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Extended gradient systems: Dimension one. (English) Zbl 1009.37004
Summary: We propose a general theory 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 translations.

##### MSC:
 37B35 Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems 37L45 Hyperbolicity, Lyapunov functions for infinite-dimensional dissipative dynamical systems 35B40 Asymptotic behavior of solutions to PDEs
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