Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source

Xie Li; Zhaoyin Xiang

doi:10.3934/dcds.2015.35.3503

Article Contents

2015, Volume 35, Issue 8: 3503-3531. Doi: 10.3934/dcds.2015.35.3503

This issue Previous Article On the partitions with Sturmian-like refinements Next Article Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation

Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source

Xie Li^1, and
Zhaoyin Xiang^1,

1.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731

Received: November 2014

Revised: December 2014

Published: May 2017

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Abstract

In this paper, we investigate the quasilinear Keller-Segel equations (q-K-S): \[ \left\{ \begin{split} &n_t=\nabla\cdot\big(D(n)\nabla n\big)-\nabla\cdot\big(\chi(n)\nabla c\big)+\mathcal{R}(n), \qquad x\in\Omega,\,t>0,\\ &\varrho c_t=\Delta c-c+n, \qquad x\in\Omega,\,t>0, \end{split} \right. \] under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$. For both $\varrho=0$ (parabolic-elliptic case) and $\varrho>0$ (parabolic-parabolic case), we will show the global-in-time existence and uniform-in-time boundedness of solutions to equations (q-K-S) with both non-degenerate and degenerate diffusions on the non-convex domain $\Omega$, which provide a supplement to the dichotomy boundedness vs. blow-up in parabolic-elliptic/parabolic-parabolic chemotaxis equations with degenerate diffusion, nonlinear sensitivity and logistic source. In particular, we improve the recent results obtained by Wang-Li-Mu (2014, Disc. Cont. Dyn. Syst.) and Wang-Mu-Zheng (2014, J. Differential Equations).

Keywords:

Global existence,
boundedness,
parabolic-elliptic Keller-Segel system,
parabolic-parabolic Keller-Segel systems.

Mathematics Subject Classification: Primary: 35K35, 92C17; Secondary: 35K59.

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