# Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation

## TL;DR

This paper establishes existence and uniqueness of solutions for nonlinear diffusion equations with degenerate, fast-decay mobility on bounded intervals, using a coordinate transformation to simplify the problem.

## Contribution

It introduces a coordinate transformation that converts nonlinear diffusion equations with degenerate fast-decay mobility into equations with linear mobility, facilitating analysis.

## Key findings

- Rescaled density $
ho$ is the unique weak solution to a linear mobility diffusion equation.
- The coordinate transformation preserves mass and incorporates the original nonlinearity.
- Results apply to original density $u$ without boundary conditions.

## Abstract

We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density $u$. In case of \emph{fast-decay} mobilities, namely mobilities functions under a Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained. We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive process. We show that the rescaled density $\rho$ is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density $\rho$ allow us to motivate the aforementioned change of variable and to state the results in terms of the original density $u$ without prescribing any boundary conditions.