Theory and Applications of Abstract Semilinear Cauchy Problems

Beschreibung

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.


Reviews the classical results on semigroup theory, Hille-Yosida Theorem, semilinear Cauchy problems with dense domain in details
Introduces the integrated semigroup theory in order to study semilinear Cauchy problems when the linear operator is nondensely defined and is not Hille-Yosida

Discusses the spectral throry for linear operators, including the spectral decomposition of the state space

Presents the center manifold theory, Hopf bifurcation theorem, and normal form theory for abstract semilinear Cauchy problems

Applies the abstract theories to functional differential equations, age-structured models, and parabolic equations arising in population dynamics and other applied subjects

Allows readers and graduate students with no background to start with the basic concepts The application-oriented readers will see how the abstract results apply to biological and physical problems Learn the fundamental theories on abstract equations

Autor*in

Pierre Magal

Themen in »Theory and Applications of Abstract Semilinear Cauchy 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

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“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)
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