  A mathematical model is presented for the Joule heating that occurs in a
ceramic powder compact during the process of flash sintering. The ceramic is
assumed to have an electrical conductivity that increases with temperature, and
this leads to the possibility of runaway heating that could facilitate and
explain the rapid sintering seen in experiments. We consider reduced models
that are sufficiently simple to enable concrete conclusions to be drawn about
the mathematical nature of their solutions. In particular we discuss how
different local and non-local reaction terms, which arise from specified
experimental conditions of fixed voltage and current, lead to thermal runaway
or to stable conditions. We identify incipient thermal runaway as a necessary
condition for the flash event, and hence identify the conditions under which
this is likely to occur.
