Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production

Mengyao Ding; Wei Wang

doi:10.3934/dcdsb.2018328

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September 2019, 24(9): 4665-4684. doi: 10.3934/dcdsb.2018328

Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production

Mengyao Ding ^1, and Wei Wang ^2,,

1.	School of Mathematical Sciences, Peking University, Beijing, 100871, China
2.	School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

^* Corresponding author: Wei Wang

Received May 2018 Published September 2019 Early access January 2019

Fund Project: This work is supported by the National Natural Science Foundation of China (11671066, 11571020, 11671021).

In this paper we develop a new and convenient technique, with fractional Gagliardo-Nirenberg type inequalities inter alia involved, to treat the quasilinear fully parabolic chemotaxis system with indirect signal production: $ u_t = \nabla\cdot(D(u)\nabla u-S(u)\nabla v) $, $ \tau_1v_t = \Delta v-a_1v+b_1w $, $ \tau_2w_t = \Delta w-a_2w+b_2u $, under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset\Bbb{R}^{n} $ ($ n\geq 1 $), where $ \tau_i,a_i,b_i>0 $ ($ i = 1,2 $) are constants, and the diffusivity $ D $ and the density-dependent sensitivity $ S $ satisfy $ D(s)\geq a_0(s+1)^{-\alpha} $ and $ 0\leq S(s)\leq b_0(s+1)^{\beta} $ for all $ s\geq 0 $ with $ a_0,b_0>0 $ and $ \alpha,\beta\in\Bbb R $. It is proved that if $ \alpha+\beta<3 $ and $ n = 1 $, or $ \alpha+\beta<4/n $ with $ n\geq 2 $, for any properly regular initial data, this problem has a globally bounded and classical solution. Furthermore, consider the quasilinear attraction-repulsion chemotaxis model: $ u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla z)+\xi\nabla\cdot(u\nabla w) $, $ z_t = \Delta z-\rho z+\mu u $, $ w_t = \Delta w-\delta w+\gamma u $, where $ \chi,\mu,\xi,\gamma,\rho,\delta>0 $, and the diffusivity $ D $ fulfills $ D(s)\geq c_0(s+1)^{M-1} $ for any $ s\geq 0 $ with $ c_0>0 $ and $ M\in\Bbb R $. As a corollary of the aforementioned assertion, it is shown that when the repulsion cancels the attraction (i.e. $ \chi\mu = \xi\gamma $), the solution is globally bounded if $ M>-1 $ and $ n = 1 $, or $ M>2-4/n $ with $ n\geq 2 $. This seems to be the first result for this quasilinear fully parabolic problem that genuinely concerns the contribution of repulsion.

Keywords: Chemotaxis, global boundedness, quasilinear, fully parabolic, indirect signal production.

Mathematics Subject Classification: Primary: 35B35, 35B40, 35K55; Secondary: 92C17.

Citation: Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328