  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 $s\leq 1/2$ certain
symmetric solutions are stable; however, when $s>1/2$ these solutions are not
stable and therefore not minimizers of the corresponding functional.
