Singular Perturbation Problem in Boundary/Fractional Combustion

Motivated by a nonlocal free boundary problem, we study uniform properties of solutions to a singular perturbation problem for a boundary-reaction-diffusion equation, where the reaction term is of combustion type. This boundary problem is related to the fractional Laplacian. After an optimal uniform Hölder regularity is shown, we pass to the limit to study the free boundary problem it leads to.