Preprint 61/2020

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

Linlin Sun and Jingyong Zhu

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Submission date: 01. Jun. 2020
Pages: 25
MSC-Numbers: 35B33, 58J35
Keywords and phrases: Kazdan-Warner equation, mean field type flow, global existence, global convergence
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Link to arXiv:See the arXiv entry of this preprint.

We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface (Σ,g)

Δgu = 8π(   heu        1   )   ∫-heudμg-− ∫-dμg-   Σ           Σ

where the prescribed function h 0 and maxΣh > 0. We prove the global existence and convergence under additional assumptions such as

Δg lnh(p0) + 8π 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 Σ. 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.

10.06.2020, 02:16