On an obstacle problem arising in large exponent asymptotics for one dimensional fully nonlinear diffusions of power type

Abstract

In this note, we discuss the limit behavior for a fully nonlinear diffusion equation of power type in one space dimension. It turns out that, when the initial value is Lipschitz and convex, the solution converges locally uniformly to a unique limit function that is independent of the time variable as the exponent tends to infinity. We characterize the limit as the minimal solution of an obstacle problem. Such asymptotic behavior is closely related to applications in mathematical models of image denoising and collapsing sandpiles.