We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and also sharp evaluation for the confinement of mass, i.e., the effective speed of propagation. We provide estimates for some moments of the solution, defined using the distance from a given vertex. Our technique relies on suitable inequalities of Faber-Krahn type, and looks at the local theory of continuous nonlinear partial differential equations. As it is known, however, not all of this approach can have a direct counterpart in graphs. A basic tool here is a result connecting the supremum of the solution at a given positive time with the measure of its level sets at previous times. We also consider the case of slowly decaying initial data, where the total mass is infinite.

中文翻译：

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初始数据衰减的无限图上p -Laplacian的渐近估计

我们考虑了演化的离散p的柯西问题-Laplacian在无限图中，初始数据在无限处衰减。当初始数据具有有限质量时，我们证明了非负解的最优上下边界，并且对质量的限制（即有效传播速度）进行了清晰的评估。我们提供解决方案某些时刻的估算值，这些估算值是使用距给定顶点的距离来定义的。我们的技术依靠适当的Faber-Krahn型不等式，并研究了连续非线性偏微分方程的局部理论。但是，众所周知，并非所有这种方法都可以在图形中具有直接对应的方法。此处的基本工具是将给定正值时解决方案的最高值与先前时间级别设置的度量联系起来的结果。我们还考虑了初始数据缓慢衰减的情况，