  We study a $K$-user coded-caching broadcast problem in a joint source-channel
coding framework. The transmitter observes a database of files that are being
generated at a certain rate per channel use, and each user has a cache, which
can store a fixed fraction of the generated symbols. In the delivery phase, the
transmitter broadcasts a message so that the users can decode their desired
files using the received signal and their cache content. The communication
between the transmitter and the receivers happens over a (deterministic)
\textit{time-varying} erasure broadcast channel, and the channel state
information is only available to the users. We characterize the maximum
achievable source rate for the $2$-user and the degraded $K$-user problems. We
provide an upper bound for any caching strategy's achievable source rates.
Finally, we present a linear programming formulation to show that the upper
bound is not a sharp characterization. Closing the gap between the achievable
rate and the optimum rate remains open.
