A heat flow with sign-changing prescribed function on finite graphs

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In this paper, the authors consider a heat flow with sign-changing prescribed function on finite graphs, which inspired by the works of Castéras (2015) [4], Lin and Yang (2021) [18], Wang and Yang (2022) [29]. To be exact, let G=(V,E) be a finite connected graph, ρ be a positive real number, Δ be the usual graph Laplacian operator with respect to the measure μ, and f:V→R be a sign-changing prescribed function satisfying maxx∈V⁡f(x)>0. Using elliptic method, the authors first show that for any initial value u0(x), the mean field type heat flow{∂∂teu=Δu+ρ(feu∫Vfeudμ−1|V|),(x,t)∈V×(0,+∞),u(x,0)=u0(x),x∈V, has a unique solution u:V×[0,+∞)→R. Moreover, the authors prove that there exists some function u∞:V→R, such that the solution of the above flow converges to u∞ uniformly on V as t→+∞, and u∞ is a solution of the following mean field equation−Δu=ρ(feu∫Vfeudμ−1|V|). Therefore, the authors apply the new method of heat flow to find a solution of the mean field equation on graphs.