分野別セミナー

概要 This talk is concerned with the Cauchy-Dirichlet problem for fast diffusion equations posed in bounded Lipschitz domains. It is well known that every energy solution vanishes in finite time and a suitably rescaled solution converges to an asymptotic profile, which is a nontrivial solution for a semilinear elliptic equation. Bonforte and Figalli (CPAM, 2021) first proved an exponential convergence to nondegenerate positive asymptotic profiles for nonnegative rescaled solutions in a weighted L^2 norm, which is weaker than the L^2 norm, for smooth (at least C^2) bounded domains by developing the so-called nonlinear entropy method. On the other hand, the speaker (ARMA, 2023) developed an energy method along with a quantitative gradient estimate and also proved the same exponential convergence in the Sobolev norm for bounded C^{1,1} domains. The optimality of the exponential rate was conjectured in view of some formal linearized analysis; however, it has never been proved so far due to some difficulty arising from nontrivial stability nature of asymptotic profiles in the fast diffusion setting. In this talk, these results are extended to possibly sign-changing asymptotic profiles as well as bounded Lipschitz domains by improving the energy method as well as quantitative gradient inequality. Moreover, a (quantitative) exponential stability result for least-energy asymptotic profiles follows as a corollary. Finally, the optimality of the exponential rate will also be proved. This talk is based on a joint work with Yasunori Maekawa (Kyoto University).

※　最新の講演題目・要旨等の情報につきましてはホームページ
http://www2.math.kyushu-u.ac.jp/FE-Seminar/
をご覧下さい．

Zoomの使い方等について：
・ご参加して頂く際には，Zoomのアプリのインストールが必要です．
下記のwebサイトから，事前に導入をお願い致します．
https://zoom.us ・改めてご連絡する招待リンクをクリックすることで参加できます．
・セキュリティ面を考慮し，本名(フルネーム)でご参加頂けると幸いです．