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2008 Harnack estimates for some non-linear parabolic equation
Masashi Mizuno
Differential Integral Equations 21(7-8): 693-716 (2008).

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.

Citation

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Masashi Mizuno. "Harnack estimates for some non-linear parabolic equation." Differential Integral Equations 21 (7-8) 693 - 716, 2008.

Information

Published: 2008
First available in Project Euclid: 20 December 2012

zbMATH: 1224.35217
MathSciNet: MR2479688

Subjects:
Primary: 35K55
Secondary: 35B25, 35B45, 35B65

Rights: Copyright © 2008 Khayyam Publishing, Inc.

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Vol.21 • No. 7-8 • 2008
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