Extended gradient systems: Dimension one

Siniša Slijepčević

doi:10.3934/dcds.2000.6.503

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2000, Volume 6, Issue 3: 503-518. Doi: 10.3934/dcds.2000.6.503

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Extended gradient systems: Dimension one

Siniša Slijepčević^1,

1.
Department of Mathemctics, Bijenička 30, 10000 Zagreb

Received: July 31, 1999

Revised: December 31, 1999

Published: June 2000

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Abstract

We propose a general theorey of formally gradient differential equations on unbounded one-dimensional domains, based on an energy-flow inequality, and on the study of the induced semiflow on the space of probability measures on the phase space. We prove that the $\omega$-limit set of each point contains an equilibrium, and that the $\omega$-limit set of $\mu$-almost every point in the phase space consists of equilibria, where $\mu$ is any Borel probability measure invariant for spatial translation.

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Mathematics Subject Classification: 37B35, 35B40.

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