2019/02/07 by Nadia Ansini, Ansini, N., Simone Fagioli +1
Mathematics · Medicine · #Mathematical Biology Tumor Growth #Nonlinear Partial Differential Equations #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.1902.02764
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 ρ is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density ρ 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.