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We show that generic linear perturbations $\psi$ solving $\Box_g \psi +\frac{\alpha}{l^2} \psi =0$ on Reissner-Nordstr\"om-AdS black holes haveinfinite local energy at the Cauchy horizon. Combined with the result ofarXiv:1812.06142 that such perturbations $\psi$ remain uniformly bounded andextend continuously across the Cauchy horizon, this settles the linear analogof the Strong Cosmic Censorship conjecture for Reissner-Nordstr\"om-AdS: the$C^0$-formulation is false but the $H^1$-formulation is true.

Blowup of the local energy of linear waves at the Reissner-Nordstr\"om-AdS Cauchy horizon