Abstract Parabolic Evolution Equations and Lojasiewicz-Simon Inequality I: Abstract Theory

This book provides a unified method to show asymptotic convergence of solutions to abstract parabolic evolution equations of the gradient form by extending the classical Łojasiewicz gradient inequality to the infinite-dimensional setting. It is then applied to a wider class of concrete nonlinear parabolic equations, including reaction-diffusion equations with discontinuous coefficients, reaction-diffusion systems, and epitaxial growth equations.

Format: Paperback / softback
Length: 61 pages
Publication date: 01 June 2021
Publisher: Springer Verlag, Singapore

The classical Łojasiewicz gradient inequality, established in 1963, was further developed by Simon in 1983, introducing the concept of the Łojasiewicz–Simon gradient inequality in the infinite-dimensional context. This comprehensive book offers a unified approach to demonstrating the asymptotic convergence of solutions to stationary solutions for abstract parabolic evolution equations of the gradient form. By leveraging this Łojasiewicz–Simon gradient inequality, the book presents a systematic method to achieve this convergence.

To apply the abstract results to a broader range 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. Furthermore, the results are extended to the renowned chemotaxis model, known as the Keller–Segel equations, even in higher-dimensional settings.

This book serves as a valuable resource for researchers and practitioners in the field of mathematical physics, providing a comprehensive framework for understanding and analyzing the behavior of complex systems governed by parabolic equations.

Weight: 454g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811618956
Edition number: 1st ed. 2021