Concentration comparison for nonlinear diffusion on model manifolds and P\'olya-Szeg\H{o} inequality

We investigate the validity of the mass concentration comparison for a class of nonlinear diffusion equations posed on Riemannian manifolds $\mathbb{M}^n$ that are spherically symmetric, that is, model manifolds. The concentration comparison states that the solution of a certain diffusion equation that takes the radially decreasing (Schwarz) rearrangement $u_0^\star$ as its initial datum is more concentrated than the original solution starting from $u_0$. This is known to hold in $\mathbb{R}^n$ as a consequence of the celebrated P\'olya-Szeg\H{o} inequality, which asserts that the $L^2$ norm of the gradient of a function $f$ (belonging to an appropriate Sobolev space) is always larger than the $L^2$ norm of the gradient of its radially decreasing rearrangement $f^\star$. However, if $\mathbb{M}^n$ is a general model manifold, it is not for granted that the P\'olya-Szeg\H{o} inequality holds; in fact, we will provide a simple condition involving the scalar curvature of $\mathbb{M}^n$ under which such an inequality actually fails. The main result we prove states that, given any continuous, nondecreasing, and nontrivial function $\phi: [0,+\infty) \to [0,+\infty)$, the filtration equation $\partial_t u = \Delta \phi(u)$ satisfies the concentration comparison in $\mathbb{M}^n \times (0,+\infty)$ if and only if $\mathbb{M}^n$ supports the P\'olya-Szeg\H{o} inequality. In particular, 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 $\mathbb{M}^n$ supports a centered isoperimetric inequality then the P\'olya-Szeg\H{o} inequality, and thus the concentration comparison, holds. This allows us to include important examples such as the hyperbolic space and the sphere.