| Abstract: |
| We discuss how concavity properties are preserved under the porous medium flows on a Riemannian manifold, clarifying the effects of diffusion nonlinearity and curvature.
In particular, curvature yields an obstruction: if the sectional curvature is negative at a given point, no concavity property is preserved. However, if the curvature is non-zero, concavity properties can only be preserved if they are stronger than those associated with the diffusion nonlinearity. |
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