in , which then yields $$\label{eq:rem_fk}
    {\Lambda_p}({d_{{\omega}}}(x))
    {d_{{\omega}}}(x)
    \le
    2{d_{{\omega}}}(x)
    \,.$$ Since ${\Lambda_p}$ is decreasing by assumption we infer $$\label{eq:rem_fk_bound}
    {d_{{\omega}}}(x)\ge {\Lambda_p}^{(-1)}(2)
    \,.$$ A remark in this connection is perhaps in order: clearly according to its definition the Faber-Krahn function ${\Lambda_p}(v)$ is defined for uniformly positive $v$ according to , so that , should be assumed for such $v$. Aiming at a technically streamlined framework, we extend ${\Lambda_p}$ to all $v>0$, while easily preserving the latter assumptions. However one can check that for large times, ${\Lambda_p}$ is evaluated at large arguments in our results, which are thus independent of this extension.

\[r:lp\_scale\] A consequence of is that any bound in $\ell^{q}(V)$ yields immediately a uniform pointwise bound: if ${v}\in\ell^{q}(V)$, $$\label{eq:lp_linf}
    {\lvert{v}(z)\rvert}^{q}
    \le
    {\lvert{v}(z)\rvert}^{q}\frac{{d_{{\omega}}}(z)}{{\Lambda_p}^{(-1)}(2)}
    \le
    \frac{1}{{\Lambda_p}^{(-1)}(2)}
    {{\lVert{v}\rVert}_{\ell^{q}(V)}}^{q}
    \,,
    \quad
    z\in V
    \,.$$ In turn this implies that $\ell^{p}(V)\subset\ell^{q}(V)$ if $p<q$, since $$\sum_{x\in V}
    {\lvertf(x)\rvert}^{q}
    {d_{{\omega}}}(x)
    \le
    M^{q-p}
    \sum_{x\in V}
    {\lvertf(x)\rvert}^{p}
    {d_{{\omega}}}(x)
    \,,$$ for a suitable $M$ as in .

\[d:sol\] We say that ${u}\in L^{\infty}(0,T; \ell^{r}(V))$ is a solution to if ${u}(x)\in C^{1}([0,T])$ for every $x\in V$ and ${u}$ satisfies in the classical pointwise sense.\
A solution to – also is required to take the initial data prescribed by , for 