Doubly Nonlinear Diffusion Equations on Metric Graphs

TL;DR

This paper investigates existence and uniqueness of solutions for a broad class of doubly nonlinear diffusion equations on metric graphs, modeling complex tubular networks with variable diffusion properties.

Contribution

It extends the analysis to include diffusion changing across edges and non-homogeneous boundary conditions, covering cases like the Porous Medium and p-Laplacian equations.

Findings

Proved existence and uniqueness of solutions for the general class of equations.

Analyzed the impact of varying diffusion properties across network edges.

Included non-homogeneous Neumann-Kirchhoff conditions at vertices.

Abstract

In this paper we study existence and uniqueness of solutions for a very general class of doubly nonlinear diffusion equations on metric graphs, which provide the appropriate mathematical framework to describe complex tubular networks in which axial diffusion is the main focus. Some important particular cases covered in our study are the Porous Medium Equation and the evolution equation for the $p$-Laplacian, but we also consider the case in that diffusion changes from one edge to another, which takes into account the influence of the properties of the tubules forming the network on axial diffusion. Furthermore, the problem is studied under non-homogeneous Neumann-Kirchhoff conditions on the vertices of the graph.