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

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

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.

Export citation