Elliptic Partial Differential Equations

I. Introduction to the theory of reaction-diffusion equations
Chapter 1. Reaction-diffusion processes, models and applications
1. Chemical physics
2. Population dynamics
3. Biomedical models
4. Mathematical analysis of reaction-diffusion equations
Chapter 2. Methods of analysis
1. Operators and spaces
2. Topological degree
3. Maximum principle, positiveness and comparison theorems
4. Spectrum and stability
Chapter 3. Reaction-diffusion problems in bounded domains
1. Existence of solutions
2. Spectrum and stability
3. Bifurcation of dissipative structures
Chapter 4. Reaction-diffusion problems on the whole axis
1. Travelling waves
2. Non-autonomous equations
3. Applications

II. Reaction-diffusion waves in cylinders
Chapter 5. Monotone systems
1. Differential-difference equations
2. Homotopy of waves
3. Minimax representation of the wave speed
4. Monostable case
Chapter 6. Reaction-diffusion problems with convection
1. Formulation of reaction-diffusion-convection problems
2. Reaction-diffusion-convection operators in unbounded cylinders
3. Bifurcations of convective waves
4. Existence of reaction-diffusion convection waves
5. Convection in reactive systems
Chapter 7. Reaction-diffusion systems with different diffusion coefficients
1. Formulation and main results
2. Integro-differential problem
3. Bifurcations of waves
Chapter 8. Nonlinear boundary conditions
1. Bounded domains
2. Travelling waves

III. Nonlocal and multi-scale models
Chapter 9. Nonlocal reaction-diffusion equations
1. Nonlocal equations in bounded domains
2. Nonlocal equations on the whole axis
3. Wave existence
4. Existence and stability of pulses
5. Evolution equations
Chapter 10. Multi-scale models in biology
1. Cell population dynamics
2. Multi-scale models with intracellular kinetics
3. Models with cell transport
4. Applications in physiology
- Bibliographical comments
- Concluding remarks
- Acknowledgements
- References
- Index

Vitaly Volpert started his scientific career in Russia and continued it in the USA and in France. He works on partial differential equations and on mathematical modelling in chemical physics, biology and medicine. He is an author of more than 200 scientific publications including three monographs.