  We study the problem of a four-dimensional brane lying in the
five-dimensional degenerate Schwarzschild-de Sitter (Nariai) black hole, in
five-dimensional $F(R)$-gravity. We show that there cannot exist the brane in
the Nariai bulk space except the case that the brane tension vanishes. We
demonstrate that the five-dimensional Nariai bulk is unstable in a large region
of the parameter space. In particular, the Nariai bulk can have classical
(anti-)evaporation instabilities. The bulk instability back-reacts on the
four-dimensional brane, in case that the brane tension vanishes, and the
unstable modes propagate in their world-volume.
