Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function. (English) Zbl 1458.35437
Summary: We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface \(\Sigma,g)\)
\[
-\Delta_gu=8\pi \left(\dfrac{he^u}{\int_\Sigma he^u\text{d}\mu_g} - \dfrac{1}{\int_\Sigma \text{d}\mu_g}\right)
\]
where the prescribed function \(h\geq 0\) and \(\max_\Sigma h>0\). We prove the global existence and convergence under additional assumptions such as
\[
\Delta_g\ln h(p_0)+8\pi -2K(p_0)>0
\]
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 Y. Yang and X. Zhu [Proc. Am. Math. Soc. 145, No. 9, 3953–3959 (2017; Zbl 1369.58010)] which generalizes existence result of W. Ding et al. [Asian J. Math. 1, No. 2, 230–248 (1997; Zbl 0955.58010)] to the non-negative prescribed function case.
MSC:
| 35R01 | PDEs on manifolds |
| 35K59 | Quasilinear parabolic equations |
| 35B33 | Critical exponents in context of PDEs |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
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