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Stability of Non-Isolated Asymptotic Profiles for Fast Diffusion

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Abstract

The stability of asymptotic profiles of solutions to the Cauchy–Dirichlet problem for fast diffusion equation (FDE, for short) is discussed. The main result of the present paper is the stability of any asymptotic profiles of least energy. It is noteworthy that this result can cover non-isolated profiles, e.g., those for thin annular domain cases. The method of proof is based on the Łojasiewicz–Simon inequality, which is usually used to prove the convergence of solutions to prescribed limits, as well as a uniform extinction estimate for solutions to FDE. Besides, local minimizers of an energy functional associated with this issue are characterized. Furthermore, the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the Łojasiewicz–Simon inequality in a different way.

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Author notes

  1. Goro Akagi

    Present address: Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

Authors and Affiliations

  1. Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe, 657-8501, Japan

    Goro Akagi

  2. Institut für Computational Biology, Helmholtz Zentrum München, Ingolstädter Landstraße 1, 85764, Neuherberg, Germany

    Goro Akagi

  3. Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, D-85748, Garching bei München, Germany

    Goro Akagi

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  1. Goro Akagi
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Correspondence to Goro Akagi.

Additional information

Communicated by C. Mouhot

G. Akagi is supported by JSPS KAKENHI Grant Number 16H03946 and by the JSPS-CNR bilateral joint research project: Innovative Variational Methods for Evolution Equations and by the Alexander von Humboldt Foundation and by the Carl Friedrich von Siemens Foundation.

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Akagi, G. Stability of Non-Isolated Asymptotic Profiles for Fast Diffusion. Commun. Math. Phys. 345, 77–100 (2016). https://doi.org/10.1007/s00220-016-2649-0

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  • DOI: https://doi.org/10.1007/s00220-016-2649-0

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