Time regularity and long-time behavior of parabolic $p$-Laplace equations on infinite graphs

TL;DR

This paper analyzes the discrete p-Laplacian on infinite graphs, establishing time regularity, long-term behavior, extinction times, and mass conservation depending on the parameter p and graph properties.

Contribution

It provides new insights into the regularity and long-time dynamics of solutions to nonlinear parabolic equations on infinite graphs, including conditions for finite extinction and mass conservation.

Findings

Solutions exhibit higher order time regularity.

Finite extinction occurs for small p under isoperimetric conditions.

Mass is conserved for large p regardless of graph.

Abstract

We consider the so-called \emph{discrete $p$-Laplacian}, a nonlinear difference operator that acts on functions defined on the nodes of a possibly infinite graph. We study the associated nonlinear Cauchy problem and identify the generator of the associated nonlinear semigroups. We prove higher order time regularity of the solutions. We investigate the long-time behavior of the solutions and discuss in particular finite extinction time and conservation of mass. Namely, on one hand, for small $p$ if an infinite graph satisfies some isoperimetric inequality, then the solution to the parabolic $p$-Laplace equation vanishes in finite time; on the other hand, for large $p,$ these parabolic $p$-Laplace equations always enjoy conservation of mass.