  In this paper, we provide a mathematical framework in studying the wave
propagation with the annihilation phenomenon in excitable media. We deal with
the existence and uniqueness of solutions to a one-dimensional free boundary
problem (called a reaction--interface system) arising from the singular limit
of a FitzHugh--Nagumo type reaction--diffusion system. Because of the presence
of the annihilation, interfaces may intersect each other. We introduce the
notion of weak solutions to study the continuation of solutions beyond the
annihilation time. Under suitable conditions, we show that the free boundary
problem is well-posed.
