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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
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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  • [1] N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015), no. 9, 1663-1763. MR 3351175,
  • [2] Piotr Biler, Waldemar Hebisch, and Tadeusz Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Anal. 23 (1994), no. 9, 1189-1209. MR 1305769,
  • [3] Adrien Blanchet, José A. Carrillo, and Philippe Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations 35 (2009), no. 2, 133-168. MR 2481820,
  • [4] Jan Burczak, Tomasz Cieślak, and Cristian Morales-Rodrigo, Global existence vs. blowup in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal. 75 (2012), no. 13, 5215-5228. MR 2927584,
  • [5] Jan Burczak and Rafael Granero-Belinchón, Critical Keller-Segel meets Burgers on $ \mathbb{S}^1$: large-time smooth solutions, Nonlinearity 29 (2016), no. 12, 3810-3836. MR 3580331,
  • [6] T. Cieślak and K. Fujie, Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. arXiV:1705.08541
  • [7] Tomasz Cieślak and Philippe Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 1, 437-446. MR 2580517,
  • [8] Tomasz Cieślak and Philippe Laurençot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system, C. R. Math. Acad. Sci. Paris 347 (2009), no. 5-6, 237-242 (English, with English and French summaries). MR 2537529,
  • [9] Tomasz Cieślak and Philippe Laurençot, Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system, Discrete Contin. Dyn. Syst. 26 (2010), no. 2, 417-430. MR 2556492,
  • [10] Tomasz Cieślak and Christian Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations 252 (2012), no. 10, 5832-5851. MR 2902137,
  • [11] Tomasz Cieślak and Michael Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity 21 (2008), no. 5, 1057-1076. MR 2412327,
  • [12] Philippe Laurençot and Noriko Mizoguchi, Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 1, 197-220. MR 3592684,
  • [13] N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system. preprint
  • [14] Toshitaka Nagai, Takasi Senba, and Kiyoshi Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. 40 (1997), no. 3, 411-433. MR 1610709
  • [15] Elissar Nasreddine, Global existence of solutions to a parabolic-elliptic chemotaxis system with critical degenerate diffusion, J. Math. Anal. Appl. 417 (2014), no. 1, 144-163. MR 3191418,
  • [16] Takasi Senba and Takasi Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal. , posted on (2006), Art. ID 23061, 21. MR 2211660,
  • [17] Yoshie Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations 19 (2006), no. 8, 841-876. MR 2263432
  • [18] Yoshie Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations 12 (2007), no. 2, 121-144. MR 2294500

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Additional Information

Tomasz Cieślak
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Warsaw, 00-656, Poland

Kentarou Fujie
Affiliation: Department of Mathematics, Tokyo University of Science, Tokyo, 162-0861, Japan

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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