A mean field type flow with sign-changing prescribed function on a symmetric Riemann surface. (English) Zbl 1485.58021
Summary: Let \((\Sigma, g)\) be a closed Riemann surface, and \(\text{G} = \{ \sigma_1, \cdots, \sigma_N \}\) be a finite isometric group acting on it. Denote a positive integer \(\ell = \min_{x \in \Sigma} I(x)\), where \(I(x)\) is the number of all distinct points of the set \(\{ \sigma_1(x), \cdots, \sigma_N(x) \} \). In this paper, we consider the following G-invariant mean field type flow
\[
\begin{cases}
\displaystyle\frac{ \partial}{ \partial t} e^u = \Delta_g u + 8 \pi \ell \left( \frac{ f e^u}{ \int_{\Sigma} f e^u d v_g} - \frac{ 1}{ |\Sigma|} \right) \\
u (\cdot, 0) = u_0,
\end{cases}
\] where \(u_0\) belongs to \(C^{2 + \alpha}(\Sigma)\) for some \(\alpha \in(0, 1), f\) is a sign-changing smooth function such that \(\int_{\Sigma} f e^{u_0} d v_g \neq 0\), both \(u_0\) and \(f\) are G-invariant, and \(|\Sigma|\) denotes the area of \((\Sigma, g)\). Such kind of flow was originally proposed by J.-B. Castéras [Pac. J. Math. 276, No. 2, 321–345 (2015; Zbl 1331.53097)]. Through a priori estimates, we prove that the flow \(u(x, t)\) exists for all time \(t \in [0, \infty)\). Moreover, by employing blow-up procedure, we obtain that under certain geometric conditions, \(u(x, t)\) converges to \(u(x)\) in \(H^2(\Sigma)\) as \(t \to \infty \), where \(u(x)\) is a solution of the mean field equation
\[
\displaystyle-\Delta_g u = 8 \pi \ell \left(\frac{ f e^u}{ \int_{\Sigma} f e^u d v_g} - \frac{1}{|\Sigma|}\right).
\] This generalizes recent results of J. Li and C. Zhu [Calc. Var. Partial Differ. Equ. 58, No. 2, Paper No. 60, 18 p. (2019; Zbl 1415.58014)] and L. Sun and J. Zhu [Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 42, 26 p. (2021; Zbl 1458.35437)].
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
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