Generic transverse blow-up conjecture for scalar waves on Reissner–Nordström interiors

Let $(\mathcal M,g)$ be the Reissner–Nordström interior background considered in the paper, let $\Sigma$ be a Cauchy hypersurface, and let $\phi$ be a solution of the wave equation arising from data on $\Sigma$. Denote the future Cauchy horizon by $\mathcal C\mathcal H^+$.

Generic transverse blow-up conjecture. For generic data on $\Sigma$, the transverse derivatives of $\phi$ blow up on $\mathcal C\mathcal H^+$; in fact, $\phi$ is not in $H^1_{\mathrm{loc}}$.

This conjecture concerns the nonlinear blueshift instability that may persist despite boundedness and continuous extendibility of the scalar wave itself at the Cauchy horizon. The cited blow-up results establish the conclusion under suitable lower bounds, but such lower bounds had not been proved for solutions arising from generic data on $\Sigma$.