Remarks on a Nonlinear Parabolic Equation

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