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

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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.

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Authors and Affiliations

  1. Department of Basic and Applied Sciences for Engineering, Sapienza University of Rome, Rome, Italy

    Daniele Andreucci

  2. South Mathematical Institute of VSC RAS, Vladikavkaz, Russian Federation

    Anatoli F. Tedeev

Authors
  1. Daniele Andreucci
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  2. Anatoli F. Tedeev
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Correspondence to Daniele Andreucci.

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The first author is member of the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM).

The second author was supported by Sapienza Grant C26V17KBT3.

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Andreucci, D., Tedeev, A.F. Asymptotic Estimates for the p-Laplacian on Infinite Graphs with Decaying Initial Data. Potential Anal 53, 677–699 (2020). https://doi.org/10.1007/s11118-019-09784-w

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  • DOI: https://doi.org/10.1007/s11118-019-09784-w

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