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Existence of self-similar solutions to a parabolic system modelling chemotaxis

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Abstract

We investigate a semilinear elliptic equation

with a parameter λ > 0 and a constant 0 < ε < 2, and obtain a structure of the pair (λ, υ) of a parameter and a solution which decays at infinity. This equation arises in the study of self-similar solutions for the Keller-Segel system. Our main results are as follows: (i) There exists a λ* > 0 such that if 0 < λ < λ*, (SE) has two distinct solutionsυ λ and guλ satisfyingυ λ < guλ, and that if λ > λ*, (SE) has no solution, (ii) If λ = λ* and 0 < ε < 1, (SE) has the unique solution υ*; (iii) The solutionsυ λ and υλ are connected through υ*.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Hiroshima University, 739-8526, Higashi-Hiroshima, Japan

    Naomi Muramoto

  2. Department of Applied Mathematics, Faculty of Engineering, Kobe University, Rokkodai, 657-8501, Kobe, Japan

    Yūki Naito

  3. Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, 739-8521, Higashi-Hiroshima, Japan

    Kiyoshi Yoshida

Authors
  1. Naomi Muramoto
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  2. Yūki Naito
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  3. Kiyoshi Yoshida
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Muramoto, N., Naito, Y. & Yoshida, K. Existence of self-similar solutions to a parabolic system modelling chemotaxis. Japan J. Indust. Appl. Math. 17, 427–451 (2000). https://doi.org/10.1007/BF03167376

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  • DOI: https://doi.org/10.1007/BF03167376

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