Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function

摘要

We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface (Sigma ,g)-Delta gu=8 pi heu integral Sigma heud mu g-1 integral Sigma d mu g where the prescribed function h >= 0 and max Sigma h>0. We prove the global existence and convergence under additional assumptions such as Delta glnh(p0)+8 pi -2K(p0)>0 for any maximum point p0 of the sum of 2lnh and the regular part of the Green function, where K is the Gaussian curvature of Sigma. In particular, this gives a new proof of the existence result by Yang and Zhu (Pro Am Math Soc 145:3953-3959, 2017) which generalizes existence result of Ding et al. (Asian J Math 1:230-248, 1997) to the non-negative prescribed function case.