Semilinear heat equations and parabolic variational inequalities on graphs

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 $\mathcal{H}=\{u\in W^{1,2}(V):u=0\mbox{ on }V\backslash\Omega^\circ\}$.