Strong Cosmic Censorship for the Spherically Symmetric Einstein–Maxwell-Charged-Klein–Gordon System with Positive $$\Lambda $$ : Stability of the Cauchy Horizon and $$H^1$$ Extensions

We investigate the interior of a dynamical black hole as described by the Einstein–Maxwell-charged-Klein–Gordon system of equations with a cosmological constant, under spherical symmetry. In particular, we consider a characteristic initial value problem where, on the outgoing initial hypersurface, interpreted as the event horizon \(\mathcal {H}^+\) of a dynamical black hole, we prescribe: (a) initial data asymptotically approaching a fixed sub-extremal Reissner–Nordström–de Sitter solution and (b) an exponential Price law upper bound for the charged scalar field. After showing local well-posedness for the corresponding first-order system of partial differential equations, we establish the existence of a Cauchy horizon \(\mathcal{C}\mathcal{H}^+\) for the evolved spacetime, extending the bootstrap methods used in the case \(\Lambda = 0\) by Van de Moortel (Commun Math Phys 360:103–168, 2018. https://doi.org/10.1007/s00220-017-3079-3). In this context, we show the existence of \(C^0\) spacetime extensions beyond \(\mathcal{C}\mathcal{H}^+\). Moreover, if the scalar field decays at a sufficiently fast rate along \(\mathcal {H}^+\), we show that the renormalized Hawking mass remains bounded for a large set of initial data. With respect to the analogous model concerning an uncharged and massless scalar field, we are able to extend the known range of parameters for which mass inflation is prevented, up to the optimal threshold suggested by the linear analyses by Costa–Franzen (Ann Henri Poincaré 18:3371–3398, 2017. https://doi.org/10.1007/s00023-017-0592-z) and Hintz–Vasy (J Math Phys 58(8):081509, 2017. https://doi.org/10.1063/1.4996575). In this no-mass-inflation scenario, which includes near-extremal solutions, we further prove that the spacetime can be extended across the Cauchy horizon with continuous metric, Christoffel symbols in \(L^2_{\text {loc}}\) and scalar field in \(H^1_{\text {loc}}\). By generalizing the work by Costa–Girão–Natário–Silva (Commun Math Phys 361:289–341, 2018. https://doi.org/10.1007/s00220-018-3122-z) to the case of a charged and massive scalar field, our results reveal a potential failure of the Christodoulou–Chruściel version of the strong cosmic censorship under spherical symmetry.