Dead-Core Problem for a Class of Infinity Laplacian

Abstract For h>1, we consider the reaction-diffusion equation: align* _ʰu (x) =f (x, u (x), Du (x) ), x, align* where _ ʰ denotes the h -degree infinity Laplacian, f C (R Rⁿ) satisfies 0 f (x, t, p) (x) ^ f (x, t, p), a positive function (x) C (), \, [0, h), \, t>0, and >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-) } -regularity along the free boundary \u>0\. 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 =h, we show that if the viscosity solution vanishes at a point, then the dead-core region must vanish.