  We study a McKean--Vlasov equation arising from a mean-field model of a
particle system with positive feedback. As particles hit a barrier they cause
the other particles to jump in the direction of the barrier and this feedback
mechanism leads to the possibility that the system can exhibit contagious
blow-ups. Using a fixed-point argument we construct a differentiable solution
up to a first explosion time. Our main contribution is a proof of uniqueness in
the class of c\`{a}dl\`{a}g functions, which confirms the validity of related
propagation-of-chaos results in the literature. We extend the allowed initial
conditions to include densities with any power law decay at the boundary, and
connect the exponent of decay with the growth exponent of the solution in small
time in a precise way. This takes us asymptotically close to the control on
initial conditions required for a global solution theory. A novel minimality
result and trapping technique are introduced to prove uniqueness.

Probability;Analysis of PDEs