Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system

Yilong  Wang; Zhaoyin Xiang

doi:10.3934/dcdsb.2016031

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2016, Volume 21, Issue 6: 1953-1973. Doi: 10.3934/dcdsb.2016031

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Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system

Yilong Wang^1, and
Zhaoyin Xiang^1,

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

Received: December 2014

Revised: March 2016

Published: August 2016

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Abstract

In this paper, we consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-parabolic type \begin{equation*} \left\{ \begin{split} &u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\qquad & x\in\Omega,\,\, t>0,\\ &v_t=\Delta v+\alpha u-\beta v,\qquad &x\in\Omega, \,\,t>0,\\ &w_t=\Delta w+\gamma u-\delta w,\qquad &x\in\Omega,\,\, t>0 \end{split} \right. \end{equation*} under homogeneous Neumann boundary conditions, where $D(u)\geq c_D (u+\varepsilon)^{m-1}$ and $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary. It is shown that whenever $m>1$, for any sufficiently smooth nonnegative initial data, the system admits a global bounded classical solution for the case of non-degenerate diffusion (i.e., $\varepsilon>0$), while the system possesses a global bounded weak solution for the case of degenerate diffusion (i.e., $\varepsilon=0$).

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Mathematics Subject Classification: Primary: 35K57, 35Q92; Secondary: 35A01, 92C17.

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