Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity

Sachiko Ishida; Tomomi Yokota

doi:10.3934/dcdss.2020012

Article Contents

2020, Volume 13, Issue 2: 211-232. Doi: 10.3934/dcdss.2020012

This issue Previous Article Global asymptotic stability in a chemotaxis-growth model for tumor invasion Next Article Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system

Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity

Sachiko Ishida^1, and
Tomomi Yokota^2, ,

1.
Department of Mathematics and Informatics, Graduate School of Science, Chiba University, 1-33, Yayoi-cho, Inage, Chiba 263-8522, Japan
2.
Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

* Corresponding author: Tomomi Yokota
* Corresponding author: Tomomi Yokota

Received: April 30, 2017

Revised: September 30, 2017

Published: February 2020

The first and second authors are supported by Grant-in-Aid for Young Scientists Research (B) (No. 15K17578) and Scientific Research (C) (No. 16K05182), JSPS, respectively.

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Abstract

This paper deals with the quasilinear Keller-Segel system

$ \begin{align*} \begin{cases} u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &x \in \Omega, \ t>0, \\ \ v_t = \Delta v - v +u, &x \in \Omega, \ t>0 \end{cases} \end{align*} $

in $ \Omega = \mathbb{R}^N $ or in a smoothly bounded domain $ \Omega\subset \mathbb{R}^N $, with nonnegative initial data $ u_0\in L^1(\Omega) \cap L^\infty(\Omega) $, and $ v_0\in L^1(\Omega) \cap W^{1, \infty}(\Omega) $; in the case that $ \Omega $ is bounded, it is supplemented with homogeneous Neumann boundary condition. The diffusivity $ D(u) $ and the sensitivity $ S(u) $ are assumed to fulfill $ D(u)\ge u^{m-1}\ (m\geq1) $ and $ S(u)\leq u^{q-1}\ (q\geq 2) $, respectively. This paper derives uniform-in-time boundedness of nonnegative solutions to the system when $ q<m+\frac{2}{N} $. In the case $ \Omega = \mathbb{R}^N $ the result says boundedness which was not attained in a previous paper (J. Differential Equations 2012; 252:1421-1440). The proof is based on the maximal Sobolev regularity for the second equation. This also simplifies a previous proof given by Tao-Winkler (J. Differential Equations 2012; 252:692-715) in the case of bounded domains.

Keywords:

Quasilinear degenerate Keller-Segel systems,
boundedness.

Mathematics Subject Classification: Primary: 35K51; Secondary: 35B35.

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References

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