Long Time Asymptotics for Fast Diffusion via Dynamical Systems Methods

For the fast diffusion equation in the mass preserving parameter range, we obtain sharp asymptotic convergence rates to the Barenblatt solution with respect to the relative L∞ norm from spectral gaps by establishing a nonlinear differentiable semiflow in Hölder spaces on a Riemannian manifold called cigar manifold. On this manifold, the equation becomes uniformly parabolic. It is possible to obtain faster rates than O(1/τ) when the reference Barenblatt solution is appropriately scaled. To this end, the interplay between weights in the function space, the spectrum of the linearized operator and growth of its (formal) eigenfunctions needs to be investigated carefully, leading to estimates in appropriately weighted relative L∞ norms.