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August 2018, 38(8): 3875-3898. doi: 10.3934/dcds.2018168

## Global weak solution and boundedness in a three-dimensional competing chemotaxis

 1 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 3 College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  September 2017 Revised  February 2018 Published  May 2018

Fund Project: The second author is partially supported by NSFC (Grant No. 11771062 and 11571062), the Fundamental Research Funds for the Central Universities (Grant No. 10611CDJXZ238826) and the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007), and the third author is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. JBK1801059) and Chongqing Scientific & Technological Talents Program (Grant No. KJXX2017006)

We consider an initial-boundary value problem for a parabolic-parabolic-elliptic attraction-repulsion chemotaxis model
 $\left\{ \begin{array}{l}u_t = Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w),&x∈ Ω,&t>0,\\v_t = Δ v-β v+α u,&x∈Ω,&t>0,\\0 = Δ w-δ w+γ u,&x∈Ω,&t>0\\\end{array} \right.$
in a bounded domain
 $Ω\subset \mathbb{R}^3$
with positive parameters
 $χ, ξ, α, β, γ$
and
 $δ$
.
It is firstly proved that if the repulsion dominates in the sense that
 $ξγ>χα$
, then for any choice of sufficiently smooth initial data
 $(u_0, v_0)$
the corresponding initial-boundary value problem is shown to possess a globally defined weak solution. To the best of our knowledge, this situation provides the first result on global existence of the above system in the three-dimensional setting when
 $ξγ>χα$
, and extends the results in Lin et al. (2016) [19] and Jin and Xiang (2017) [14] to more general case.
Secondly, if the initial data is appropriately small or the repulsion is enough strong in the sense that
 $ξγ$
is suitable large as related to
 $χα$
, then the classical solutions to the above system are uniformly-in-time bounded.
Citation: Hua Zhong, Chunlai Mu, Ke Lin. Global weak solution and boundedness in a three-dimensional competing chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3875-3898. doi: 10.3934/dcds.2018168
##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Commun. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅱ, Commun. Pure Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104. [3] 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. [4] T. Cieślak, Ph. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, In Parabolic and Navier-Stokes equations, Banach Center Publ. Polish Acad. Sci. Inst. Math., 81 (2008), 105-117. doi: 10.4064/bc81-0-7. [5] N. Dunford and J. T. Schwartz, Linear Operators. Ⅰ. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. [6] E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007. [7] K. Fujie and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148. doi: 10.1016/j.jde.2017.02.031. [8] H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106. [9] 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. [10] 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. [11] H. Y. 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. [12] H. Y. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457. doi: 10.1002/mma.3080. [13] H. Y. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196. doi: 10.1016/j.jde.2015.08.040. [14] H. Y. Jin and T. Xiang, Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions, Discrete Contin. Dyn. Syst. B, 22 (2017), 1-15. [15] 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. [16] Y. Li and Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real World Appl., 30 (2016), 170-183. doi: 10.1016/j.nonrwa.2015.12.003. [17] Y. H. Li, K. Lin and C. L. Mu, Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system, Electron. J. Differential Equations, 146 (2015), 1-13. [18] K. Lin and C. L. Mu, Global existence and convergence to steady states for an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 31 (2016), 630-642. doi: 10.1016/j.nonrwa.2016.03.012. [19] K. Lin, C. L. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, J. Differential Equations, 261 (2016), 4524-4572. doi: 10.1016/j.jde.2016.07.002. [20] K. Lin, C. L. Mu and L. C. 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. [21] K. Lin, C. L. Mu and D. Q. Zhou, Stabilization in a higher-dimensional attraction-repulsion chemotaxis system if repulsion dominates over attraction, preprint, (2018). [22] J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722. [23] P. Liu, J. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion keller-segel system, Discrete Contin. Dyn. Syst. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. [24] M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimer's disease senile plague: is there a connection?, Bull. Math. Biol., 65 (2003), 673-730. [25] N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system, preprint, 2016. [26] M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal., 5 (1974), 597-612. doi: 10.1137/0505061. [27] M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl., 49 (1975), 215-225. doi: 10.1016/0022-247X(75)90172-9. [28] 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. [29] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [30] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469. [31] K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [32] C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X. [33] 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. [34] 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. [35] 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. [36] Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Methods Appl. Sci., 27 (2017), 1645-1683. doi: 10.1142/S0218202517500282. [37] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. [38] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. [39] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426. [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. [41] H. Yu, Q. Guo and S. N. Zheng, Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 34 (2017), 335-342. doi: 10.1016/j.nonrwa.2016.09.007. [42] Q. S. Zhang and Y. X. Li, An attraction-repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584. doi: 10.1002/zamm.201400311.

show all references

##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Commun. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅱ, Commun. Pure Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104. [3] 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. [4] T. Cieślak, Ph. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, In Parabolic and Navier-Stokes equations, Banach Center Publ. Polish Acad. Sci. Inst. Math., 81 (2008), 105-117. doi: 10.4064/bc81-0-7. [5] N. Dunford and J. T. Schwartz, Linear Operators. Ⅰ. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. [6] E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007. [7] K. Fujie and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148. doi: 10.1016/j.jde.2017.02.031. [8] H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106. [9] 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. [10] 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. [11] H. Y. 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. [12] H. Y. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457. doi: 10.1002/mma.3080. [13] H. Y. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196. doi: 10.1016/j.jde.2015.08.040. [14] H. Y. Jin and T. Xiang, Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions, Discrete Contin. Dyn. Syst. B, 22 (2017), 1-15. [15] 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. [16] Y. Li and Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real World Appl., 30 (2016), 170-183. doi: 10.1016/j.nonrwa.2015.12.003. [17] Y. H. Li, K. Lin and C. L. Mu, Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system, Electron. J. Differential Equations, 146 (2015), 1-13. [18] K. Lin and C. L. Mu, Global existence and convergence to steady states for an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 31 (2016), 630-642. doi: 10.1016/j.nonrwa.2016.03.012. [19] K. Lin, C. L. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, J. Differential Equations, 261 (2016), 4524-4572. doi: 10.1016/j.jde.2016.07.002. [20] K. Lin, C. L. Mu and L. C. 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. [21] K. Lin, C. L. Mu and D. Q. Zhou, Stabilization in a higher-dimensional attraction-repulsion chemotaxis system if repulsion dominates over attraction, preprint, (2018). [22] J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722. [23] P. Liu, J. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion keller-segel system, Discrete Contin. Dyn. Syst. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. [24] M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimer's disease senile plague: is there a connection?, Bull. Math. Biol., 65 (2003), 673-730. [25] N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system, preprint, 2016. [26] M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal., 5 (1974), 597-612. doi: 10.1137/0505061. [27] M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl., 49 (1975), 215-225. doi: 10.1016/0022-247X(75)90172-9. [28] 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. [29] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [30] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441-469. [31] K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [32] C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X. [33] 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. [34] 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. [35] 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. [36] Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Methods Appl. Sci., 27 (2017), 1645-1683. doi: 10.1142/S0218202517500282. [37] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. [38] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. [39] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426. [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. [41] H. Yu, Q. Guo and S. N. Zheng, Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 34 (2017), 335-342. doi: 10.1016/j.nonrwa.2016.09.007. [42] Q. S. Zhang and Y. X. Li, An attraction-repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584. doi: 10.1002/zamm.201400311.
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