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Abstract Parabolic Evolution Equations and their Applications

2009-11-03 Mathematics
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Release : 2009-11-03
Genre : Mathematics
Kind : eBook
Book Rating : 312/5 ( reviews)

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Book Synopsis Abstract Parabolic Evolution Equations and their Applications by : Atsushi Yagi

Download or read book Abstract Parabolic Evolution Equations and their Applications written by Atsushi Yagi. This book was released on 2009-11-03. Available in PDF, EPUB and Kindle. Book excerpt: 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 Łojasiewicz–Simon Inequality II

2021-08-12 Mathematics
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Release : 2021-08-12
Genre : Mathematics
Kind : eBook
Book Rating : 634/5 ( reviews)

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Book Synopsis Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality II by : Atsushi Yagi

Download or read book Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality II written by Atsushi Yagi. This book was released on 2021-08-12. Available in PDF, EPUB and Kindle. Book excerpt: 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 II

2021
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Release : 2021
Genre :
Kind : eBook
Book Rating : 647/5 ( reviews)

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Book Synopsis Abstract Parabolic Evolution Equations and Łojasiewicz-Simon Inequality II by : Atsushi Yagi

Download or read book Abstract Parabolic Evolution Equations and Łojasiewicz-Simon Inequality II written by Atsushi Yagi. This book was released on 2021. Available in PDF, EPUB and Kindle. Book excerpt: 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

2021-05-31 Mathematics
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Release : 2021-05-31
Genre : Mathematics
Kind : eBook
Book Rating : 968/5 ( reviews)

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Book Synopsis Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I by : Atsushi Yagi

Download or read book Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I written by Atsushi Yagi. This book was released on 2021-05-31. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Nonlocal and Abstract Parabolic Equations and Their Applications

2009 Differential equations, Parabolic
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Release : 2009
Genre : Differential equations, Parabolic
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis Nonlocal and Abstract Parabolic Equations and Their Applications by : Piotr Mucha

Download or read book Nonlocal and Abstract Parabolic Equations and Their Applications written by Piotr Mucha. This book was released on 2009. Available in PDF, EPUB and Kindle. Book excerpt:

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