on infinite graphs\
    with decaying initial data
---

\[1994/06/01\]

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Introduction {#s:intro}
============

We consider nonnegative solutions to the Cauchy problem for discrete degenerate parabolic equations $$\begin{aligned}
{2}
  \label{eq:pde}
  {\frac{\partial {u}}{\partial t}}(x,t)
  -
  \operatorname{\Delta}_{p}
  {u}(x,t)
  &=
  0
  \,,
  &\qquad&
  x\in V\,, t>0
  \,,
  \\
  \label{eq:init}
  {u}(x,0)
  &=
  {u}_{0}(x)
  \ge 0
  \,,
  &\qquad&
  x\in V
  \,.\end{aligned}$$ Here $V$ is the set of vertices of the graph $G(V,E)$ with edge set $E\subset V\times V$ and weight ${\omega}$, and $$\operatorname{\Delta}_{p} u(x,t)
  =
  \frac{1}{{d_{{\omega}}}(x)}
  \sum_{y\in V}
  {\lvert{u}(y)-{u}(x)\rvert}^{p-2}
  ({u}(y)-{u}(x))
  {\omega}(x,y)
  \,.$$ We assume that the graph $G$ is simple, undirected, infinite, connected with locally finite degree $${d_{{\omega}}}(x)
  =
  \sum_{y\sim x}
  {\omega}(x,y)
  \,,$$ where we write $y\sim x$ if and only if $\{x,y\}\in E$. Here the weight ${\omega}:V\times V\to [0,+\infty)$ is symmetric, i.e., ${\omega}(x,y)={\omega}(y,x)$, and is strictly positive if and only if $y\sim x$; then ${\omega}(x,x)=0$ for $x\in V$.

We assume also that $p>2$ and that ${u}_{0}$ is nonnegative; further assumptions on ${u}_{0}$ will be stated below.

We prove sharp sup bounds for large times of solutions corresponding to finite mass initial data; in order to prove the bound from below we find an optimal estimate for the effective speed of propagation of mass. We 