Rate of convergence to asymptotic profiles for fast diffusion on domains via energy methodsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

This talk is concerned with the Cauchy-Dirichlet problem for fast diffusion equations on bounded domains. It is well known that every weak solution vanishes in finite time at the unique power rate, and therefore, asymptotic profiles for such vanishing solutions are defined as a limit of rescaled solutions, which solve the Cauchy-Dirichlet problem for a fast diffusion equation with a blow-up reaction. Asymptotic profiles are characterized as nontrivial equilibria of the rescaled problem (see pioneer works of Berryman and Holland in 1980s and subsequent results for qualitative results). Recently, Bonforte and Figalli (CPAM, 2021) established a quantitative result on the convergence of rescaled solutions to nondegenerate positive asymptotic profiles. More precisely, they proved an exponential convergence of nonnegative rescaled solutions to nondegenerate positive asymptotic profiles in a weighted L2 space with a sharp rate (in view of some linearized analysis) by developing a nonlinear entropy method. In this talk, we present a different approach to prove exponential convergence with rates for nondegenerate asymptotic profiles. In particular, we can directly verify an H1 0 convergence with the sharp rate. Our method of proof is based on an energy method rather than entropic one, and a key ingredient is a quantitative gradient inequality established based on an eigenvalue problem with weights