each $x\in V$.

We refer the reader to [@Hua:Mugnolo:2015] for existence and uniqueness of solutions. To make this paper more self-contained however we sketch in Section \[s:prelim\] a proof of existence in Proposition \[p:ex\] (in $\ell^{q}$, $q>1$, see Theorem \[t:l1\] for $q=1$), and of uniqueness via comparison in Proposition \[p:compare\].

Our first two results are typical of the local approach we pursue. All solutions we consider below are nonnegative.

\[p:linf\_meas\] Let ${u}:V\to {{\boldsymbol}{R}}$ be a solution to , with ${u}\in L^{\infty}(0,T;\ell^{r}(V))$ for some $r\ge 1$. Then for all $x\in V$, $0<t<T$ $$\label{eq:linf_meas_n}
    {u}(x,t)
    \le
    k
    \,,$$ provided $k>0$ satisfies for a suitable $\gamma_{0}(p,{N})$ $$\label{eq:linf_meas_nn}
    k^{-1}
    t^{-\frac{1}{p-2}}
    {\Lambda_p}\Big(\sup_{\frac{t}{4}<\tau<t}{\mu_{{\omega}}}(\{x\in V\mid {u}(x,\tau)> {k}/{2}\})\Big)^{-\frac{1}{p-2}}
    \le
    \gamma_{0}
    \,.$$

\[co:linf\_int\] Under the assumptions in Proposition \[p:linf\_meas\], we have $$\label{eq:linf_int_m}
    {u}(x,t)
    \le
    \gamma
    \sup_{0<\tau<t}{{\lVert{u}(\tau)\rVert}_{\ell^{r}(V)}}
    \big[{\psi}_{r}^{(-1)}
    \big(
    t^{-1}
    \sup_{0<\tau<t}{{\lVert{u}(\tau)\rVert}_{\ell^{r}(V)}}^{-(p-2)}
    \big)
    \big]^{\frac{1}{r}}
    \,,$$ for all $x\in V$, $0<t<T$. Here ${\psi}_{r}^{(-1)}$ is the inverse function of ${\psi}_{r}$ as defined in .

\[r:dcf\] One can check easily using the fact that ${\Lambda_p}$ is nonincreasing that $$a\mapsto a{\psi}_{r}^{(-1)}(sa^{-(p-2)})^{\frac{1}{r}}$$ is nondecreasing in $a>0$ for each fixed $s>0$.

Next Theorem follows directly from the estimates we stated above. Note that conservation of mass in was proved also in [@Hua:Mugnolo:2015], while the other estimates are 