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September  2017, 22(7): 2717-2729. doi: 10.3934/dcdsb.2017132

Boundedness in a two-species chemotaxis parabolic system with two chemicals

1. 

College of Mathematic & Information, China West Normal University, Nanchong, 637002, China

2. 

School of Sciences, Southwest Petroleum University, Chengdu, 610500, China

* Corresponding author: Xie Li

Received  August 2016 Revised  February 2017 Published  April 2017

This paper is devoted to the chemotaxis system
$\left\{\begin{aligned}&u_t=Δ u-χ\nabla·(u\nabla v), &x∈Ω,\,t>0,\\& τ v_t=Δ v-v+w, &x∈Ω,\,t>0,\\&w_t=Δ w-ξ\nabla·(w\nabla z), &x∈Ω,\,t>0,\\& τ z_t=Δ z-z+u, &x∈Ω,\,t>0,\end{aligned}\right.$
which models the interaction between two species in presence of two chemicals, where
$χ, \, ξ∈\mathbb{R}$
,
$Ω\subset\mathbb{R}^2$
are bounded domains with smooth boundary. It is shown that under the homogeneous Neumann boundary conditions the system possesses a unique global classical solution which is bounded whenever both
$\int_Ω u_0dx$
and
$\int_Ω w_0dx$
are appropriately small. In particular, we extend the recent results obtained by Tao and Winkler (2015, Disc. Cont. Dyn. Syst. B) to the fully parabolic case, i.e., the case of
$τ=1$
.
Citation: Xie Li, Yilong Wang. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2717-2729. doi: 10.3934/dcdsb.2017132
References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.  Google Scholar

[2]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[3]

P. BilerE. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.  doi: 10.3934/cpaa.2013.12.89.  Google Scholar

[4]

V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pure Appl., 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar

[5]

T. CieslakP. LaurenÇot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117.  doi: 10.4064/bc81-0-7.  Google Scholar

[6] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.   Google Scholar
[7]

M. A. GatesV. M. CoupeE. M. TorresR. A. Fricker-Gares and S. B. Dunnett, Spatially and temporally restricted chemoattractant and chemorepulsive cues direct the formation of the nigro-striatal circuit, Euro. J. Neurosci., 19 (2004), 831-844.   Google Scholar

[8]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.   Google Scholar

[9]

M. E. HibbingC. FuquaM. R. Parsek and S. B. Peterson, Bacterial competition: surviving and thriving in the microbial jungle, Nature Reviews Microbiology., 8 (2010), 15-25.  doi: 10.1038/nrmicro2259.  Google Scholar

[10]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

[11]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Ⅰ, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103-165.   Google Scholar

[12]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[13]

H. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Meth. Appl. Sci., 38 (2015), 444-457.  doi: 10.1002/mma.3080.  Google Scholar

[14]

H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 206 (2016), 162-196.  doi: 10.1016/j.jde.2015.08.040.  Google Scholar

[15]

H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.  Google Scholar

[16]

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

[17]

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

[18]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimers disease senile plague: Is there a connection?, Bull.Math. Biol., 65 (2003), 673-730.   Google Scholar

[19]

X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301.  doi: 10.1002/mma.3477.  Google Scholar

[20]

X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Disc. Cont. Dyn. System -A, 35 (2015), 3503-3531.  doi: 10.3934/dcds.2015.35.3503.  Google Scholar

[21]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with logistic source, IMA J. of Appl. Math., 81 (2016), 165-198.  doi: 10.1093/imamat/hxv033.  Google Scholar

[22]

K. LinC. Mu and L. Wang, Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.  doi: 10.1016/j.jmaa.2014.12.052.  Google Scholar

[23]

T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.  Google Scholar

[24]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.  Google Scholar

[25]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Ser. A: Theory Methods and Applications, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[26]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543.   Google Scholar

[27]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, Journal of Theoretical Biology, 225 (2003), 327-339.  doi: 10.1016/S0022-5193(03)00258-3.  Google Scholar

[28]

C. StinnerJ. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biology, 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar

[29]

Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[30]

Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discr. Cont. Dyn. Syst. B, 18 (2013), 2705-2722.  doi: 10.3934/dcdsb.2013.18.2705.  Google Scholar

[31]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.  Google Scholar

[32]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[33]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.  Google Scholar

[34]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discr. Cont. Dyn. Syst. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar

[35]

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

[36]

Y. Wang, A quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type with logistic source, J. Math. Anal. Appl., 441 (2016), 259-292.  doi: 10.1016/j.jmaa.2016.03.061.  Google Scholar

[37]

Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Comput. Math. Appl., 72 (2016), 2226-2240.  doi: 10.1016/j.camwa.2016.08.024.  Google Scholar

[38]

Y. Wang, Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Probl., 2016 (2016), 1-22.  doi: 10.1186/s13661-016-0518-6.  Google Scholar

[39]

Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discr. Cont. Dyn. Syst. B, 21 (2016), 1953-1973.  doi: 10.3934/dcdsb.2016031.  Google Scholar

[40]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[41]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[42]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[43]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.  Google Scholar

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.  Google Scholar

[2]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[3]

P. BilerE. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.  doi: 10.3934/cpaa.2013.12.89.  Google Scholar

[4]

V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pure Appl., 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar

[5]

T. CieslakP. LaurenÇot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117.  doi: 10.4064/bc81-0-7.  Google Scholar

[6] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.   Google Scholar
[7]

M. A. GatesV. M. CoupeE. M. TorresR. A. Fricker-Gares and S. B. Dunnett, Spatially and temporally restricted chemoattractant and chemorepulsive cues direct the formation of the nigro-striatal circuit, Euro. J. Neurosci., 19 (2004), 831-844.   Google Scholar

[8]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.   Google Scholar

[9]

M. E. HibbingC. FuquaM. R. Parsek and S. B. Peterson, Bacterial competition: surviving and thriving in the microbial jungle, Nature Reviews Microbiology., 8 (2010), 15-25.  doi: 10.1038/nrmicro2259.  Google Scholar

[10]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

[11]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Ⅰ, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103-165.   Google Scholar

[12]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[13]

H. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Meth. Appl. Sci., 38 (2015), 444-457.  doi: 10.1002/mma.3080.  Google Scholar

[14]

H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 206 (2016), 162-196.  doi: 10.1016/j.jde.2015.08.040.  Google Scholar

[15]

H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.  Google Scholar

[16]

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

[17]

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

[18]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimers disease senile plague: Is there a connection?, Bull.Math. Biol., 65 (2003), 673-730.   Google Scholar

[19]

X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301.  doi: 10.1002/mma.3477.  Google Scholar

[20]

X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Disc. Cont. Dyn. System -A, 35 (2015), 3503-3531.  doi: 10.3934/dcds.2015.35.3503.  Google Scholar

[21]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with logistic source, IMA J. of Appl. Math., 81 (2016), 165-198.  doi: 10.1093/imamat/hxv033.  Google Scholar

[22]

K. LinC. Mu and L. Wang, Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.  doi: 10.1016/j.jmaa.2014.12.052.  Google Scholar

[23]

T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.  Google Scholar

[24]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.  doi: 10.1137/140971853.  Google Scholar

[25]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Ser. A: Theory Methods and Applications, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[26]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543.   Google Scholar

[27]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, Journal of Theoretical Biology, 225 (2003), 327-339.  doi: 10.1016/S0022-5193(03)00258-3.  Google Scholar

[28]

C. StinnerJ. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biology, 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar

[29]

Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[30]

Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discr. Cont. Dyn. Syst. B, 18 (2013), 2705-2722.  doi: 10.3934/dcdsb.2013.18.2705.  Google Scholar

[31]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.  doi: 10.1137/100802943.  Google Scholar

[32]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[33]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.  Google Scholar

[34]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discr. Cont. Dyn. Syst. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar

[35]

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

[36]

Y. Wang, A quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type with logistic source, J. Math. Anal. Appl., 441 (2016), 259-292.  doi: 10.1016/j.jmaa.2016.03.061.  Google Scholar

[37]

Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Comput. Math. Appl., 72 (2016), 2226-2240.  doi: 10.1016/j.camwa.2016.08.024.  Google Scholar

[38]

Y. Wang, Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Probl., 2016 (2016), 1-22.  doi: 10.1186/s13661-016-0518-6.  Google Scholar

[39]

Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discr. Cont. Dyn. Syst. B, 21 (2016), 1953-1973.  doi: 10.3934/dcdsb.2016031.  Google Scholar

[40]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[41]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[42]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[43]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.  Google Scholar

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