We study a nonlocal version of the one-phase Stefan problem which develops
mushy regions, even if they were not present initially, a model which can be of
interest at the mesoscopic scale. The equation involves a convolution with a
compactly supported kernel. The created mushy regions have the size of the
support of this kernel. If the kernel is suitably rescaled, such regions
disappear and the solution converges to the solution of the usual local version
of the one-phase Stefan problem. We prove that the model is well posed, and
give several qualitative properties. In particular, the long-time behavior is
identified by means of a nonlocal mesa solving an obstacle problem
