is a finite system of ordinary differential equations, uniquely solvable in the class $C^{1}(0,T)$ at least as long as the solution stays bounded over $(0,T)$.

In this connection, we rewrite , as $${u}_{n}(x,t)^{q-1}
    {\frac{\partial {u}_{n}}{\partial t}}(x,t)
    =
    {u}_{n}(x,t)^{q-1}
    \operatorname{\Delta}_{p}{u}_{n}(x,t)
    \,,
    \qquad
    x\in V
    \,,
    t>0
    \,,$$ where we stress that the equality holds for all $x\in V$. In this Subsection we denote $s^{q-1}={\lverts\rvert}^{q-1}\textup{sign}(s)$ for all $s\in {{\boldsymbol}{R}}$. Thus, summing over $x\in V$ and integrating by parts both in $t$ and in $x$ (in the suitable sense) we see that the elliptic part of the equation yields a nonnegative contribution, so that $$\label{eq:exist_energy_n}
    \sum_{x\in V}
    {\lvert{u}_{n}(x,t)\rvert}^{q}
    {d_{{\omega}}}(x)
    \le
    \sum_{x\in B_{n}}
    {\lvert{u}_{0}(x)\rvert}^{q}
    {d_{{\omega}}}(x)
    \le
    {{\lVert{u}_{0}\rVert}_{\ell^{q}(V)}}^{q}
    \,.$$ In turn, as explained in Remark \[r:lp\_scale\], this implies stable sup bounds for ${u}_{n}$ which, together with the discrete character of the $p$-laplacian and with the equation , also give stable sup bounds for the time derivative $\partial {u}_{n}/\partial t$, for each fixed $x$. However $V$ is countable, so that this is enough to enable us to extract a subsequence, still denoted by ${u}_{n}$ such that $$\label{eq:exist_unif_conv}
    {u}_{n}(x,t)
    \to
    {u}(x,t)
    \,,
    \quad
    {\frac{\partial {u}_{n}}{\partial t}}(x,t)
    \to
    {\frac{\partial {u}}{\partial t}}(x,t)$$ for each $x\in V$, uniformly for $t\in 