Existence of the heat flow with sign-changing prescribed function

Abstract

Let (Σ,g) be a closed Riemann surface, u0=u(⋅,0)∈C2+β(Σ) with β∈(0,1) and h be a sign-changing smooth function satisfying ∫Σheu0dvg≠0. In this paper, we consider the following mean field type flow∂eu∂t=−Δgu+α(u−u¯)+8π(heu∫Σheudvg−1), where u¯=1|Σ|∫Σudvg, 0≤α<λ1(Σ), and λ1(Σ) is the first eigenvalue of the Laplace-Beltrami operator Δg. We prove that there exists a unique solution u to the above heat flow. Moreover, if the corresponding elliptic equation has no solution, then we obtain the related functional has a low bound. We use the blow-up analysis to derive our results and we also need to overcome the difficulty arising from the term α(u−u¯).