Abstract Parabolic Evolution Equations and Lojasiewicz-Simon Inequality I (eBook)

Abstract Theory

Atsushi Yagi (Autor)

eBook Download: PDF

2021 | 1. Auflage
X, 61 Seiten
Springer Singapore (Verlag)
978-981-16-1896-3 (ISBN)

Infos

The classical ?ojasiewicz gradient inequality (1963) was extended by Simon (1983) to the infinite-dimensional setting, now called the ?ojasiewicz-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 ?ojasiewicz-Simon gradient inequality.

In order to apply the abstract results to a wider class of concrete nonlinear parabolic equations, the usual ?ojasiewicz-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.

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The classical Lojasiewicz gradient inequality (1963) was extended by Simon (1983) to the infinite-dimensional setting, now called the Lojasiewicz-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 Lojasiewicz-Simon gradient inequality.In order to apply the abstract results to a wider class of concrete nonlinear parabolic equations, the usual Lojasiewicz-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.

Produkt-Details

• Erscheint lt. Verlag: 31.5.2021
• Reihe/Serie: SpringerBriefs in Mathematics
• Zusatzinfo: X, 61 p. 17 illus.
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Schlagworte: abstract parabolic evolution equations • Asymptotic Convergence of Solutions • Keller--Segel Equations • Łojasiewicz--Simon Inequality • Partial differential equations • Semilinear parabolic equations
• ISBN-10: 981-16-1896-8 / 9811618968
• ISBN-13: 978-981-16-1896-3 / 9789811618963