# American Institute of Mathematical Sciences

July  2000, 6(3): 503-518. doi: 10.3934/dcds.2000.6.503

## Extended gradient systems: Dimension one

 1 Department of Mathemctics, Bijenička 30, 10000 Zagreb, Croatia

Received  August 1999 Revised  January 2000 Published  April 2000

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.
Citation: Siniša Slijepčević. Extended gradient systems: Dimension one. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 503-518. doi: 10.3934/dcds.2000.6.503
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