the support of ${u}_{0}$, and $R$ is chosen so that $$\label{eq:bbl_nn}
    R
    \ge
    \varGamma
    t^{\frac{1}{p}}
    {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}^{\frac{p-2}{p}}
    {\psi}_{1}^{(-1)}(t^{-1}{{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}^{-(p-2)})^{\frac{p-2}{p}}
    \ge 8
    \,.$$ In addition, provided $R$ is chosen as in , for ${\varepsilon}=1/2$, and $\alpha\in (0,1)$, $$\label{eq:bbl_p}
    \sum_{x\in V}
    d(x,x_{0})^{\alpha}
    {u}(x,t)
    {d_{{\omega}}}(x)
    \le
    \gamma
    R^{\alpha}
    {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}
    \,,
    \qquad
    t>0
    \,.$$

Next we exploit the estimate – in order to show that up to a change in the constant we can reverse the inequality in , proving at once the optimality of both results.

\[p:bbl2\] Under the assumptions in Theorem \[p:bbl\], let in addition ${\Lambda_p}$ satisfy . Then $$\label{eq:bbl_nnn}
    {{\lVert{u}(t)\rVert}_{\ell^{\infty}(V)}}
    \ge
    \frac{{{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}}{2{\mu_{{\omega}}}(B_{R})}
    \ge
    \gamma_{0}
    {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}
    {\psi}_{1}^{(-1)}
    \big(
    t^{-1}
    {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}^{-(p-2)}
    \big)
    \,,$$ where $R$ is as in , for ${\varepsilon}=1/2$.

Clearly, owing to the comparison principle of Proposition \[p:compare\], results like those in and may be proved even dropping the assumption that ${u}_{0}$ is finitely supported; for the sake of brevity we omit the details.

In order to state our last result we need to introduce the following function, which essentially gives the correct scaling between time and space in the case of slow decay initial data: for ${u}_{0}\in\ell^{q}(V)\setminus\ell^{1}(V)$ for some $q>1$ set $$\label{eq:decay_fn}
  {T}_{{u}_{0}}(R,x_{0})
  =
  \Big[
  \frac{{{\lVert{u}_{0}\rVert}_{\ell^{1}(B_{R}(x_{0}))}}}{{{\lVert{u}_{0}\rVert}_{\ell^{q}(V\setminus B_{R}(x_{0}))}}^{q}}
  \Big]^{\frac{p-2}{q-1}}
  \,
  {\Lambda_p}\bigg(
  \Big(
  \frac{
    