Abstract Parabolic Evolution Equations and Lojasiewicz-Simon Inequality I: Abstract Theory (SpringerBriefs in Mathematics)

Atsushi Yagi

ISBN 10: 981161895X / ISBN 13: 9789811618956

Editore: Springer 2021-06-01, 2021

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Codice articolo 6666-IUK-9789811618956

Dati bibliografici

• Titolo: Abstract Parabolic Evolution Equations and Lojasiewicz-Simon Inequality I: Abstract Theory (SpringerBriefs in Mathematics)
• Editore: Springer 2021-06-01
• Data di pubblicazione: 2021
• Lingua: Inglese
• ISBN 10: 981161895X
• ISBN 13: 9789811618956
• Rilegatura: Paperback
• Condizione: New

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Riassunto

The classical Lojasiewicz gradient inequality (1963) was extended by Simon (1983) to the infinite-dimensional setting, now called the Lojasiewicz–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 Lojasiewicz–Simon gradient inequality.

In order to apply the abstract results to a wider class of concrete nonlinear parabolic equations, the usual Lojasiewicz–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.