  We prove existence and uniqueness of a reaction-diffusion equation whose
diffusivity is a non-linear functional of the boundary temperature. We do this
by studying systems of one-dimensional reflecting diffusions whose noise is a
function of the reflection local-time of the system, and by characterizing the
large-scale (hydrodynamic) behavior by showing propagation of chaos. In
addition, we analyze the one-particle case by computing the distribution of the
hitting times of its reflection local-time. This work is the noise analog of
work done by Frank Knight (2001).
