A mean field type flow with sign-changing prescribed function on a symmetric Riemann surface.

Let (Σ , g) be a closed Riemann surface, and G = { σ 1 , ⋯ , σ N } be a finite isometric group acting on it. Denote a positive integer ℓ = min x ∈ Σ ⁡ I (x) , where I (x) is the number of all distinct points of the set { σ 1 (x) , ⋯ , σ N (x) }. In this paper, we consider the following G -invariant mean field type flow { ∂ ∂ t e u = Δ g u + 8 π ℓ (f e u ∫ Σ f e u d v g − 1 | Σ |) u (⋅ , 0) = u 0 , where u 0 belongs to C 2 + α (Σ) for some α ∈ (0 , 1) , f is a sign-changing smooth function such that ∫ Σ f e u 0 d v g ≠ 0 , both u 0 and f are G -invariant, and | Σ | denotes the area of (Σ , g). Such kind of flow was originally proposed by Castéras [6]. Through a priori estimates, we prove that the flow u (x , t) exists for all time t ∈ [ 0 , ∞). Moreover, by employing blow-up procedure, we obtain that under certain geometric conditions, u (x , t) converges to u (x) in H 2 (Σ) as t → ∞ , where u (x) is a solution of the mean field equation − Δ g u = 8 π ℓ (f e u ∫ Σ f e u d v g − 1 | Σ |). This generalizes recent results of Li-Zhu [27] and Sun-Zhu [37]. [ABSTRACT FROM AUTHOR]