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This paper is concerned with the analysis of blow-ups for two McKean-Vlasovequations involving hitting times. Let $(B(t); \, t \ge 0)$ be standardBrownian motion, and $\tau:= \inf\{t \ge 0: X(t) \le 0\}$ be the hitting timeto 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 $\alpha$ andthe distribution of $X(0)$ such that the corresponding Fokker-Planck equationhas no blow-up, and thus the McKean-Vlasov dynamics is well-defined for alltime $t \ge 0$. Our approach relies on a connection between the McKean-Vlasovequation and the supercooled Stefan problem, as well as several comparisonprinciples. 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 thatfor $\beta > 0$ sufficiently large and $\alpha$ no greater than a sufficientlysmall positive constant, there is no blow-up and the McKean-Vlasov dynamics iswell-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.

McKean-Vlasov equations involving hitting times: blow-ups and global solvability