  We consider an evolution problem associated to the Kazdan-Warner equation on
a closed Riemann surface $(\Sigma,g)$
  \begin{align*}
  -\Delta_{g}u=8\pi\left(\frac{he^{u}}{\int_{\Sigma}he^{u}{\rm
d}\mu_{g}}-\frac{1}{\int_{\Sigma}{\rm d}\mu_{g}}\right)
  \end{align*} where the prescribed function $h\geq0$ and $\max_{\Sigma}h>0$.
We prove the global existence and convergence under additional assumptions such
as
  \begin{align*}
  \Delta_{g}\ln h(p_0)+8\pi-2K(p_0)>0 \end{align*} 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 $\Sigma$. 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.
