Leibenson's equation on graphs

Abstract

In this paper we study on infinite graphs the Leibenson equation $${\partial }_{t}u={\Delta }_p{u}^q,$$ where $p>1$, $q>0$ and ${\Delta }_p$ denotes the discrete $p$-Laplacian. We prove, for any integrable initial data $u_0$, the existence of a global solution, which is unique for a certain range of $p$ and $q$. Assuming a Faber--Krahn inequality, we obtain sharp ${\ell }^1$-${\ell }^{\infty }$ smoothing estimates and quantitative bounds on the propagation of solutions with initially finite support. Under certain assumptions on $p$ and $q$, we also prove finite-time extinction results for solutions when the graph satisfies an \textit{isoperimetric inequality}. In particular, on Cayley graphs with polynomial volume growth, we establish the optimal large-time decay rate of the ${\ell }^{\infty }$-norm for nonnegative finite-mass solutions when $q(p-1)>1$, and demonstrate a sharp dichotomy regarding the finite-time extinction of exhaustion solutions.

32 pages, no figures