  We study a free boundary problem which arises as the continuum version of a
stochastic particles system in the context of Fourier law. Local existence and
uniqueness of the classical solution are well known in the literature of free
boundary problems. We introduce the notion of generalized solutions (which
extends that of classical solutions when the latter exist) and prove global
existence and uniqueness of generalized solutions for a large class of initial
data. The proof is obtained by characterizing a generalized solution as the
unique element which separates suitably defined lower and upper barriers in the
sense of mass transport inequalities.
