Nonlinear evolution equation associated with hypergraph Laplacian

Let V $$ be a finite set, E ⊂ 2 E\subset {2}^V set of hyperedges, and w : → ( 0 , ∞ ) w:E\to \left(0,\infty \right) an edge weight. On the (wighted) hypergraph G = G=\left(V,E,w\right) we can define multivalued nonlinear operator L p {L}_{G,p} ∈ [ 1 p\in \left[1,\infty as subdifferential convex function on ℝ {\mathbb{R}}^V which is called “hypergraph -Laplacian.” In this article, first introduce inequality for resembles Poincaré–Wirtinger in PDEs. Next, consider ordinary differential equation governed by referred “heat” graph used to study geometric structure recent researches. With aid type inequality, discuss existence large time behavior solutions ODE procedures similar those standard heat PDEs with zero Neumann boundary condition.