Advances in Differential Equations Volume 1, Number 1, January 1996, pp. 21 – 50 THE FAST DIFFUSION EQUATION WITH LOGARITHMIC NONLINEARITY AND THE EVOLUTION OF CONFORMAL METRICS IN THE PLANE Juan L. Vazquez, Juan R. Esteban Departamento de Matem´ aticas, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain Ana Rodriguez Departamento de Matem´ atica Aplicada, E.T.S. Arquitectura Universidad Polit´ ecnica de Madrid, 28040 Madrid, Spain Abstract. We consider the nonlinear equation ut = Δ 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) ∈ L 1 (R 2 ) there exists a unique maximal solution u ∈ C ∞ (R 2 ×(0,T )) of the Cauchy problem, characterized by the additional property Z R 2 u(x, t) dx = Z R 2 u(x, 0) dx - 4π t, (*) and, accordingly, the existence time is T = R u(x, 0) dx/4π . We then interpret the solutions as the conformal factor of a metric in 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 ≤ . We thus obtain infinitely many nonmaximal solutions of the Cauchy problem having different behaviors (more precisely fluxes) at r =+∞ . One of these options, namely the solution corresponding to formula (*) with last term -8πt, describes the evolution of a complete compact surface under Ricci flow. For data u(x, 0) with infinite integral solutions are unique. Introduction. This paper is devoted to studying the remarkable properties of the nonlinear diffusion equation u t = r · ( ru u )= Δ log u (0.1) posed in two space dimensions. For definiteness we consider the Cauchy problem for equation (0.1) with initial data u(x, 0) = u 0 (x) , x 2 R 2 . (0.2) Let us call this Cauchy problem (CP). We assume throughout that the initial data are nonnegative, locally integrable and radially symmetric, u 0 = u 0 (r) with r = |x| , and look for a positive solution u(x, t) defined in a strip Q T = R 2 ⇥ (0,T ) for some T> 0 which will be also radially symmetric in x . Some times we can take T =+1 , global Received for publication February 1995. AMS Subject Classifications: 35K55, 53C21. 21