page 15
2.1. Existence of the Parabolic Flow......Page 16
2.2. Invariant Sets......Page 18
2.3. Invariant Order Relations......Page 20
2.4. Blow- Up in Finite Time and A Priori Bounds......Page 21
3. Existence of Equilibria and Connecting Orbits......Page 22
3.1. Invariant Manifolds and the Comparison Principle......Page 23
3.2. The Definite Homogeneous Case......Page 27
3.3. Invariant Manifolds and the Zero Number......Page 29
3.4. Invariant Manifolds and Linking Theorems......Page 30
3.5. Order Intervals, Sub- and Supersolutions......Page 32
Bibliography......Page 35
1. Introduction......Page 40
2. Stability Criteria......Page 42
3. Applications of Theorem 1 and Theorem 2......Page 43
4.1.......Page 48
4.2.......Page 52
References......Page 55
1. Introduction......Page 58
2.1. Preliminaries......Page 64
2.2. Main results......Page 67
3. The Holling type II predator-prey model......Page 71
3.1. Protection zone above critical size......Page 73
3.2. Protection zone below critical size......Page 75
4. The Leslie predator-prey model......Page 76
References......Page 79
1. De Giorgi conjecture......Page 83
1.1. (Crude) physical motivation......Page 84
1.2. Possible motivation for the conjecture......Page 85
2. Available results......Page 86
2.1. The case n = 2......Page 87
2.1.1. First proof of Conjecture 1.1 for n = 2......Page 89
2.1.2. Second proof of Conjecture 1.1 for n = 2......Page 90
2.2. The case n = 3......Page 91
2.3. The case 4 n 8......Page 93
2.4. The case of the quasiminima......Page 96
2.5. The case in which the level sets are global graphs......Page 97
2.6. The fully nonlinear case......Page 98
2.8. The Heisenberg group case......Page 99
References......Page 101
1. Introduction......Page 106
2. Existence of periodic solutions......Page 108
3. Constancy on Aubry sets......Page 111
References......Page 127
1. Introduction......Page 129
2. Proof of Theorem......Page 133
3. Discussions......Page 137
References......Page 138
1. Introduction......Page 140
2. Preliminaries......Page 143
3.1. Linear growth (i.e. (1) and (2))......Page 147
3.2. Superlinear growth (i.e. (3) and (4))......Page 150
3.3. Linear and superlinear growth......Page 153
4. Parabolic PDEs......Page 155
4.1. Bounded coefficients (i.e. (1))......Page 156
4.2. Linear growth (i.e. (2) and (3))......Page 157
4.3. Superlinear growth (i.e. (4) and (5))......Page 158
4.4. Linear and superlinear growth......Page 159
References......Page 161
1. Introduction......Page 163
2. Cellular or subcellular level modeling (microscopic): Ordinary differential equations......Page 164
3. Tissue level modeling (mesoscopic)......Page 166
4.1. Buildup of geometric model of heart......Page 167
4.2. Numerical methods......Page 168
4.3. EGG computing......Page 170
6. Conclusion......Page 172
References......Page 173
1. Introduction: basic problems, results and some history......Page 179
1.1. Equations on bounded domains......Page 181
1.2. Equations on ]RN......Page 185
1.3. Cooperative systems......Page 187
1.4. Applications......Page 188
3. Fully nonlinear equations on bounded domains......Page 189
3.1. Solutions on (0,00)......Page 192
3.2. Solutions: (-00, T) and linearized problems......Page 194
3.3. Asymptotically symmetric equations......Page 197
4. Quasilinear equations on ]RN......Page 199
4.1. Solutions on (0, )......Page 200
4.2. Solutions on (- , T) and linearized problems......Page 201
5. Cooperative systems......Page 203
6. On the proofs: a comparison of bounded and unbounded domains......Page 206
7. Applications and some open problems......Page 210
References......Page 212
1. Introduction and main result......Page 218
2. Existence of travelling pulse waves......Page 225
2.1. Homoclinic orbits......Page 226
2.2. Small and Large pulse waves ¢S,L......Page 227
3. Spectral property of pulse wave solutions......Page 228
References......Page 232
1. Introduction......Page 234
2. Expansions, Truncations, and Thresholding......Page 237
2.1. Expansions Near the Curve......Page 238
2.2.1. Willmore with Lower Order Terms......Page 239
2.2.2. Surface Diffusion......Page 240
2.3. Signed Distance Function to a Smooth Curve and Redistancing......Page 241
2.3.1. General Curvature Motions......Page 243
3. Numerical Implementations and Experiments......Page 244
3.1. Expanding Circle......Page 245
3.3. Surface Diffusion Flow......Page 246
3.4. Shape Reconstruction......Page 248
3.5. Junctions......Page 249
3.6. General Curvature Motion Using Signed Distance Functions and Redistancing......Page 251
References......Page 252
Blow-Up Problems for Partial Differential Equations and Systems of Parabolic Type y. Chen and M. Wang......Page 254
1. Introduction......Page 255
2. Simultaneous blow-up......Page 256
3. Systems with homogeneous Dirichlet boundary conditions......Page 259
4. Nonlinear Boundary Value Problem......Page 261
5. Initial and boundary value problems with localized terms......Page 268
6. Degenerate problems not in divergence form......Page 277
7. Non-local problems......Page 280
References......Page 285
1. Problems......Page 291
2. Global existence results......Page 294
3. A priori estimate......Page 296
1. Idea of proof of Theorem 2.3......Page 300
II. Outline of proofs of Theorems 2.4 and 2.5......Page 301
III. Idea of proof of Theorem 2.6......Page 302
IV. Idea of proof of Theorem 2.7......Page 304
5. Concluding remarks......Page 305
References......Page 306
1. Introduction......Page 309
2.1. Critical exponents......Page 311
2.2. Asymptotic behavior of steady states......Page 313
3.1. Stability of steady states......Page 314
3.2. Approach of two solutions......Page 315
3.3. Rate of approach......Page 316
3.4. Anisotropically decaying solutions......Page 317
4.1. Non-stabilizing solutions......Page 318
4.2. Quasi-convergence......Page 320
4.3. Birth-and-death of peaks......Page 322
5.1. A sufficient condition for grow-up......Page 323
5.2. Grow-up rate......Page 324
5.3. Grow-up in the critical case......Page 326
6.1. Sufficient conditions for the converyence to zero......Page 327
6.2. lJecay rate......Page 328
6.3. Slowly decaying solutions......Page 329
7.1. Forward self-similar solutions......Page 330
7.2. Convergence rote......Page 332
8.1. Linear behavior of solutions......Page 334
8.2. Convergence to self-similar solutions of the linear heat equation......Page 335
References......Page 338
1. Introduction......Page 341
2.1. Monostable case......Page 344
2.2. Bistable case......Page 349
3. A class of non-monotone systems......Page 352
4.1. A model with a quiescent stage......Page 356
4.2. A nonlocallattice differential system......Page 357
4.3. A multi-type SIS epidemic model......Page 358
4.4. A vector disease model with spatial spread......Page 359
4.5. A nonlocal and periodic model with dispersal......Page 361
References......Page 363