  Self-interacting dark matter (SIDM) naturally gives rise to a cored
isothermal density profile, which is favored in observations of many dwarf
galaxies. The dark matter distribution in the presence of a central black hole
in an isothermal halo develops a density spike with a power law of $r^{-3/2}$,
which is shallower than $r^{-7/3}$ as expected for collisionless dark matter
(CDM). Thus, indirect detection constraints on dark matter annihilations from
the density spike could be significantly relaxed in SIDM. Taking the most dense
satellite galaxy of the Milky Way Draco as an example, we derive upper limits
on the annihilation cross section and the black hole mass for both SIDM and CDM
halos. For the former case, Draco could host an intermediate mass black hole
even if dark matter is composed of thermal relics. Even for the M87 galaxy,
which hosts a supermassive black hole with a mass of $6.5\times10^9~{\rm
M_\odot}$, we estimate the constraints on the annihilation cross section are a
factor of $10^5$ weaker in SIDM than in CDM. Our results further indicate that
the Event Horizon Telescope may provide a unique probe of the SIDM spike near
the supermassive black hole of M87 for light dark matter.
