Advances in Differential Equations

The maximal solution of the logarithmic fast diffusion equation in two space dimensions

Juan R. Esteban, Ana Rodriguez, and Juan L. Vazquez

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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.

Article information

Adv. Differential Equations, Volume 2, Number 6 (1997), 867-894.

First available in Project Euclid: 22 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc.


Rodriguez, Ana; Vazquez, Juan L.; Esteban, Juan R. The maximal solution of the logarithmic fast diffusion equation in two space dimensions. Adv. Differential Equations 2 (1997), no. 6, 867--894.

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