Asymptotic Estimates for the p-Laplacian on Infinite Graphs with Decaying Initial Data

Abstract

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.

Qualitative Properties of Solutions of Degenerate Parabolic Equations via Energy Approaches

“…In particular, we don't use here the cylindrical parabolic embedding, which may be restrictive, for example in a Riemannian setting or in some other problems. In addition, this approach allows us to prove the optimal finite speed of propagation for degenerate parabolic equations even for initial data measures (see e.g., the two-sided estimate ð3:21Þ), and it works also in graphs [15]. See also [49] for an approach to the Cauchy problem for doubly nonlinear equations, and [19] for the property of finite speed of propagation.…”

“…References for this section: [7], [8], [9], [10], [11], [13], [15], [16], [17], [19], [28], [33], [34], [35], [41], [43], [44], [45], [49], [51], [52], [53], [54], [58], [61], [66], [68], [70].…”