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Stability and Instability of Group Invariant Asymptotic Profiles for Fast Diffusion Equations

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Book cover Geometric Properties for Parabolic and Elliptic PDE's

Part of the book series: Springer INdAM Series ((SINDAMS,volume 2))

Abstract

This paper is concerned with group invariant solutions for fast diffusion equations in symmetric domains. First, it is proved that the group invariance of weak solutions is inherited from initial data. After briefly reviewing previous results on asymptotic profiles of vanishing solutions and their stability, the notions of stability and instability of group invariant profiles are introduced under a similarly invariant class of perturbations, and moreover, some stability criteria are exhibited and applied to symmetric domain (e.g., annulus) cases.

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Acknowledgements

This work is partially supported by KAKENHI #22740093 and Hyogo Science and Technology Association.

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Authors and Affiliations

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

    Goro Akagi

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  1. Goro Akagi
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Correspondence to Goro Akagi .

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Editors and Affiliations

  1. Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, Firenze, 50124, Italy

    Rolando Magnanini

  2. Graduate School of Information Sciences, Division of Mathematics, Tohoku University, Sendai, 980-8579, Japan

    Shigeru Sakaguchi

  3. , Dipartimento di Matematica ed Applicazio, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, Napoli, 80126, Italy

    Angelo Alvino

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Akagi, G. (2013). Stability and Instability of Group Invariant Asymptotic Profiles for Fast Diffusion Equations. In: Magnanini, R., Sakaguchi, S., Alvino, A. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2841-8_1

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