## Differential and Integral Equations

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

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

#### Abstract

We consider the following nonlinear parabolic equation \begin{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{#} \end{equation} 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

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356038619

**Mathematical Reviews number (MathSciNet)**

MR2479688

**Zentralblatt MATH identifier**

1224.35217

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35B25: Singular perturbations 35B45: A priori estimates 35B65: Smoothness and regularity of solutions

#### Citation

Mizuno, Masashi. Harnack estimates for some non-linear parabolic equation. Differential Integral Equations 21 (2008), no. 7-8, 693--716. https://projecteuclid.org/euclid.die/1356038619