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No critical nonlinear diffusion in 1D quasilinear fully parabolic chemotaxis system


Authors: Tomasz Cieślak and Kentarou Fujie
Journal: Proc. Amer. Math. Soc. 146 (2018), 2529-2540
MSC (2010): Primary 35B45, 35K45, 35Q92, 92C17
DOI: https://doi.org/10.1090/proc/13939
Published electronically: January 12, 2018
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Abstract: This paper deals with the fully parabolic 1d chemotaxis system with diffusion $ 1/(1+u)$. We prove that the above mentioned nonlinearity, despite being a natural candidate, is not critical. It means that for such a diffusion any initial condition, independently on the magnitude of mass, generates the global-in-time solution. In view of our theorem one sees that the one-dimensional Keller-Segel system is essentially different from its higher-dimensional versions. In order to prove our theorem we establish a new Lyapunov-like functional associated to the system. The information we gain from our new functional (together with some estimates based on the well-known classical Lyapunov functional) turns out to be rich enough to establish global existence for the initial-boundary value problem.


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Additional Information

Tomasz Cieślak
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Warsaw, 00-656, Poland
Email: cieslak@impan.pl

Kentarou Fujie
Affiliation: Department of Mathematics, Tokyo University of Science, Tokyo, 162-0861, Japan
Email: fujie@rs.tus.ac.jp

DOI: https://doi.org/10.1090/proc/13939
Keywords: Chemotaxis, global existence, Lyapunov functional
Received by editor(s): May 29, 2017
Received by editor(s) in revised form: August 14, 2017, and August 24, 2017
Published electronically: January 12, 2018
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society

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