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MiS Preprint
61/2020
Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function
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(\dfrac{he^{u}}{\int_{\Sigma}he^{u}\textnormal{d}\mu_{g}}-\dfrac{1}{\int_{\Sigma}\textnormal{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.
