The generalized porous medium equation on graphs: existence and uniqueness of solutions with $\ell^1$ data

TL;DR

This paper investigates the generalized porous medium equation on infinite graphs, establishing existence and uniqueness of solutions for various data types under specific conditions, thereby extending the mathematical understanding of such equations on graph structures.

Contribution

It provides new existence and uniqueness results for solutions of the generalized porous medium equation on infinite graphs, including cases with changing sign data under additional assumptions.

Findings

Existence and uniqueness for nonnegative or nonpositive data.

Existence and uniqueness for sign-changing data under extra conditions.

Results applicable to any infinite graph structure.

Abstract

We study solutions of the generalized porous medium equation on infinite graphs. For nonnegative or nonpositive integrable data, we prove the existence and uniqueness of mild solutions on any graph. For changing sign integrable data, we show existence and uniqueness under extra assumptions such as local finiteness or a uniform lower bound on the node measure.