$W^{2,\infty }$ regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension

Abstract

In this paper we provide and analyze a nonlinear degenerate parabolic equation in one space dimension with the following smoothing property: If the initial data is only uniformly continuous, at positive times, the solution has bounded second derivatives (it belongs to W 2, ∞). We call this surprising phenomenon a W 2, ∞ regularizing effect. So far, such phenomena have only been observed in uniformly parabolic equations.