Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I
Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I

Author: Atsushi Yagi

Publisher: Springer Nature

Published: 2021-05-31

Total Pages: 68

ISBN-13: 9811618968

DOWNLOAD EBOOK

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.

Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality II
Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality II

Author: Atsushi Yagi

Publisher: Springer Nature

Published: 2021-08-12

Total Pages: 128

ISBN-13: 9811626634

DOWNLOAD EBOOK

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 Łojasiewicz-Simon Inequality I
Abstract Parabolic Evolution Equations and Łojasiewicz-Simon Inequality I

Author: Atsushi Yagi

Publisher:

Published: 2021

Total Pages: 68

ISBN-13: 9789811618970

DOWNLOAD EBOOK

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.

Nonlinear Evolution Equations and Related Topics
Nonlinear Evolution Equations and Related Topics

Author: Wolfgang Arendt

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 803

ISBN-13: 3034879245

DOWNLOAD EBOOK

Philippe Bénilan was a most original and charismatic mathematician who had a deep and decisive impact on the theory of Nonlinear Evolution Equations. Dedicated to him, Nonlinear Evolution Equations and Related Topics contains research papers written by highly distinguished mathematicians. They are all related to Philippe Benilan's work and reflect the present state of this most active field. The contributions cover a wide range of nonlinear and linear equations.

Abstract Parabolic Evolution Equations and their Applications
Abstract Parabolic Evolution Equations and their Applications

Author: Atsushi Yagi

Publisher: Springer Science & Business Media

Published: 2009-11-03

Total Pages: 594

ISBN-13: 3642046312

DOWNLOAD EBOOK

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

The Cahn–Hilliard Equation: Recent Advances and Applications
The Cahn–Hilliard Equation: Recent Advances and Applications

Author: Alain Miranville

Publisher: SIAM

Published: 2019-09-09

Total Pages: 231

ISBN-13: 1611975921

DOWNLOAD EBOOK

This is the first book to present a detailed discussion of both classical and recent results on the popular Cahn–Hilliard equation and some of its variants. The focus is on mathematical analysis of Cahn–Hilliard models, with an emphasis on thermodynamically relevant logarithmic nonlinear terms, for which several questions are still open. Initially proposed in view of applications to materials science, the Cahn–Hilliard equation is now applied in many other areas, including image processing, biology, ecology, astronomy, and chemistry. In particular, the author addresses applications to image inpainting and tumor growth. Many chapters include open problems and directions for future research. The Cahn-Hilliard Equation: Recent Advances and Applications is intended for graduate students and researchers in applied mathematics, especially those interested in phase separation models and their generalizations and applications to other fields. Materials scientists also will find this text of interest.