Semilinear heat equations and parabolic variational inequalities on graphs

TL;DR

This paper investigates the existence of solutions for semilinear heat equations and parabolic variational inequalities on graphs using Rothe's method, extending PDE analysis to discrete graph structures.

Contribution

It introduces a novel approach to analyze parabolic equations and inequalities on graphs, applying Rothe's method to establish solution existence in this discrete setting.

Findings

Existence results for semilinear heat equations on graphs.

Existence results for parabolic variational inequalities on graphs.

Extension of PDE techniques to graph-based models.

Abstract

Let $G=(V,E)$ be a locally finite connected weighted graph, and $\Omega$ be an unbounded subset of $V$. Using Rothe's method, we study the existence of solutions for the semilinear heat equation $\partial_tu+|u|^{p-1}\cdot u=\Delta u~(p\ge1)$ and the parabolic variational inequality \begin{eqnarray*} \int_{\Omega^\circ} \partial_tu\cdot(v-u)\,d\mu\ge \int_{\Omega^\circ}(\Delta u+f)\cdot(v-u)\,d\mu \qquad\mbox{for any }v\in \mathcal{H}, \end{eqnarray*} where $\mathcal{H}=\{u\in W^{1,2}(V):u=0\mbox{ on }V\backslash\Omega^\circ\}$.