  This paper is concerned with the analysis of blow-ups for two McKean-Vlasov
equations involving hitting times. Let $(B(t); \, t \ge 0)$ be standard
Brownian motion, and $\tau:= \inf\{t \ge 0: X(t) \le 0\}$ be the hitting time
to zero of a given process $X$. The first equation is $X(t) = X(0) + B(t) -
\alpha \mathbb{P}(\tau \le t)$. We provide a simple condition on $\alpha$ and
the distribution of $X(0)$ such that the corresponding Fokker-Planck equation
has no blow-up, and thus the McKean-Vlasov dynamics is well-defined for all
time $t \ge 0$. Our approach relies on a connection between the McKean-Vlasov
equation and the supercooled Stefan problem, as well as several comparison
principles. The second equation is $X(t) = X(0) + \beta t + B(t) + \alpha \log
\mathbb{P}(\tau > t)$, whose Fokker-Planck equation is non-local. We prove that
for $\beta > 0$ sufficiently large and $\alpha$ no greater than a sufficiently
small positive constant, there is no blow-up and the McKean-Vlasov dynamics is
well-defined for all time $t \ge 0$. The argument is based on a new transform,
which removes the non-local term, followed by a relative entropy analysis.
