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

Semilinear heat equations and parabolic variational inequalities on graphs