Remarks on a Nonlinear Parabolic Equation

Abstract

The equation u t = Δ u + μ | ∇ u | u_{t} =\Delta u +\mu |\nabla u | , μ ∈ R \mu \in \mathbb {R} , is studied in R n \mathbb {R}^{n} and in the periodic case. It is shown that the equation is well-posed in L 1 L^{1} and possesses regularizing properties. For nonnegative initial data and μ > 0 \mu >0 the solution decays in L 1 ( R n ) L^{1}(\mathbb {R}^{n}) as t → ∞ t\to \infty . In the periodic case it tends uniformly to a limit. A consistent difference scheme is presented and proved to be stable and convergent.