Abstract Parabolic Evolution Equations and ?ojasiewicz-Simon Inequality I

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.