NONLINEAR DIFFUSION EQUATIONS WITH DEGENERATE FAST-DECAY MOBILITY BY COORDINATE TRANSFORMATION

摘要

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 resealed 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.