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

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 an 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

nonlinear diffusion equations, degenerate mobility, gradient flows, minimising movement

2010 Mathematics Subject Classification

35A01, 35D30, 35Q84, 35Q92

Full Text (PDF format)

Received 14 February 2019

Accepted 19 October 2019

Published 20 June 2022