Remarks on a Nonlinear Parabolic Equation

Ben-Artzi, Matania; Goodman, Jonathan; Levy, Arnon

doi:10.1090/S0002-9947-99-02336-3

Remarks on a Nonlinear Parabolic Equation
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by Matania Ben-Artzi, Jonathan Goodman and Arnon Levy PDF

Trans. Amer. Math. Soc. 352 (2000), 731-751 Request permission

Abstract:

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.

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