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1997 The maximal solution of the logarithmic fast diffusion equation in two space dimensions
Juan R. Esteban, Ana Rodriguez, Juan L. Vazquez
Adv. Differential Equations 2(6): 867-894 (1997).

Abstract

We consider the problem $$ \begin{alignat}{2} &u_t = \Delta\log u\quad && x\in\mathbb{R}^2, 0<t<T,\\ &u(x,0)=u_0(x)\quad && x\in\mathbb{R}^2, \end{alignat} $$ with nonnegative and integrable data $u_0(x)$, which appears in Riemannian geometry. We construct a class of maximal solutions of this problem and prove that they generate a semigroup of contractions in $L^1(\mathbb R^2)$, characterized by the property of area loss $$ \dfrac{d}{dt}\int u(x,t)\,dx= -4\pi, $$ which holds until $u(x,t)$ vanishes identically at the time $T=\int u_0(x)\, dx/(4\pi)$. Several constructions of the maximal solutions are proposed and alternative characterizations proved. Sharp uniqueness criteria are obtained.

Citation

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Juan R. Esteban. Ana Rodriguez. Juan L. Vazquez. "The maximal solution of the logarithmic fast diffusion equation in two space dimensions." Adv. Differential Equations 2 (6) 867 - 894, 1997.

Information

Published: 1997
First available in Project Euclid: 22 April 2013

zbMATH: 1023.35515
MathSciNet: MR1606339

Subjects:
Primary: 35K55
Secondary: 35B05

Rights: Copyright © 1997 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.2 • No. 6 • 1997
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