## Differential and Integral Equations

### Harnack estimates for some non-linear parabolic equation

Masashi Mizuno

#### Abstract

We consider the following nonlinear parabolic equation \left\{ \begin{aligned} \partial_tu-\Delta u+\frac{u}{\varepsilon}(|\nabla u|^2-1) & =0, \quad(t,x)\in(0,\infty)\times{\mathbb{R}}^n, \\ u(0,x) & =u_0(x),\quad x\in{\mathbb{R}}^n, \end{aligned} \right. \label{eq:16} \tag{#} which is derived by Goto-K. Ishii-Ogawa [6] to show the convergence of some numerical algorithms for the motion by mean curvature. They assumed that the solution of (#) is sufficiently regular. In this paper, we study the regularity of solutions of (#) from the Harnack estimate. We show the explicit dependence of a constant in the Harnack inequality using the De Giorgi-Nash-Moser method. We employ the Cole-Hopf transform to treat the nonlinear term.

#### Article information

Source
Differential Integral Equations, Volume 21, Number 7-8 (2008), 693-716.

Dates
First available in Project Euclid: 20 December 2012