Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian

The existence, uniqueness and regularity of viscosity solutions to the Cauchy-Dirichlet problem are proved for a degenerate nonlinear parabolic equation of the form ut =Δ∞u, where Δ∞ denotes the so-called infinity-Laplacian given by Δ∞u = 〈D²uDu, Du〉. To do so, a coercive regularization of the equation is introduced and barrier function arguments are also employed to verify the equi-continuity of approximate solutions. Furthermore, the Cauchy problem is also studied by using the preceding results on the Cauchy-Dirichlet problem.