Theory and Applications of Abstract Semilinear Cauchy Problems

1. Front Matter

   Pages i-xxii

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2. Introduction

   • Pierre Magal, Shigui Ruan

   Pages 1-55

3. Semigroups and Hille-Yosida Theorem

   • Pierre Magal, Shigui Ruan

   Pages 57-99

4. Integrated Semigroups and Cauchy Problems with Non-dense Domain

   • Pierre Magal, Shigui Ruan

   Pages 101-164

5. Spectral Theory for Linear Operators

   • Pierre Magal, Shigui Ruan

   Pages 165-216

6. Semilinear Cauchy Problems with Non-dense Domain

   • Pierre Magal, Shigui Ruan

   Pages 217-248

7. Center Manifolds, Hopf Bifurcation, and Normal Forms

   • Pierre Magal, Shigui Ruan

   Pages 249-308

8. Functional Differential Equations

   • Pierre Magal, Shigui Ruan

   Pages 309-356

9. Age-Structured Models

   • Pierre Magal, Shigui Ruan

   Pages 357-449

10. Parabolic Equations

    • Pierre Magal, Shigui Ruan

    Pages 451-521

11. Back Matter

    Pages 523-543

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Several types of differential equations, such as functional differential equation, age-structured models, transport equations, reaction-diffusion equations, and partial differential equations with delay, can be formulated as abstract Cauchy problems with non-dense domain. This monograph provides a self-contained and comprehensive presentation of the fundamental theory of non-densely defined semilinear Cauchy problems and their applications. Starting from the classical Hille-Yosida theorem, semigroup method, and spectral theory, this monograph introduces the abstract Cauchy problems with non-dense domain, integrated semigroups, the existence of integrated solutions, positivity of solutions, Lipschitz perturbation, differentiability of solutions with respect to the state variable, and time differentiability of solutions. Combining the functional analysis method and bifurcation approach in dynamical systems, then the nonlinear dynamics such as the stability of equilibria, center manifoldtheory, Hopf bifurcation, and normal form theory are established for abstract Cauchy problems with non-dense domain. Finally applications to functional differential equations, age-structured models, and parabolic equations are presented. This monograph will be very valuable for graduate students and researchers in the fields of abstract Cauchy problems, infinite dimensional dynamical systems, and their applications in biological, chemical, medical, and physical problems.

• abstract semilinear cauchy problems
• densely and non-densely defined cauchy problems
• integrated semigroup theory
• spectral theory of linear operators
• center manifold theory
• Hopf bifurcation
• normal form theory
• functional differential equations
• age-structured models
• parabolic equations
• ordinary differential equations
• partial differential equations

“This interesting monograph can be a useful tool for researchers interested in the theory of abstract differential equations along with their applications, especially in age-structured models. However, it can be also used by graduate students as well as PhD students who are willing to get familiar with this theory. … Remarks and Notes appearing at the end of each chapter are a good hint for further reading. The monograph is worth to be recommended.” (Dariusz Bugajewski, zbMATH 1447.34002, 2020)

“This book will be of great interest for researchers studying abstract ODEs and their applications, especially for those with interest in nonlinear population dynamics, particularly in age-structured models.” (Paul Georgescu, Mathematical Reviews, August, 2019)

• Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence, France

  Pierre Magal

• Department of Mathematics, University of Miami, Coral Gables, USA

  Shigui Ruan

Dr.  Pierre Magal is a professor in the Institut de Mathématiques de Bordeaux  at the University of Bordeaux, France. His research interests are Differential Equations, Dynamical Systems, and Mathematical Biology.  He studies nonlinear dynamics of abstract semilinear equations, functional differential equations, age-structured models, and parabolic systems. He is also interested in modeling some biological, epidemiological, and medical problems and studying the nonlinear dynamics of these models.

Shigui Ruan is a professor in the Department of Mathematics at the University of Miami, Coral Gables, Florida, USA. His research interests are Differential Equations, Dynamical Systems, and Mathematical Biology.  He studies nonlinear dynamics of some types of differential equations, such as the center manifold theory and Hopf bifurcation in semilinear evolution equations, multiple-parameter bifurcations in delay equations, and traveling waves in nonlocal reaction-diffusion systems. He is also interested in modeling and studying transmission dynamics of some infectious diseases (malaria, Rift Valley Fever, Hepatitis B virus, schistosomiasis, human rabies, SARS, West Nile virus, etc.) and nonlinear population dynamics.

 

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