  Phase transitions and critical behaviour of a class of MV-SDEs, whose
concomitant non-local Fokker-Planck equation includes the Granular Media
equation with quadratic interaction potential as a special case, is studied. By
careful analysis of an implicit auxiliary integral equation, it is shown for a
wide class of potentials that below a certain `critical threshold' there are
exactly as many stationary measures as extrema of the potential, while above
another the stationary measure is unique, and consequently phase transition(s)
between. For symmetric bistable potentials, these critical thresholds are
proven to be equal and a strictly increasing function of the aggregation
parameter. Additionally, a simple condition is provided for symmetric
multi-well potentials with an arbitrary number of extrema to demonstrate
analogous behaviour. This answers, with considerably more generality, a
conjecture of Tugaut [Stochastics, 86:2, 257-284]. To the best of our knowledge
many of these results are novel. Others simplify the proofs of known results
whilst greatly increasing their applicability.
