# American Institute of Mathematical Sciences

September  2017, 37(9): 5021-5036. doi: 10.3934/dcds.2017216

## Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions

 Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China

* Corresponding author. QW acknowledges the support from National Natural Science Foundation of China (Grant No. 11501460). All authors thank the two anonymous referees for their helpful suggestions, which improve the presentation of this paper

Currently at Department of Mathematics, University of Central Florida, Orlando, USA. yfeng@knights.ucf.edu.

Received  June 2016 Revised  April 2017 Published  June 2017

We consider the following fully parabolic Keller-Segel system
 $\left\{\begin{array}{ll}u_t=\nabla · (D(u) \nabla u-S(u) \nabla v)+u(1-u^γ),&x ∈ Ω,t>0, \\v_t=Δ v-v+u,&x ∈ Ω,t>0, \\\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0,&x∈\partial Ω,t>0\end{array}\right.$
over a multi-dimensional bounded domain
 $Ω \subset \mathbb R^N$
,
 $N≥2$
. Here
 $D(u)$
and
 $S(u)$
are smooth functions satisfying:
 $D(0)>0$
,
 $D(u)≥ K_1u^{m_1}$
and
 $S(u)≤ K_2u^{m_2}$
,
 $\forall u≥0$
, for some constants
 $K_i∈\mathbb R^+$
,
 $m_i∈\mathbb R$
,
 $i=1, 2$
. It is proved that, when the parameter pair
 $(m_1, m_2)$
lies in some specific regions, the system admits global classical solutions and they are uniformly bounded in time. We cover and extend [22,28], in particular when
 $N≥3$
and
 $γ≥1$
, and [3,29] when
 $m_1>γ-\frac{2}{N}$
if
 $γ∈(0, 1)$
or
 $m_1>γ-\frac{4}{N+2}$
if
 $γ∈[1, ∞)$
. Moreover, according to our results, the index
 $\frac{2}{N}$
is, in contrast to the model without cellular growth, no longer critical to the global existence or collapse of this system.
Citation: Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216
##### References:

show all references

##### References:
An illustration of parameter regions for the global existence of (1.1) in the $m_1$-$m_2$ plane as $\gamma$ varies. For each pair of $(m_1,m_2)$ in the shaded region, (1.1) over bounded domain $\Omega\subset \mathbb R^N$, $N\geq2$, admits global bounded classical solutions. The slopes of $L_1(L^*_1)$ and $L_2(L^*_2)$ are 1 and the slope of $L_3(L^*_3)$ is $\frac{1}{2}$. Note that in the lower right plot, we use $\frac{1}{N}+\frac{2}{N+2}$, instead of $\frac{3N+2}{N(N+2)}$, for better illustration
An illustration of up-to-date summary results on global existence of the nonlinear diffusion system (1.1). For each pair of $(m_1,m_2)$ in the shaded region, (1.1) over bounded domain $\Omega\subset \mathbb R^N$, $N\geq2$, admits global bounded classical solutions. The slope of $L_1(L^*_1)$ is 1 and of $L^*_2$ is $\frac{1}{2}$; $L_4$ is the horizontal line $m_2=\gamma$
 [1] 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 [2] Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035 [3] Marcel Freitag. Global existence and boundedness in a chemorepulsion system with superlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5943-5961. doi: 10.3934/dcds.2018258 [4] 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 [5] Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069 [6] Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262 [7] Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737 [8] 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 - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437 [9] 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 [10] 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 [11] Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 321-328. doi: 10.3934/dcdss.2020018 [12] Tong Li, Jeungeun Park. Traveling waves in a chemotaxis model with logistic growth. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6465-6480. doi: 10.3934/dcdsb.2019147 [13] Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789 [14] Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026 [15] Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125 [16] Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463 [17] E. Trofimchuk, Sergei Trofimchuk. Global stability in a regulated logistic growth model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 461-468. doi: 10.3934/dcdsb.2005.5.461 [18] Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199 [19] Youshan Tao, Michael Winkler. Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1901-1914. doi: 10.3934/dcds.2012.32.1901 [20] Masaki Kurokiba, Toshitaka Nagai, T. Ogawa. The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Communications on Pure & Applied Analysis, 2006, 5 (1) : 97-106. doi: 10.3934/cpaa.2006.5.97

2018 Impact Factor: 1.143