On an exponentially decaying diffusive chemotaxis system with indirect signals

Pan Zheng; Jie Xing

doi:10.3934/cpaa.2022044

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May 2022, 21(5): 1735-1753. doi: 10.3934/cpaa.2022044

On an exponentially decaying diffusive chemotaxis system with indirect signals

Pan Zheng ^a,b,, and Jie Xing ^a,

a.	College of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China
b.	School of Mathematics and Statistics, Yunnan University, Kunming 650091, China

^* Corresponding author

Received August 2021 Revised January 2022 Published May 2022 Early access February 2022

Fund Project: The work is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042), Natural Science Foundation of Chongqing (Grant No. cstc2019jcyj-msxmX0082), China-South Africa Young Scientist Exchange Project in 2020, The Hong Kong Scholars Program (Grant Nos: XJ2021042, 2021-005) and Young Hundred Talents Program of CQUPT in 2022-2024

This paper deals with an exponentially decaying diffusive chemotaxis system with indirect signal production or consumption

$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &(x,t)\in \Omega\times (0,\infty), \\ &v_t = \Delta v+h(v,w), &(x,t)\in \Omega\times (0,\infty), \\ &w_t = \Delta w- w+u, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} $

under homogeneous Neumann boundary conditions in a smoothly bounded domain

$ \Omega\subset \mathbb{R}^{n} $

$ n\geq2 $

, where the nonlinear diffusivity

$ D $

and chemosensitivity

$ S $

are supposed to satisfy

$ K_{1}e^{-\beta^{-}s}\leq D(s) \leq K_{2}e^{-\beta^{+}s} \;\;\;{\rm{and}}\;\;\;\frac{D(s)}{S(s)}\geq K_{3}s^{-\alpha}+\gamma, $

with the constants

$ \beta^{-}\geq \beta^{+}>0 $

$ K_{1},K_{2},K_{3}>0 $

and

$ \alpha,\gamma\geq0 $

. When

$ h(v,w) = -v+w $

, we study the global existence and boundedness of solutions for the above system provided that

$ \alpha\in[0,\frac{2}{n}) $

$ \beta^{-}\geq \beta^{+}>\frac{n}{2} $

$ \gamma>1 $

and the initial mass of

$ u_{0} $

is small enough. Moreover, it is proved that the global bounded solution

$ (u,v,w) $

converges to

$ (\overline{u_{0}},\overline{u_{0}},\overline{u_{0}}) $

in the

$ L^{\infty} $

-norm as

$ t\rightarrow \infty $

, where

$ \overline{u_{0}} = \frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)dx $

. When

$ h(v,w) = -vw $

, it is shown that this system possesses a unique uniformly bounded classical solution if

$ \alpha\geq0 $

$ \gamma>0 $

and

$ \beta^{-}\geq \beta^{+}>\frac{n}{2} $

. Furthermore, if

$ n = 2 $

$ \alpha\geq0 $

$ \gamma\geq0 $

, and

$ \beta^{-}\geq \beta^{+}>\varepsilon $

with some

$ \varepsilon>0 $

, we only obtain the global existence of solutions for the above system.

Keywords: Global existence, boundedness, large-time behavior, exponentially decaying diffusivity, indirect signals.

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

Citation: Pan Zheng, Jie Xing. On an exponentially decaying diffusive chemotaxis system with indirect signals. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1735-1753. doi: 10.3934/cpaa.2022044