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" Theory and applications of abstract semilinear cauchy problems / "
Pierre Magal, Shigui Ruan.
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BL
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| Record Number
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859501
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| Main Entry
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Magal, Pierre
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| Title & Author
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Theory and applications of abstract semilinear cauchy problems /\ Pierre Magal, Shigui Ruan.
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| Publication Statement
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Cham, Switzerland :: Springer,, [2018]
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, ©2018
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| Series Statement
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Applied mathematical sciences ;; volume 201
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1 online resource
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| ISBN
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3030015068
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: 9783030015060
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303001505X
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9783030015053
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Includes bibliographical references and index.
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| Contents
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Intro; Foreword; Preface; Contents; Acronyms; 1 Introduction; 1.1 Ordinary Differential Equations; 1.1.1 Spectral Properties of Matrices; 1.1.2 State Space Decomposition; 1.1.3 Semilinear Systems; 1.2 Retarded Functional Differential Equations; 1.2.1 Existence and Uniqueness of Solutions; 1.2.2 Linearized Equation at an Equilibrium; 1.2.3 Characteristic Equations; 1.2.4 Center Manifolds; 1.3 Age-Structured Models; 1.3.1 Volterra Formulation; 1.3.2 Age-Structured Models Without Birth; 1.3.3 Age-Structured Models with Birth; 1.3.4 Equilibria and Linearized Equations
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1.3.5 Age-Structured Models Reduce to DDEs and ODEs1.4 Abstract Semilinear Formulation; 1.4.1 Functional Differential Equations; 1.4.2 Age-Structured Models; 1.4.3 Size-Structured Models; 1.4.4 Partial Functional Differential Equations; 1.5 Remarks and Notes; 2 Semigroups and Hille-Yosida Theorem; 2.1 Semigroups; 2.1.1 Bounded Case; 2.1.2 Unbounded Case; 2.2 Resolvents; 2.3 Infinitesimal Generators; 2.4 Hille-Yosida Theorem; 2.5 Nonhomogeneous Cauchy Problem; 2.6 Examples; 2.7 Remarks and Notes; 3 Integrated Semigroups and Cauchy Problems with Non-dense Domain; 3.1 Preliminaries
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3.2 Integrated Semigroups3.3 Exponentially Bounded Integrated Semigroups; 3.4 Existence of Mild Solutions; 3.5 Bounded Perturbation; 3.6 The Hille-Yosida Case; 3.7 The Non-Hille-Yosida Case; 3.8 Applications to a Vector Valued Age-Structured Model in Lp; 3.9 Remarks and Notes; 4 Spectral Theory for Linear Operators; 4.1 Basic Properties of Analytic Maps; 4.2 Spectra and Resolvents of Linear Operators; 4.3 Spectral Theory of Bounded Linear Operators; 4.4 Essential Growth Bound of Linear Operators; 4.5 Spectral Decomposition of the State Space
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4.6 Asynchronous Exponential Growth of Linear Operators4.7 Remarks and Notes; 5 Semilinear Cauchy Problems with Non-dense Domain; 5.1 Introduction; 5.2 Existence and Uniqueness of a Maximal Semiflow: The Blowup Condition; 5.3 Positivity; 5.4 Lipschitz Perturbation; 5.5 Differentiability with Respect to the State Variable; 5.6 Time Differentiability and Classical Solutions; 5.7 Stability of Equilibria; 5.8 Remarks and Notes; 6 Center Manifolds, Hopf Bifurcation, and Normal Forms; 6.1 Center Manifold Theory; 6.1.1 Existence of Center Manifolds; 6.1.2 Smoothness of Center Manifolds
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6.2 Hopf Bifurcation6.2.1 State Space Decomposition; 6.2.2 Hopf Bifurcation Theorem; 6.3 Normal Form Theory; 6.3.1 Nonresonant Type Results; 6.3.2 Normal Form Computation; 6.4 Remarks and Notes; 7 Functional Differential Equations; 7.1 Retarded Functional Differential Equations; 7.1.1 Integrated Solutions and Spectral Analysis; 7.1.2 Projectors on the Eigenspaces; 7.1.3 Hopf Bifurcation; 7.2 Neutral Functional Differential Equations; 7.2.1 Spectral Theory; 7.2.2 Projectors on the Eigenspaces; 7.3 Partial Functional Differential Equations
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| Abstract
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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 manifold theory, 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.--
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Cauchy problem.
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Differential equations, Parabolic.
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Differential equations.
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Evolution equations.
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Semigroup algebras.
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Cauchy problem.
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Differential calculus equations.
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Differential equations, Parabolic.
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Differential equations.
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Evolution equations.
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MATHEMATICS-- Essays.
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MATHEMATICS-- Pre-Calculus.
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MATHEMATICS-- Reference.
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Semigroup algebras.
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| Dewey Classification
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510
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| LC Classification
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QA1.A647 v. 201eb
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QA377.M27 2018
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| Added Entry
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Ruan, Shigui,1963-
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