assumed to be natural numbers. Let $R_{2}\ge R_{1}+1$, $R_{1}$, $R_{2}>0$; we define the cutoff function $\zeta$ in $B_{R_{2}}(x_{0})$ by means of $$\begin{aligned}
{2}
  \zeta(x)
  &=
  1
  \,,
  &\qquad&
  x\in B_{R_{1}}(x_{0})
  \,,
  \\
  \zeta(x)
  &=
  \frac{
    R_{2}
    -
    d(x,x_{0})
  }{
    R_{2}
    -
    R_{1}
  }
  \,,
  &\qquad&
  x\in B_{R_{2}}\setminus B_{R_{1}}(x_{0})
  \,,
  \\
  \zeta(x)
  &=
  0
  \,,
  &\qquad&
  x\not \in B_{R_{2}}(x_{0})
  \,.\end{aligned}$$ The function $\zeta$ is chosen so that $${\lvert{D}_{y}\zeta(x)\rvert}
  =
  {\lvert
    \zeta(y)
    -
    \zeta(x)
  \rvert}
  \le
  \frac{1}{R_{2}-R_{1}}
  \,,
  \qquad
  x\sim y
  \,.$$ For $\tau_{1}>\tau_{2}>0$ we also define the standard nonnegative cutoff function $\eta\in C^{1}({{\boldsymbol}{R}})$ such that $$\eta(t)
  =0
  \,,
  \,\,\,
  t\ge \tau_{1}
  \,;
  \quad
  \eta(t)
  =
  0
  \,,
  \,\,\,
  t\le \tau_{2}
  \,;
  \quad
  0\le\eta'(t)\le \frac{2}{\tau_{1}-\tau_{2}}
  \,,
  \,\,\,
  t\in{{\boldsymbol}{R}}\,.$$

Our next Lemma is not used in the sequel; we present it here to substantiate our claim made in the Introduction that suitable local Caccioppoli type inequalities are available in the nonlinear setting, and also for its possible independent interest. The proof is somehow more complex than in the continuous case.

\[l:cacc\] Let ${u}$ be a solution of in $V\times (0,T)$, $x_{0}\in V$. Then for $T>\tau_{1}>\tau_{2}>0$, $R_{2}>R_{1}+1$, $R_{1}>0$, $h>k>0$, $1>\theta>0$ we have $$\label{eq:cacc_n}
    \begin{split}
      &\sup_{\tau_{1}<\tau<t}
      \sum_{x\in B_{R_{1}}(x_{0})}
      {{\ifthenelse{\equal{*}{*}}{({u}(x,\tau)-h)_{+}}{\left({u}(x,\tau)-h\right)_{+}}}}^{\theta+1}
      \zeta(x)^{p}
      {d_{{\omega}}}(x)
      \\
      &\quad+
      \int_{\tau_{1}}^{t}
      \sum_{x\in 