# Asymptotic behavior of the solution of the space dependent variable order fractional diffusion equation: ultra-slow anomalous aggregation

## TL;DR

This paper derives the long-time asymptotic behavior of solutions to space-dependent variable order fractional diffusion equations, revealing a new advection mechanism causing ultra-slow particle aggregation, supported by experiments and simulations.

## Contribution

It introduces the first asymptotic representation for solutions to space-dependent variable order fractional diffusion equations, uncovering a novel advection term responsible for ultra-slow aggregation.

## Key findings

- Identification of a new advection term causing ultra-slow aggregation
- Excellent agreement between asymptotic solutions, simulations, and experiments
- Demonstration of anomalous distribution mechanisms in subdiffusive systems

## Abstract

We find for the first time the asymptotic representation of the solution to the space dependent variable order fractional diffusion and Fokker-Planck equations. We identify a new advection term that causes ultra-slow spatial aggregation of subdiffusive particles due to dominance over the standard advection and diffusion terms, in the long-time limit. This uncovers the anomalous mechanism by which non-uniform distributions can occur. We perform experiments on intracellular lysosomal distributions and Monte Carlo simulations and find excellent agreement between the asymptotic solution, particle histograms and experiments.