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In a recent letter, we presented numerical relativity simulations, solvingthe full Einstein--Maxwell--Klein-Gordon equations, of superradiantly unstableReissner-Nordstr\"om black holes (BHs), enclosed in a cavity. Low frequency,spherical perturbations of a charged scalar field, trigger this instability.The system's evolution was followed into the non-linear regime, until itrelaxed into an equilibrium configuration, found to be a $\textit{hairy}$ BH: acharged horizon in equilibrium with a scalar field condensate, whose phase isoscillating at the (final) critical frequency. Here, we investigate the impactof adding self-interactions to the scalar field. In particular, we findsufficiently large self-interactions suppress the exponential growth phase,known from linear theory, and promote a non-monotonic behaviour of the scalarfield energy. Furthermore, we discuss in detail the influence of the variousparameters in this model: the initial BH charge, the initial scalarperturbation, the scalar field charge, mass, and the position of the cavity'sboundary (mirror). We also investigate the "explosive" non-linear regimepreviously reported to be akin to a bosenova. A mode analysis shows that the"explosions" can be interpreted as the decay into the BH of modes that exit thesuperradiant regime.

Dynamical formation of a Reissner-Nordstr\"om black hole with scalar hair in a cavity