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

A nonlocal one-phase Stefan problem that develops mushy regions