Abstract Parabolic Evolution Equations And Their Applications
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Abstract Parabolic Evolution Equations And Their Applications
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Author : Atsushi Yagi
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-11-03
Abstract Parabolic Evolution Equations And Their Applications written by Atsushi Yagi and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-11-03 with Mathematics categories.
This monograph is intended to present the fundamentals of the theory of abstract parabolic evolution equations and to show how to apply to various nonlinear dif- sion equations and systems arising in science. The theory gives us a uni?ed and s- tematic treatment for concrete nonlinear diffusion models. Three main approaches are known to the abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. In order to keep the volume of the monograph in reasonable length, we will focus on the semigroup methods. For other two approaches, see the related references in Bibliography. The semigroup methods, which go back to the invention of the analytic se- groups in the middle of the last century, are characterized by precise formulas representing the solutions of the Cauchy problem for evolution equations. The ?tA analytic semigroup e generated by a linear operator ?A provides directly a fundamental solution to the Cauchy problem for an autonomous linear e- dU lution equation, +AU =F(t), 0
Abstract Parabolic Evolution Equations And Ojasiewicz Simon Inequality Ii
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Author : Atsushi Yagi
language : en
Publisher: Springer Nature
Release Date : 2021-08-12
Abstract Parabolic Evolution Equations And Ojasiewicz Simon Inequality Ii written by Atsushi Yagi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-08-12 with Mathematics categories.
This second volume continues the study on asymptotic convergence of global solutions of parabolic equations to stationary solutions by utilizing the theory of abstract parabolic evolution equations and the Łojasiewicz–Simon gradient inequality. In the first volume of the same title, after setting the abstract frameworks of arguments, a general convergence theorem was proved under the four structural assumptions of critical condition, Lyapunov function, angle condition, and gradient inequality. In this volume, with those abstract results reviewed briefly, their applications to concrete parabolic equations are described. Chapter 3 presents a discussion of semilinear parabolic equations of second order in general n-dimensional spaces, and Chapter 4 is devoted to treating epitaxial growth equations of fourth order, which incorporate general roughening functions. In Chapter 5 consideration is given to the Keller–Segel equations in one-, two-, and three-dimensional spaces. Some of these results had already been obtained and published by the author in collaboration with his colleagues. However, by means of the abstract theory described in the first volume, those results can be extended much more. Readers of this monograph should have a standard-level knowledge of functional analysis and of function spaces. Familiarity with functional analytic methods for partial differential equations is also assumed.
Abstract Parabolic Evolution Equations And Ojasiewicz Simon Inequality Ii
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Author : Atsushi Yagi
language : en
Publisher:
Release Date : 2021
Abstract Parabolic Evolution Equations And Ojasiewicz Simon Inequality Ii written by Atsushi Yagi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with categories.
This second volume continues the study on asymptotic convergence of global solutions of parabolic equations to stationary solutions by utilizing the theory of abstract parabolic evolution equations and the Łojasiewicz-Simon gradient inequality. In the first volume of the same title, after setting the abstract frameworks of arguments, a general convergence theorem was proved under the four structural assumptions of critical condition, Lyapunov function, angle condition, and gradient inequality. In this volume, with those abstract results reviewed briefly, their applications to concrete parabolic equations are described. Chapter 3 presents a discussion of semilinear parabolic equations of second order in general n-dimensional spaces, and Chapter 4 is devoted to treating epitaxial growth equations of fourth order, which incorporate general roughening functions. In Chapter 5 consideration is given to the Keller-Segel equations in one-, two-, and three-dimensional spaces. Some of these results had already been obtained and published by the author in collaboration with his colleagues. However, by means of the abstract theory described in the first volume, those results can be extended much more. Readers of this monograph should have a standard-level knowledge of functional analysis and of function spaces. Familiarity with functional analytic methods for partial differential equations is also assumed.
Abstract Parabolic Evolution Equations And Ojasiewicz Simon Inequality I
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Author : Atsushi Yagi
language : en
Publisher: Springer Nature
Release Date : 2021-05-31
Abstract Parabolic Evolution Equations And Ojasiewicz Simon Inequality I written by Atsushi Yagi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-05-31 with Mathematics categories.
The classical Łojasiewicz gradient inequality (1963) was extended by Simon (1983) to the infinite-dimensional setting, now called the Łojasiewicz–Simon gradient inequality. This book presents a unified method to show asymptotic convergence of solutions to a stationary solution for abstract parabolic evolution equations of the gradient form by utilizing this Łojasiewicz–Simon gradient inequality. In order to apply the abstract results to a wider class of concrete nonlinear parabolic equations, the usual Łojasiewicz–Simon inequality is extended, which is published here for the first time. In the second version, these abstract results are applied to reaction–diffusion equations with discontinuous coefficients, reaction–diffusion systems, and epitaxial growth equations. The results are also applied to the famous chemotaxis model, i.e., the Keller–Segel equations even for higher-dimensional ones.
Analytic Semigroups And Optimal Regularity In Parabolic Problems
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Author : Alessandra Lunardi
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-13
Analytic Semigroups And Optimal Regularity In Parabolic Problems written by Alessandra Lunardi and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-13 with Mathematics categories.
The book shows how the abstract methods of analytic semigroups and evolution equations in Banach spaces can be fruitfully applied to the study of parabolic problems. Particular attention is paid to optimal regularity results in linear equations. Furthermore, these results are used to study several other problems, especially fully nonlinear ones. Owing to the new unified approach chosen, known theorems are presented from a novel perspective and new results are derived. The book is self-contained. It is addressed to PhD students and researchers interested in abstract evolution equations and in parabolic partial differential equations and systems. It gives a comprehensive overview on the present state of the art in the field, teaching at the same time how to exploit its basic techniques. - - - This very interesting book provides a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and how this theory may be used in the study of parabolic partial differential equations; it takes into account the developments of the theory during the last fifteen years. (...) For instance, optimal regularity results are a typical feature of abstract parabolic equations; they are comprehensively studied in this book, and yield new and old regularity results for parabolic partial differential equations and systems. (Mathematical Reviews) Motivated by applications to fully nonlinear problems the approach is focused on classical solutions with continuous or Hölder continuous derivatives. (Zentralblatt MATH)
Nonlocal And Abstract Parabolic Equations And Their Applications
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Author : Piotr Mucha
language : en
Publisher:
Release Date : 2009
Nonlocal And Abstract Parabolic Equations And Their Applications written by Piotr Mucha and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Differential equations, Parabolic categories.
Abstract Evolution Equations Periodic Problems And Applications
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Author : D Daners
language : en
Publisher: Chapman and Hall/CRC
Release Date : 1992-12-29
Abstract Evolution Equations Periodic Problems And Applications written by D Daners and has been published by Chapman and Hall/CRC this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-12-29 with Mathematics categories.
Part of the Pitman Research Notes in Mathematics series, this text covers: linear evolution equations of parabolic type; semilinear evolution equations of parabolic type; evolution equations and positivity; semilinear periodic evolution equations; and applications.
Abstract Parabolic Evolution Equations And Ojasiewicz Simon Inequality I
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Author : Atsushi Yagi
language : en
Publisher:
Release Date : 2021
Abstract Parabolic Evolution Equations And Ojasiewicz Simon Inequality I written by Atsushi Yagi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with Differential equations, Parabolic categories.
The classical ojasiewicz gradient inequality (1963) was extended by Simon (1983) to the infinite-dimensional setting, now called the ojasiewiczSimon gradient inequality. This book presents a unified method to show asymptotic convergence of solutions to a stationary solution for abstract parabolic evolution equations of the gradient form by utilizing this ojasiewiczSimon gradient inequality. In order to apply the abstract results to a wider class of concrete nonlinear parabolic equations, the usual ojasiewiczSimon inequality is extended, which is published here for the first time. In the second version, these abstract results are applied to reactiondiffusion equations with discontinuous coefficients, reactiondiffusion systems, and epitaxial growth equations. The results are also applied to the famous chemotaxis model, i.e., the KellerSegel equations even for higher-dimensional ones.
Evolution Equations And Their Applications In Physical And Life Sciences
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Author : G Lumer
language : en
Publisher: CRC Press
Release Date : 2019-04-24
Evolution Equations And Their Applications In Physical And Life Sciences written by G Lumer and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-04-24 with Medical categories.
This volume presents a collection of lectures on linear partial differntial equations and semigroups, nonlinear equations, stochastic evolutionary processes, and evolution problems from physics, engineering and mathematical biology. The contributions come from the 6th International Conference on Evolution Equations and Their Applications in Physica
Abstract Evolution Equations Periodic Problems And Applications
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Author : Daniel Daners
language : en
Publisher: Longman Sc & Tech
Release Date : 1992
Abstract Evolution Equations Periodic Problems And Applications written by Daniel Daners and has been published by Longman Sc & Tech this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Mathematics categories.