2023/12/04 by Goro Akagi, Akagi, Goro, Yasunori Maekawa +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2312.01685
openalex publication_date 2023/12/04 · openalex created_date 2023/12/06 · openalex updated_date 2026/07/28
This paper is concerned with the Cauchy-Dirichlet problem for fast diffusion equations posed in bounded domains, where every energy solution vanishes in finite time and a suitably rescaled solution converges to an asymptotic profile. Bonforte and Figalli (CPAM, 2021) first proved an exponential convergence to nondegenerate positive asymptotic profiles for nonnegative rescaled solutions in a weighted L2 norm for smooth bounded domains by developing a nonlinear entropy method. However, the optimality of the rate remains open to question. In the present paper, their result is fully extended to possibly sign-changing asymptotic profiles as well as general bounded domains by improving an energy method along with a quantitative gradient inequality developed by the first author (ARMA, 2023). Moreover, a (quantitative) exponential stability result for least-energy asymptotic profiles follows as a corollary, and it is further employed to prove the optimality of the exponential rate.