# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021064

## Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production

 School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

* Corresponding author

Received  October 2020 Revised  March 2021 Published  April 2021

Fund Project: This work is supported by the Chongqing Research and Innovation Project of Graduate Students (No. CYS20271) and the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN202000618)

This paper deals with the following quasilinear two-species chemotaxis system
 $\begin{equation*} \begin{cases} \partial_{t} u_1 = \nabla \cdot (D_1(u_{1})\nabla u_{1} - S_1(u_{1})\nabla v) + f_{1}(u_{1}),\quad &x\in\Omega,\quad t>0,\\ \partial_{t} u_2 = \nabla \cdot (D_2(u_{2})\nabla u_{2} - S_2(u_{2})\nabla v) + f_{2}(u_{2}),\quad &x\in\Omega,\quad t>0,\\ \partial_{t} v = \Delta v-v+g_1(u_{1})+g_2(u_{2}),\quad &x\in\Omega,\quad t>0 \end{cases} \end{equation*}$
under homogeneous Neumann boundary conditions in a bounded domain
 $\Omega\subset \mathbb{R}^{n}$
 $(n\geq2)$
. The diffusivity and the density-dependent sensitivity are given by
 $D_{i}(s) \geq C_{d_{i}} (1+s)^{-\alpha_i}$
and
 $S_{i}(s) \leq C_{s_{i}} s (1+s)^{\beta_{i}-1}$
for all
 $s\geq0$
, respectively, where
 $C_{d_{i}},C_{s_{i}}>0$
and
 $\alpha_i,\beta_{i} \in \mathbb{R}$
; the logistic source and the signal productions are given by
 $f_{i}(s) \leq r_{i}s - \mu_{i} s^{k_{i}}$
and
 $g_{i}(s)\leq s^{\gamma_{i}}$
for all
 $s\geq0$
respectively, where
 $r_{i} \in \mathbb{R}$
,
 $\mu_{i},\gamma_{i} > 0$
and
 $k_{i} > 1$
 $(i = 1,2)$
. It is proved that this system possesses a global bounded smooth solution under some specific conditions with or without the logistic functions
 $f_{i}(s)$
, which partially improves the results in [25]. Moreover, in case
 $r_{i}>0$
, if
 $\mu_{i}$
are sufficiently large, it is shown that the global bounded solution exponentially converges to
 $((\frac{r_{1}}{\mu_{1}})^{\frac{1}{k_{1}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{1}{k_{2}-1}}, (\frac{r_{1}}{\mu_{1}})^{\frac{\gamma_{1}}{k_{1}-1}} + (\frac{r_{2}}{\mu_{2}})^{\frac{\gamma_{2}}{k_{2}-1}})$
as
 $t\rightarrow \infty$
.
Citation: Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021064
##### References:
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Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar [25] M. Tian and S. Zheng, Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species, Commun. Pure Appl. Anal., 15 (2016), 243-260.   Google Scholar [26] J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar [27] L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 481 (2020), 123705. doi: 10.1016/j.jmaa.2019.123705.  Google Scholar [28] L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.   Google Scholar [29] L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114.  Google Scholar [30] L. Wang, J. Zhang, C. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 191-221.  doi: 10.3934/dcdsb.2019178.  Google Scholar [31] M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e.  Google Scholar [32] T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.  doi: 10.1016/j.jmaa.2017.11.022.  Google Scholar [33] L. Xie and Y. Wang, On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.   Google Scholar

show all references

##### References:
 [1] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar [2] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar [3] T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.  Google Scholar [4] T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.   Google Scholar [5] M. Ding, W. Wang, S. Zhou and S. Zheng, Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production, J. Diff. Equations., 268 (2020), 6729-6777.  doi: 10.1016/j.jde.2019.11.052.  Google Scholar [6] E. Espejo, K. Vilches and C. Conca, A simultaneous blow-up problem arising in tumor modeling, J. Math. Biol., 79 (2019), 1357-1399.   Google Scholar [7] D. D. Haroske, H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.  Google Scholar [8] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar [9] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equ., 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar [10] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [11] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.  Google Scholar [12] D. Liu and Y. Tao, Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chin. Univ. Ser. B., 31 (2016), 379-388.  doi: 10.1007/s11766-016-3386-z.  Google Scholar [13] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire., 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar [14] M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 269-278.  doi: 10.3934/dcdss.2020015.  Google Scholar [15] M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differ. Equ., 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.  Google Scholar [16] K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $R^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606.   Google Scholar [17] X. Pan and L. Wang, Boundedness in a two-species chemotaxis system with nonlinear sensitivity and signal secretion, J. Math. Anal. Appl., (2021), 125078. Google Scholar [18] X. Pan and L. Wang, Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Math., 359 (2021), 161-168.  doi: 10.5802/crmath.148.  Google Scholar [19] X. Pan and L. Wang, On a quasilinear fully parabolic two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., (2021). doi: 10.3934/dcdsb.2021047.  Google Scholar [20] X. Pan, L. Wang and J. Zhang, Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Method. Appl. Sci., 43 (2020), 9529-9542.  doi: 10.1002/mma.6621.  Google Scholar [21] X. Pan, L. Wang, J Zhang and J Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020), 15pp. doi: 10.1007/s00033-020-1248-2.  Google Scholar [22] C. Stinner, J.I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar [23] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar [24] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar [25] M. Tian and S. Zheng, Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species, Commun. Pure Appl. Anal., 15 (2016), 243-260.   Google Scholar [26] J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar [27] L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 481 (2020), 123705. doi: 10.1016/j.jmaa.2019.123705.  Google Scholar [28] L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.   Google Scholar [29] L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114.  Google Scholar [30] L. Wang, J. Zhang, C. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 191-221.  doi: 10.3934/dcdsb.2019178.  Google Scholar [31] M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e.  Google Scholar [32] T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.  doi: 10.1016/j.jmaa.2017.11.022.  Google Scholar [33] L. Xie and Y. Wang, On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.   Google Scholar
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