English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

This paper is concerned with the Cauchy-Dirichlet problem for fast diffusionequations posed in bounded domains, where every energy solution vanishes infinite time and a suitably rescaled solution converges to an asymptoticprofile. Bonforte and Figalli (CPAM, 2021) first proved an exponentialconvergence to nondegenerate positive asymptotic profiles for nonnegativerescaled solutions in a weighted $L^2$ norm for smooth bounded domains bydeveloping a nonlinear entropy method. However, the optimality of the rateremains open to question. In the present paper, their result is fully extendedto possibly sign-changing asymptotic profiles as well as general boundeddomains by improving an energy method along with a quantitative gradientinequality developed by the first author (ARMA, 2023). Moreover, a(quantitative) exponential stability result for least-energy asymptoticprofiles follows as a corollary, and it is further employed to prove theoptimality of the exponential rate.

Optimal rate of convergence to nondegenerate asymptotic profiles for fast diffusion in domains