Viscosity solutions to the inhomogeneous reaction-diffusion equation involving the infinity Laplacian

In this paper, we study the inhomogeneous reaction–diffusion equation involving the infinity Laplacian: Formula: see text where the continuous function Formula: see text satisfies Formula: see text a positive function Formula: see text Formula: see text Formula: see text and Formula: see text. Such a model permits existence of solutions with dead core zones, i.e. a priori unknown regions where non-negative solutions vanish identically. For Formula: see text and the non-positive inhomogeneous term Formula: see text we establish the existence, uniqueness and stability of the viscosity solution of the corresponding continuous Dirichlet problem. Under additional structure conditions on Formula: see text and Formula: see text we obtain the optimal Formula: see text regularity across the free boundary Formula: see text Moreover, we establish the porosity of the free boundary and Liouville type theorem for entire solutions. Finally, we prove that the dead core vanishes in the limit case Formula: see text