Order Structure and Topological Methods in Nonlinear Partial Differential Equations PDF Download
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Author: Yihong Du
Publisher: World Scientific
ISBN: 9812774440
Category : Mathematics
Languages : en
Pages : 202
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Book Description
The maximum principle induces an order structure for partial differential equations, and has become an important tool in nonlinear analysis. This book is the first of two volumes to systematically introduce the applications of order structure in certain nonlinear partial differential equation problems. The maximum principle is revisited through the use of the Krein-Rutman theorem and the principal eigenvalues. Its various versions, such as the moving plane and sliding plane methods, are applied to a variety of important problems of current interest. The upper and lower solution method, especially its weak version, is presented in its most up-to-date form with enough generality to cater for wide applications. Recent progress on the boundary blow-up problems and their applications are discussed, as well as some new symmetry and Liouville type results over half and entire spaces. Some of the results included here are published for the first time. Sample Chapter(s). Chapter 1: Krein-Rutman Theorem and the Principal Eigenvalue (128 KB). Contents: KreinOCoRutman Theorem and the Principal Eigenvalue; Maximum Principles Revisited; The Moving Plane Method; The Method of Upper and Lower Solutions; The Logistic Equation; Boundary Blow-Up Problems; Symmetry and Liouville Type Results Over Half and Entire Spaces. Readership: Researchers and postgraduate students in partial differential equations."
Author: Yihong Du
Publisher: World Scientific
ISBN: 9812774440
Category : Mathematics
Languages : en
Pages : 202
Get Book Here
Book Description
The maximum principle induces an order structure for partial differential equations, and has become an important tool in nonlinear analysis. This book is the first of two volumes to systematically introduce the applications of order structure in certain nonlinear partial differential equation problems. The maximum principle is revisited through the use of the Krein-Rutman theorem and the principal eigenvalues. Its various versions, such as the moving plane and sliding plane methods, are applied to a variety of important problems of current interest. The upper and lower solution method, especially its weak version, is presented in its most up-to-date form with enough generality to cater for wide applications. Recent progress on the boundary blow-up problems and their applications are discussed, as well as some new symmetry and Liouville type results over half and entire spaces. Some of the results included here are published for the first time. Sample Chapter(s). Chapter 1: Krein-Rutman Theorem and the Principal Eigenvalue (128 KB). Contents: KreinOCoRutman Theorem and the Principal Eigenvalue; Maximum Principles Revisited; The Moving Plane Method; The Method of Upper and Lower Solutions; The Logistic Equation; Boundary Blow-Up Problems; Symmetry and Liouville Type Results Over Half and Entire Spaces. Readership: Researchers and postgraduate students in partial differential equations."
Author: Yihong Du
Publisher: World Scientific
ISBN: 9814478857
Category : Mathematics
Languages : en
Pages : 202
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Book Description
The maximum principle induces an order structure for partial differential equations, and has become an important tool in nonlinear analysis. This book is the first of two volumes to systematically introduce the applications of order structure in certain nonlinear partial differential equation problems.The maximum principle is revisited through the use of the Krein-Rutman theorem and the principal eigenvalues. Its various versions, such as the moving plane and sliding plane methods, are applied to a variety of important problems of current interest. The upper and lower solution method, especially its weak version, is presented in its most up-to-date form with enough generality to cater for wide applications. Recent progress on the boundary blow-up problems and their applications are discussed, as well as some new symmetry and Liouville type results over half and entire spaces. Some of the results included here are published for the first time.
Author: Michel Chipot
Publisher: Springer Nature
ISBN: 3031541235
Category :
Languages : en
Pages : 393
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Book Description
Author: Enayet U Tarafdar
Publisher: World Scientific
ISBN: 9814476218
Category : Mathematics
Languages : en
Pages : 627
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Book Description
This book provides a comprehensive overview of the authors' pioneering contributions to nonlinear set-valued analysis by topological methods. The coverage includes fixed point theory, degree theory, the KKM principle, variational inequality theory, the Nash equilibrium point in mathematical economics, the Pareto optimum in optimization, and applications to best approximation theory, partial equations and boundary value problems.Self-contained and unified in presentation, the book considers the existence of equilibrium points of abstract economics in topological vector spaces from the viewpoint of Ky Fan minimax inequalities. It also provides the latest developments in KKM theory and degree theory for nonlinear set-valued mappings.
Author:
Publisher: Elsevier
ISBN: 0080871550
Category : Mathematics
Languages : en
Pages : 339
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Book Description
Nonlinear Partial Differential Equations
Author: King-Yeung Lam
Publisher: Springer Nature
ISBN: 3031204220
Category : Mathematics
Languages : en
Pages : 316
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Book Description
This book introduces some basic mathematical tools in reaction-diffusion models, with applications to spatial ecology and evolutionary biology. It is divided into four parts. The first part is an introduction to the maximum principle, the theory of principal eigenvalues for elliptic and periodic-parabolic equations and systems, and the theory of principal Floquet bundles. The second part concerns the applications in spatial ecology. We discuss the dynamics of a single species and two competing species, as well as some recent progress on N competing species in bounded domains. Some related results on stream populations and phytoplankton populations are also included. We also discuss the spreading properties of a single species in an unbounded spatial domain, as modeled by the Fisher-KPP equation. The third part concerns the applications in evolutionary biology. We describe the basic notions of adaptive dynamics, such as evolutionarily stable strategies and evolutionary branching points, in the context of a competition model of stream populations. We also discuss a class of selection-mutation models describing a population structured along a continuous phenotypical trait. The fourth part consists of several appendices, which present a self-contained treatment of some basic abstract theories in functional analysis and dynamical systems. Topics include the Krein-Rutman theorem for linear and nonlinear operators, as well as some elements of monotone dynamical systems and abstract competition systems. Most of the book is self-contained and it is aimed at graduate students and researchers who are interested in the theory and applications of reaction-diffusion equations.
Author: Frithjof Lutscher
Publisher: Springer Nature
ISBN: 3030292940
Category : Mathematics
Languages : en
Pages : 390
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Book Description
This book is the first thorough introduction to and comprehensive treatment of the theory and applications of integrodifference equations in spatial ecology. Integrodifference equations are discrete-time continuous-space dynamical systems describing the spatio-temporal dynamics of one or more populations. The book contains step-by-step model construction, explicitly solvable models, abstract theory and numerical recipes for integrodifference equations. The theory in the book is motivated and illustrated by many examples from conservation biology, biological invasions, pattern formation and other areas. In this way, the book conveys the more general message that bringing mathematical approaches and ecological questions together can generate novel insights into applications and fruitful challenges that spur future theoretical developments. The book is suitable for graduate students and experienced researchers in mathematical ecology alike.
Author: Mingxin Wang
Publisher: CRC Press
ISBN: 1000353915
Category : Mathematics
Languages : en
Pages : 298
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Book Description
The parabolic partial differential equations model one of the most important processes in the real-world: diffusion. Whether it is the diffusion of energy in space-time, the diffusion of species in ecology, the diffusion of chemicals in biochemical processes, or the diffusion of information in social networks, diffusion processes are ubiquitous and crucial in the physical and natural world as well as our everyday lives. This book is self-contained and covers key topics such as the Lp theory and Schauder theory, maximum principle, comparison principle, regularity and uniform estimates, initial-boundary value problems of semilinear parabolic scalar equations and weakly coupled parabolic systems, the upper and lower solutions method, monotone properties and long-time behaviours of solutions, convergence of solutions and stability of equilibrium solutions, global solutions and finite time blowup. It also touches on periodic boundary value problems, free boundary problems, and semigroup theory. The book covers major theories and methods of the field, including topics that are useful but hard to find elsewhere. This book is based on tried and tested teaching materials used at the Harbin Institute of Technology over the past ten years. Special care was taken to make the book suitable for classroom teaching as well as for self-study among graduate students. About the Author: Mingxin Wang is Professor of Mathematics at Harbin Institute of Technology, China. He has published ten monographs and textbooks and 260 papers. He is also a supervisor of 30 PhD students.
Author: Julian Lopez-gomez
Publisher: World Scientific Publishing Company
ISBN: 9814440264
Category : Mathematics
Languages : en
Pages : 356
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Book Description
The main goal of the book is to provide a comprehensive and self-contained proof of the, relatively recent, theorem of characterization of the strong maximum principle due to Molina-Meyer and the author, published in Diff. Int. Eqns. in 1994, which was later refined by Amann and the author in a paper published in J. of Diff. Eqns. in 1998. Besides this characterization has been shown to be a pivotal result for the development of the modern theory of spatially heterogeneous nonlinear elliptic and parabolic problems; it has allowed us to update the classical theory on the maximum and minimum principles by providing with some extremely sharp refinements of the classical results of Hopf and Protter-Weinberger. By a celebrated result of Berestycki, Nirenberg and Varadhan, Comm. Pure Appl. Maths. in 1994, the characterization theorem is partially true under no regularity constraints on the support domain for Dirichlet boundary conditions.Instead of encyclopedic generality, this book pays special attention to completeness, clarity and transparency of its exposition so that it can be taught even at an advanced undergraduate level. Adopting this perspective, it is a textbook; however, it is simultaneously a research monograph about the maximum principle, as it brings together for the first time in the form of a book, the most paradigmatic classical results together with a series of recent fundamental results scattered in a number of independent papers by the author of this book and his collaborators.Chapters 3, 4, and 5 can be delivered as a classical undergraduate, or graduate, course in Hilbert space techniques for linear second order elliptic operators, and Chaps. 1 and 2 complete the classical results on the minimum principle covered by the paradigmatic textbook of Protter and Weinberger by incorporating some recent classification theorems of supersolutions by Walter, 1989, and the author, 2003. Consequently, these five chapters can be taught at an undergraduate, or graduate, level. Chapters 6 and 7 study the celebrated theorem of Krein-Rutman and infer from it the characterizations of the strong maximum principle of Molina-Meyer and Amann, in collaboration with the author, which have been incorporated to a textbook by the first time here, as well as the results of Chaps. 8 and 9, polishing some recent joint work of Cano-Casanova with the author. Consequently, the second half of the book consists of a more specialized monograph on the maximum principle and the underlying principal eigenvalues.
Author: Julián López-Gómez
Publisher: CRC Press
ISBN: 1482238993
Category : Mathematics
Languages : en
Pages : 372
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Book Description
Metasolutions of Parabolic Equations in Population Dynamics explores the dynamics of a generalized prototype of semilinear parabolic logistic problem. Highlighting the author's advanced work in the field, it covers the latest developments in the theory of nonlinear parabolic problems. The book reveals how to mathematically determine if a species maintains, dwindles, or increases under certain circumstances. It explains how to predict the time evolution of species inhabiting regions governed by either logistic growth or exponential growth. The book studies the possibility that the species grows according to the Malthus law while it simultaneously inherits a limited growth in other regions. The first part of the book introduces large solutions and metasolutions in the context of population dynamics. In a self-contained way, the second part analyzes a series of very sharp optimal uniqueness results found by the author and his colleagues. The last part reinforces the evidence that metasolutions are also categorical imperatives to describe the dynamics of huge classes of spatially heterogeneous semilinear parabolic problems. Each chapter presents the mathematical formulation of the problem, the most important mathematical results available, and proofs of theorems where relevant.
