MiS Preprint Repository

We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(\mathrm{\Sigma },g)$$$\begin{array}{r}-{\mathrm{\Delta }}_{g}u=8\pi ({\displaystyle \frac{h{e}^u}{{\int }_{\mathrm{\Sigma }}h{e}^u\text{d}{\mu }_g}}-{\displaystyle \frac{1}{{\int }_{\mathrm{\Sigma }}\text{d}{\mu }_g}})\end{array}$$ where the prescribed function $h\ge 0$ and $\underset{\mathrm{\Sigma }}{max}h>0$. We prove the global existence and convergence under additional assumptions such as $$\begin{array}{r}{\mathrm{\Delta }}_g\ln h(p_0)+8\pi -2K(p_0)>0\end{array}$$for any maximum point $p_0$ of the sum of $2\ln h$ and the regular part of the Green function, where $K$ is the Gaussian curvature of $\mathrm{\Sigma }$. In particular, this gives a new proof of the existence result by Yang and Zhu [Proc. Amer. Math. Soc. 145(2017), no. 9, 3953-3959] which generalizes existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), no. 2, 230-248] to the non-negative prescribed function case.