{{\lVert{u}_{0}\rVert}_{\ell^{1}(B_{R}(x_{0}))}}
  }{
    {{\lVert{u}_{0}\rVert}_{\ell^{q}(V\setminus B_{R}(x_{0}))}}
  }
  \Big)^{\frac{q}{q-1}}
  \bigg)^{-1}
  \,,$$ for $R\in {{\boldsymbol}{N}}$, $x_{0}\in V$. Clearly for each fixed $x_{0}$ the function ${T}_{{u}_{0}}$ is nondecreasing in $R$ and ${T}_{{u}_{0}}(R,x_{0})\to +\infty$ as $R\to\infty$. Conversely, ${T}_{{u}_{0}}(0,x_{0})$ may be positive. However it can be easily seen that for any given ${\varepsilon}>0$ there exists $x_{0}$ such that ${T}_{{u}_{0}}(0,x_{0})<{\varepsilon}$.

\[t:decay\] Let ${u}_{0}\in\ell^{q}(V)\setminus\ell^{1}(V)$ for some $q>1$. Then for all $t>0$, $x_{0}\in V$ $$\label{eq:decay_n}
    {{\lVert{u}(t)\rVert}_{\ell^{\infty}(V)}}
    \le
    \gamma
    {{\lVert{u}_{0}\rVert}_{\ell^{1}(B_{R}(x_{0}))}}
    {\psi}_{1}^{(-1)}\big(t^{-1} {{\lVert{u}_{0}\rVert}_{\ell^{1}(B_{R}(x_{0}))}}^{-(p-2)}\big)
    \,,$$ provided $R$ is chosen so that $$\label{eq:decay_nn}
    t\le
    {T}_{{u}_{0}}(R,x_{0})
    \,,$$ the optimal choice being of course the minimum $R=R(t)$ such that holds true.

Let us comment briefly on the existing literature on the non-linear $p$-Laplacian in graphs. The papers [@Mugnolo:2013], [@Hua:Mugnolo:2015], deal with the Cauchy problem applying techniques inspired from the theory of semigroups of continuous differential operators. They consider a more general variety of weighted graphs and operators than we do here, dealing e.g., with existence, uniqueness, time regularity, possible extinction in a finite time. However our results do not seem to be easily reached by this approach. We also quote [@Keller:Mugnolo:2016] where a connection between Cheeger constants and the eigenvalues of the $p$-laplacian is drawn in a very flexible setting.\
Boundary problems on finite subgraphs are 