Dead-Core Problem for a Class of Infinity Laplacian

For $h>1,$ we consider the reaction-diffusion equation: align* Δ_∞ʰu(x)=f(x,u(x),Du(x)), x∈Ω, align* where Δ _∞ ʰ denotes the h -degree infinity Laplacian, f∈ C(Ω × R× Rⁿ) satisfies 0≤ f(x,δ t,p)≤ Λ (x)δ γf(x,t,p), a positive function Λ (x)∈ C(Ω ̄), \, γ ∈ [0,h), \,t>0 , and δ>0 is small enough. Such an equation may cause a dead-core region, that is, an unknown region where the nonnegative solution vanishes completely. We establish a flattening estimate for the viscosity solution and obtain sharp C^(h+1)/(h-γ ) -regularity along the free boundary ∂ >0\∩ Ω . Using the sharp regularity, we prove Liouville-type theorems for the global solution and give the porosity of the free boundary. In the end, for the limit case γ =h, we show that if the viscosity solution vanishes at a point, then the dead-core region must vanish.