Abstract Parabolic Evolution Equations and Łojasiewicz-Simon Inequality I : Abstract Theory / Atsushi Yagi.

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Abstract Parabolic Evolution Equations and Łojasiewicz-Simon Inequality I : Abstract Theory / Atsushi Yagi.

9789811618963 (electronic book)
9811618968 (electronic book)
9789811618970 (print)
9811618976
981161895X (print)
9789811618956 (print)

Singapore : Springer Singapore Pte. Limited, [2021]

English

1 online resource (68 pages).

10.1007/978-981-16-1896-3 doi

QA377.3 .Y34 2021

515/.35

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.

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