Extended gradient systems: Dimension one
Siniša Slijepčević · Discrete and Continuous Dynamical Systems · 2000
We propose a general theorey of formally gradient differentialequations on unbounded one-dimensional domains, based on anenergy-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 containsan equilibrium, and that the $\omega$-limit set of $\mu$-almostevery point in the phase space consists of equilibria,where $\mu$ is any Borel probability measure invariantfor spatial translation.