Axisymmetric solutions and singular parabolic equations in the theory of viscosity solutions

We extend the theory of viscosity solutions for singular parabolic equa­tions including, for example, axisymmetrized level set equation for mean curvature flow equation. We establish a comparison principle for viscosity solutions of singular degenerate parabolic equations including such an equation . We discuss the rela­tion between axisymmetric viscosity solutions of original level set equation for mean curvature flow equation and the viscosity solution of axisymmetrized one

Singular degenerate parabolic equations with applications to the p$B!>(Blaplace diffusion equation

We consider singular degenerate parabolic equations including the p-Laplace diffusion equation. We establish a comparison priciple which is a natural extension of the paper [12] by Ishii and Souganidis. Once we get a comparison priciple we can construct the unique global-in-time solution to the Cauchy problem for the p-Laplace diffution equation. The solution is bounded, uniformly continuous [0,T）× R N if the initial data is bounded, unformaly continuous on R N