## Advances in Differential Equations

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

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

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

#### 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

**Permanent link to this document**

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

**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