June  2021, 20(6): 2211-2236. 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  June 2021 Early access  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 and Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064
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.

[2]

N. BellomoA. BellouquidY. 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.

[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.

[4]

T. BlackJ. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876. 

[5]

M. DingW. WangS. 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.

[6]

E. EspejoK. Vilches and C. Conca, A simultaneous blow-up problem arising in tumor modeling, J. Math. Biol., 79 (2019), 1357-1399. 

[7]

D. D. Haroske, H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.

[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.

[9]

S. IshidaK. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $R^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606. 

[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.

[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.

[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.

[20]

X. PanL. 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.

[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.

[22]

C. StinnerJ.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.

[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.

[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.

[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. 

[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.

[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.

[28]

L. WangY. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802. 

[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.

[30]

L. WangJ. ZhangC. 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.

[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.

[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.

[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. 

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.

[2]

N. BellomoA. BellouquidY. 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.

[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.

[4]

T. BlackJ. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876. 

[5]

M. DingW. WangS. 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.

[6]

E. EspejoK. Vilches and C. Conca, A simultaneous blow-up problem arising in tumor modeling, J. Math. Biol., 79 (2019), 1357-1399. 

[7]

D. D. Haroske, H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.

[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.

[9]

S. IshidaK. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $R^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606. 

[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.

[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.

[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.

[20]

X. PanL. 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.

[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.

[22]

C. StinnerJ.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.

[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.

[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.

[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. 

[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.

[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.

[28]

L. WangY. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802. 

[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.

[30]

L. WangJ. ZhangC. 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.

[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.

[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.

[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. 

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