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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 22 April 2013

Mathematical Reviews number (MathSciNet)
MR1606339

Zentralblatt MATH identifier
1023.35515

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

#### Citation

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. https://projecteuclid.org/euclid.ade/1366638676