## 1 Introduction
Even more remarkably, strong experimental evidence for [PERSON] radiation in BEC analogue systems was recently presented [28; 29; 30; 31; 32], largely confirming the theoretical expectations (at least in some suitable regime of the analogue black hole dynamics) and quite convincingly settling the issue of the robustness against possible high-energy modifications of the effective spacetime. In particular, in BECs this can be deduced from the realization that the analogue Hawking spectrum retains its standard form, as long as the typical wavelengths of the Hawking quanta are sufficiently larger than the healing length, which plays the role of the quantum gravity scale in these effective spacetimes. These results have of course stimulated new studies, also regarding the interplay of the microscopical (the analogue quantum gravitational) dynamics and the emergent phenomenology of spacetime and quantum fields. In particular, it was argued by the present authors that the other big issue related to [PERSON] radiation, the information loss problem--the apparent loss of unitary evolution implied by this phenomenon--can be understood using again the analogue gravity insight [33]. Indeed, in an isolated BEC system unitarity must be preserved and it is so once the evaporation process is considered at the microscopical level, tracking down the generation of correlations between the atoms underlying the emergent spacetime and the [PERSON] quanta. Although in analogue gravity the back-reaction of the quantum fields on the acoustic geometry would not follow the [PERSON] equations, the observation of evaporation in analogue black holes could provide information over the interplay of quantum fields and classical geometry within a semiclassical scheme in a broad class of quantum gravity scenarios. If one sees this analogue gravity system as a toy model for emergent gravity, it could then give a physical intuition of how [PERSON] radiation can be pictured as a feature emerging from an underlying full quantum theory. Therefore, seeking this kind of theoretical insights, as well as the possibility to test this understanding of the back-reaction in experimental realizations, attracted an increasing interest on the nature of the back-reaction in analogue [PERSON] radiation [34; 35; 36; 33]. For example, in recent experiments with surface waves on a draining vortex water flow [36], it has been shown that the back-reaction is indeed observable, and it is possible to measure the exchange of energy and angular momentum between the background flow and the waves incident on the horizon. In Bose-Einstein condensates, a next step in this direction should be achievable as it should be possible to observe the back-reaction of the Hawking Bogoliubov quasi-particles on the acoustic geometry--i.e., on the substratum of condensate atoms--and interpret it through the lens of the underlying atomic theory. With this aim in mind in this paper we focus our attention on the back-reaction of the [PERSON] radiation on a canonical analogue black hole within a BEC analogue system characterized by a \(\lambda\phi^{4}\) interaction [37; 38; 39]. The canonical analogue black hole is for our study a remarkable geometry because it is the spherically symmetric stationary solution of the Gross-Pitaevskii equation with homogeneous atom number density. The spherical symmetry allows reduction of the problem to a time-radius problem that presents a horizon at the radius where the velocity of the stationary ingoing superfluid equals the speed of sound. In what follows we first describe the acoustic metric in this system and how the solutions of the Klein-Gordon equation behave at the horizon, and we calculate the [PERSON] radiation. These are the basis for showing how the back-reaction exerted by the [PERSON] radiation can be studied and how it affects the geometry of the black hole. We then propose to study a regime in which the [PERSON] radiation would lead to a sizeable evaporation of the analogue black hole.