Książki

If we had to formulate in one sentence what this book is about, it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equations and new topics such as nonlocal equations and multi-scale models in biology will be considered.

Elliptic Partial Differential Equations: Volume 2: Reaction-Diffusion 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.