Abstract parabolic evolution equations and Łojasiewicz-Simon inequality. 1, Abstract theory / Atsushi Yagi.

Format:

Book

Author/Creator:

Yagi, Atsushi, 1951- author.

Series:

SpringerBriefs in mathematics.

SpringerBriefs in mathematics

Language:

English

Subjects (All):

Differential equations, Partial.

Physical Description:

1 online resource (68 pages)

Edition:

1st ed.

Place of Publication:

Gateway East, Singapore : Springer, [2021]

Summary:

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.

Contents:

Intro

Preface

Contents

1 Preliminaries

1.1 Basic Materials in Complex Functional Analysis

1.1.1 Dual Spaces

1.1.2 Adjoint Spaces

1.1.3 Interpolation of Spaces

1.1.4 Triplets of Spaces

1.1.5 Sectorial Operators

1.1.6 Abstract Parabolic Evolution Equations

1.2 Real Banach Spaces and Hilbert Spaces

1.2.1 Conjugated Spaces

1.2.2 Interpolation in Conjugated Spaces

1.2.3 Triplets of Conjugated Spaces

1.2.4 Real Sobolev-Lebesgue Spaces

1.3 Real Sectorial Operators

1.3.1 Real Operators in Conjugated Spaces

1.3.2 Sectorial Operators in Conjugated Spaces

1.4 Operators Associated with Sesquilinear Forms

1.4.1 Real Sesquilinear Forms

1.4.2 Realization of Elliptic Operators

1.5 Differentiation of Operators

1.5.1 Fréchet Differentiation

1.5.2 Gâteaux Differentiation

1.6 Some Other Materials

1.6.1 Fredholm Operators

1.6.2 Łojasiewicz Gradient Inequality

1.7 Notes

2 Asymptotic Convergence

2.1 Semilinear Heat Equation

2.2 General Settings

2.3 Structural Assumptions

2.4 Comments for Applications

2.5 Notes and Future Studies

3 Łojasiewicz-Simon Gradient Inequality

3.1 Favorable Case

3.2 General Settings and Structural Assumptions

3.2.1 Some Properties of L in X

3.2.2 Some Properties of L in Y

3.3 Critical Manifold

3.3.1 Definition

3.3.2 Decomposition of Y into S+L(Y)

3.4 Gradient Inequality in L(Y)

3.5 Gradient Inequality on S

3.6 Main Theorem

3.7 Gradient Inequality with Respect to "026B30D ·"026B30D Z

3.7.1 Case Where Z Y

3.7.2 Case Where Y Z

3.8 Notes and Future Studies

Bibliography

Symbol Index

Subject Index.

Notes:

Includes bibliographical references and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

981-16-1896-8

OCLC:

1255228475