  We study regularity properties of solutions to reaction-diffusion equations
ruled by the infinity laplacian operator. We focus our analysis in models
presenting plateaus, i.e. regions where a non-negative solution vanishes
identically. We obtain sharp geometric regularity estimates for solutions along
the boundary of plateaus sets. In particular we show that the
$(n-\epsilon)$-Hausdorff measure of the plateaus boundary is finite, for a
universal number $\epsilon>0$.
