Large exponent asymptotics for one dimensional fully nonlinear diffusions of power type (Analysis on Shapes of Solutions to Partial Differential Equations)

In this note, we discuss limit behavior for fully nonlinear diffusion of power type in one space dimension. It turns out that, as the exponent tends to infinity, the solution converges locally uniformly to a unique limit function that is independent of the time variable. We rescale the time variable to characterize the limit as a unique viscosity solution of a fully nonlinear singular parabolic equation with jump discontinuity. Such asymptotic behavior is closely related to applications in math models of image denoising and collapsing sandpiles