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arXiv · 2312.01685

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

Goro Akagi·Yasunori Maekawa
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Abstract

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 $L^2$ 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.

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Goro Akagi, Yasunori Maekawa. 2023-12-04. Optimal rate of convergence to nondegenerate asymptotic profiles for fast diffusion in domains. https://arxiv.org/abs/2312.01685

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