Chain recurrence and structure of ω-limit sets of multivalued semiflows

The paper proves that when the set of equilibria R is finite, every weak solution’s ω-limit set reduces to a single equilibrium, using multivalued semiflow/attractor machinery and a Lyapunov function on the attractor. The candidate gives a classical gradient-system proof via energy identity, Aubin–Lions asymptotic compactness, and constancy of the energy on complete limit trajectories. The conclusions coincide; the approaches differ. The candidate’s outline is standard and essentially correct but glosses some technical steps (e.g., passing the energy to the limit on ω(ϕ) and justifying the strict Lyapunov identity for weak solutions).