We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation [ (-Delta)^s u = chi_{{u>c}} ] for $0<s<1$ and an arbitrary constant $c$. Our main object of study is the free boundary $partial{u>c}$. We study the behavior of the free boundary and prove an upper bound for the Hausdorff dimension of the singular set. We also show that when $sleq 1/2$ certain symmetric solutions are stable; however, when $s>1/2$ these solutions are not stable and therefore not minimizers of the corresponding functional.
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