# American Institute of Mathematical Sciences

January  2019, 18(1): 255-284. doi: 10.3934/cpaa.2019014

## Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion

 Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

* Corresponding author

Received  November 2017 Revised  March 2018 Published  August 2018

In this work, we are concerned with a class of parabolic-elliptic chemotaxis systems with the prototype given by
 $\left\{ \begin{array}{lll}&u_t = \nabla\cdot(\nabla u-\chi u\nabla v)+au-bu^\theta, &x\in \Omega, t>0, \\&0 = \Delta v -v+u^\kappa, & x\in \Omega, t>0 \end{array}\right.$
with nonnegative initial condition for
 $u$
and homogeneous Neumann boundary conditions in a smooth bounded domain
 $Ω\subset \mathbb{R}^n(n≥ 2)$
, where
 $χ, b, κ>0$
,
 $a∈ \mathbb{R}$
and
 $θ>1$
.
First, using different ideas from [9,11], we re-obtain the boundedness and global existence for the corresponding initial-boundary value problem under, either
 $κ+1<\max\{θ, 1+\frac{2}{n}\}$
or
 $θ = κ+1, \ \ b≥ \frac{(κ n-2)}{κ n}χ.$
Next, carrying out bifurcation from "old multiplicity", we show that the corresponding stationary system exhibits pattern formation for an unbounded range of chemosensitivity
 $χ$
and the emerging patterns converge weakly in
 $L^θ(Ω)$
to some constants as
 $χ \to ∞$
. This provides more details and also fills up a gap left in Kuto et al. [13] for the particular case that
 $θ = 2$
and
 $κ = 1$
. Finally, for
 $θ = κ+1$
, the global stabilities of the equilibria
 $((a/b)^{\frac{1}{κ}}, a/b)$
and
 $(0,0)$
are comprehensively studied and explicit convergence rates are computed out, which exhibits chemotaxis effects and logistic damping on long time dynamics of solutions. These stabilization results indicate that no pattern formation arises for small
 $χ$
or large damping rate
 $b$
; on the other hand, they cover and extend He and Zheng's [6,Theorems 1 and 2] for logistic source and linear secretion (
 $θ = 2$
and
 $κ = 1$
) (where convergence rate estimates were shown) to generalized logistic source and nonlinear secretion.
Citation: Tian Xiang. Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion. Communications on Pure &amp; Applied Analysis, 2019, 18 (1) : 255-284. doi: 10.3934/cpaa.2019014
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##### References:
 [1] Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220 [2] Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180 [3] Ke Lin, Chunlai Mu. Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5025-5046. doi: 10.3934/dcds.2016018 [4] Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050 [5] Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061 [6] Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011 [7] Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108 [8] Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064 [9] Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262 [10] Guoqiang Ren, Heping Ma. Global existence in a chemotaxis system with singular sensitivity and signal production. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021045 [11] Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324 [12] Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150 [13] Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170 [14] Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063 [15] Messoud Efendiev, Etsushi Nakaguchi, Wolfgang L. Wendland. Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system. Conference Publications, 2007, 2007 (Special) : 334-343. doi: 10.3934/proc.2007.2007.334 [16] Belkacem Said-Houari, Flávio A. Falcão Nascimento. Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction. Communications on Pure & Applied Analysis, 2013, 12 (1) : 375-403. doi: 10.3934/cpaa.2013.12.375 [17] Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341 [18] T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125 [19] Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437 [20] Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 165-176. doi: 10.3934/dcdss.2020009

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