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

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.