First-singularity conjecture for nonlinear waves on extremal Reissner–Nordström spacetimes

Let $(\mathcal{M},g)$ be an extremal Reissner–Nordström spacetime with event horizon $\mathcal{H}^{+}$, and consider spherically symmetric solutions $\psi$ of

$$\Box_g\psi=\psi^{2n}+\chi(T\psi)^{2n}+\chi(Y\psi)^{2n},$$

where $n\in\mathbb{N}$, $n\geqslant 2$, and $\chi=\chi(r)\in C^{\infty}_0([M,\infty))$ is a smooth cutoff equal to $1$ on $[M,M+c/2]$ and equal to $0$ on $[M+c,\infty)$ for some finite constant $c$.

First-singularity conjecture. Spherically symmetric solutions of this equation have their first singularities on the event horizon $\mathcal{H}^{+}$.

This conjecture proposes an interpretation of the Aretakis instability as a shock-formation mechanism: although the corresponding equation with only the term $\psi^{2n}$ has well-behaved spherically symmetric solutions for sufficiently large $n$, the additional derivative nonlinearities are expected to force the first singularity to form on the event horizon.