| Abstract: |
| We investigate the validity of the mass concentration comparison for a class of nonlinear diffusion equations, commonly known as filtration equations, posed on Riemannian manifolds that are spherically symmetric, that is, model manifolds. Our main result states that, given any continuous bijection $\varphi : [0,+\infty) \rightarrow [0,+\infty)$, the filtration equation $\partial_{t}u=\Delta\varphi(u)$ satisfies the concentration comparison if and only if the underlying model manifold supports the P\`olya-Szeg\H{o} inequality. As a simple corollary, the validity of such a comparison for the heat equation is sufficient to guarantee that the same holds for all filtration equations. Moreover, we prove that if the manifold supports a centered isoperimetric inequality then the P\`olya-Szeg\H{o} inequality holds, allowing us to include important examples such as the hyperbolic space and the sphere. This is a joint work with Matteo Muratori. |
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