Non-Local Partial Differential Equations for Engineering and Biology (eBook)

Preface 7
References 13
Acknowledgements 15
Contents 16
Part I Applications in Engineering 19
1 Micro-Electro-Mechanical-Systems (MEMS) 20
1.1 Derivation of the Basic Model and Its Variations 20
1.1.1 The Elastic Problem 21
1.1.2 The Electric Problem 23
1.1.3 An Uncoupled Local Model 24
1.1.4 An Uncoupled Non-local Model 26
1.2 Mathematical Analysis 29
1.2.1 A Non-local Parabolic Problem 29
1.2.2 A Non-local Hyperbolic Problem 64
References 78
2 Ohmic Heating Phenomena 81
2.1 Ohmic Heating of Foods 81
2.1.1 Derivation of the Basic Model and Its Variations 81
2.1.2 Local Existence and Monotonicity 85
2.1.3 Stationary Problem 89
2.1.4 Stability 95
2.1.5 Finite-Time Blow-Up 101
2.2 A Non-local Thermistor Problem 108
2.2.1 Neumann Problem 108
2.2.2 Robin Problem 111
2.2.3 Dirichlet Problem 114
References 122
3 Linear Friction Welding 125
3.1 Derivation of the Model 125
3.2 The Exponential Case 130
3.3 Numerical Results 140
3.3.1 The Soft Material Case 140
3.3.2 The Hard Material Case 142
References 144
4 Resistance Spot Welding 146
4.1 Derivation of the Non-local Model 146
4.2 The Mathematical Problem 151
4.3 The Numerical Scheme 152
4.4 Stability 154
4.5 Error Estimates 158
4.6 Numerical Experiments 165
References 173
Part II Applications in Biology 175
5 Gierer--Meinhardt System 176
5.1 Derivation of the Non-local Model 176
5.2 Mathematical Analysis 181
5.2.1 Global-in-time Existence 181
5.2.2 ODE Type Blow-Up 189
5.2.3 Diffusion Driven Blow-Up 192
5.2.4 Blow-Up Rate and Blow-Up Pattern 202
References 205
6 A Non-local Model Illustrating Replicator Dynamics 207
6.1 Derivation of the Non-local Model 207
6.2 Mathematical Analysis 212
6.2.1 Local Existence and Extendability of Weak Solutions 213
6.2.2 Global Existence Versus Blow-Up 232
References 237
7 A Non-local Model Arising in Chemotaxis 240
7.1 Derivation of the Non-local Model 240
7.2 Mathematical Analysis 242
7.2.1 Preliminaries 242
7.2.2 Blow-Up Results 244
7.3 An Associated Competition-Diffusion System 251
7.4 Miscellanea 257
References 259
8 A Non-local Reaction-Diffusion System Illustrating Cell Dynamics 261
8.1 Derivation of the Non-local Reaction-Diffusion System 261
8.2 Mathematical Analysis 265
8.2.1 Preliminary Results 265
8.2.2 Phase Separation 269
8.2.3 Long-Time Behavior 276
8.2.4 Decay Rate Towards the Steady States 281
References 299
Appendix Appendix 301
A.1 Kirchhoff Equation 301
A.2 Equilibrium and Relaxation States of Point Vortices 302
A.3 Normalized Ricci Flow on Surfaces 305
References 307
Index 309