Elliptic Partial Differential Equations

1. Front Matter

   Pages i-xviii

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2. Introduction to the Theory of Reaction-diffusion Equations

   1. Front Matter

      Pages 1-1

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   2. Reaction-diffusion Processes, Models and Applications

      • Vitaly Volpert

      Pages 3-78

   3. Methods of Analysis

      • Vitaly Volpert

      Pages 79-121

   4. Reaction-diffusion Problems in Bounded Domains

      • Vitaly Volpert

      Pages 123-200

   5. Reaction-diffusion Problems on the Whole Axis

      • Vitaly Volpert

      Pages 201-324

3. Reaction-diffusion Waves in Cylinders

   1. Front Matter

      Pages 325-326

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   2. Monotone Systems

      • Vitaly Volpert

      Pages 327-390

   3. Reaction-diffusion Problems with Convection

      • Vitaly Volpert

      Pages 391-451

   4. Reaction-diffusion Systems with Different Transport Coefficients

      • Vitaly Volpert

      Pages 453-490

   5. Nonlinear Boundary Conditions

      • Vitaly Volpert

      Pages 491-517

4. Nonlocal and Multi-scale Models

   1. Front Matter

      Pages 519-519

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   2. Nonlocal Reaction-diffusion Equations

      • Vitaly Volpert

      Pages 521-626

   3. Multi-scale Models in Biology

      • Vitaly Volpert

      Pages 627-696

5. Back Matter

   Pages 697-784

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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.

• bifurcations of solutions
• classical theory
• existence of solutions
• population dynamics
• reaction-diffusion equations
• stability of solutions
• partial differential equations

“This volume is concerned with the mathematical analysis of reaction-diffusion partial differential equations in relationship with their multiple applications. … This volume is useful to researchers in pure and applied nonlinear analysis and to graduate students in mathematics and mathematical physics.” (Vicenţiu D. Rădulescu, Mathematical Reviews, November, 2015)

• Institut Camille Jordan, CNRS, Université Claude Bernard Lyon 1, Villeurbanne, France

  Vitaly Volpert

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.

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