Differential and Integral Equations

Harnack estimates for some non-linear parabolic equation

Masashi Mizuno

Full-text: Open access


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

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

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B25: Singular perturbations 35B45: A priori estimates 35B65: Smoothness and regularity of solutions


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

Export citation