Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I

Beschreibung

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 utilizing this Łojasiewicz–Simon gradient inequality.
In order to apply the abstract results to a wider class 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. The results are also applied to the famous chemotaxis model, i.e., the Keller–Segel equations even for higher-dimensional ones.

Makes an extended version of the Lojasiewicz–Simon 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 Keller–Segel equations

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Atsushi Yagi

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Abstract Parabolic Evolution Equations Łojasiewicz--Simon Inequality Asymptotic Convergence of Solutions Semilinear Parabolic Equations Keller--Segel Equations partial differential equations

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