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