The fractional unstable obstacle problem

We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation (−Δ)su=χ{u>c}for 0c}. 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≤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.