## Advances in Differential Equations

- Adv. Differential Equations
- Volume 1, Number 1 (1996), 21-50.

### The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane

Juan R. Esteban, Ana Rodríguez, and Juan L. Vázquez

#### Abstract

We consider the nonlinear equation $$ u_t = \Delta \log u $$ posed in two space
dimensions. For the Cauchy problem with radially symmetric data, we investigate the
existence of solutions, both global and local in time, as well as the question of
uniqueness/multiplicity. The most striking result is as follows: for every radial
$u(x,0)\in L^1(\mathbf{R}^2)$ there exists a unique *maximal* solution $u\in
C^\infty(\mathbf{R}^2\times (0,T))$ of the Cauchy problem, characterized by the additional
property \begin{equation} \int_{\mathbf{R}^2} u(x,t)\,dx= \int_{\mathbf{R}^2} u(x,0)\,dx
-4\pi\,t\,, \tag{*} \end{equation} and, accordingly, the existence time is $T=\int
u(x,0)\,dx/4\pi\,$. We then interpret the solutions as the conformal factor of a metric in
$\mathbf{R}^2$ evolving by Ricci flow; formula (*) is a version of Gauss-Bonnet's Theorem.
The solution here described is not unique if one weakens the equality (*) into an
inequality $\le\,$. We thus obtain infinitely many nonmaximal solutions of the Cauchy
problem having different behaviors (more precisely *fluxes*) at $r=+\infty\,$. One of
these options, namely the solution corresponding to formula (*) with last term $-8\pi t$,
describes the evolution of a complete compact surface under Ricci flow. For data $u(x,0)$
with infinite integral solutions are unique.

#### Article information

**Source**

Adv. Differential Equations, Volume 1, Number 1 (1996), 21-50.

**Dates**

First available in Project Euclid: 25 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366896313

**Mathematical Reviews number (MathSciNet)**

MR1357953

**Zentralblatt MATH identifier**

0844.35056

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 53A30: Conformal differential geometry 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

#### Citation

Vázquez, Juan L.; Esteban, Juan R.; Rodríguez, Ana. The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane. Adv. Differential Equations 1 (1996), no. 1, 21--50. https://projecteuclid.org/euclid.ade/1366896313