Abstract Parabolic Evolution Equations and Lojasiewicz-Simon Inequality I

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.

Makes an extended version of the LojasiewiczSimon inequality more available to certain concrete problems Offers a unified method to show asymptotic convergence of solutions for nonlinear parabolic equations and systems Covers a range of applications of concrete nonlinear parabolic equations, including the famous KellerSegel equations

Klappentext

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.

Inhalt
1.Preliminary.- 2.Asymptotic Convergence.- 3.Extended ojasiewiczSimon Gradient Inequality.