Calculus of Variations and Geometric Measure Theory

N. Ansini - S. Fagioli

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

created by ansini on 15 Feb 2019
modified on 20 Jan 2021


Published Paper

Inserted: 15 feb 2019
Last Updated: 20 jan 2021

Journal: Comm Math Sci.
Volume: 18
Number: 2
Pages: 459-486
Year: 2020


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 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.

Keywords: Gradient flows, nonlinear diffusion equations, degenerate mobility, minimising movement