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

Abstract

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]