# On Fokker-Planck Equations with In- and Outflow of Mass

## TL;DR

This paper studies nonlinear Fokker-Planck equations with non-conserved mass, modeling neuron growth, proving exponential decay to equilibrium despite the lack of mass conservation, using entropy methods and numerical validation.

## Contribution

It provides new analytical results on decay to equilibrium for Fokker-Planck equations with mass in- and outflow, overcoming challenges due to non-conservation of mass.

## Key findings

- Proved exponential decay towards equilibrium in various scenarios.
- Developed entropy-based methods applicable without classical Sobolev inequalities.
- Validated results through extensive numerical simulations.

## Abstract

Motivated by modeling transport processes in the growth of neurons, we present results on (nonlinear) Fokker-Planck equations where the total mass is not conserved. This is either due to in- and outflow boundary conditions or to spatially distributed reaction terms. We are able to prove exponential decay towards equilibrium using entropy methods in several situations. As there is no conservation of mass it is difficult to exploit the gradient flow structure of the differential operator which renders the analysis more challenging. In particular, classical logarithmic Sobolev inequalities are not applicable any more. Our analytic results are illustrated by extensive numerical studies.