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June  2020, 40(6): 3509-3527. doi: 10.3934/dcds.2020027

## Global stabilization of the full attraction-repulsion Keller-Segel system

 1 Department of Mathematics, South China University of Technology, Guangzhou 510640, China 2 Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China

* Corresponding author: Zhi-An Wang

To Professor Wei-Ming Ni on the occasion of his 70th birthday, with our best wishes

Received  May 2018 Revised  March 2019 Published  October 2019

Fund Project: The research of H.Y. Jin was supported by the NSF of China No. 11871226, and the Fundamental Research Funds for the Central Universities. The research of Z.A. Wang acknowledges partial supports from an internal grant No. ZZHY in Hong Kong Polytechnic University and from NSFC grant 11571086.

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system
 $\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \nabla \cdot (\chi u\nabla v) + \nabla \cdot (\xi u\nabla w),}&{x \in \Omega ,t > 0,}\\{{v_t} = {D_1}\Delta v + \alpha u - \beta v,}&{x \in \Omega ,t > 0,}\\{{w_t} = {D_2}\Delta w + \gamma u - \delta w,}&{x \in \Omega ,t > 0,}\\{u(x,0) = {u_0}(x),v(x,0) = {v_0}(x),w(x,0) = {w_0}(x)}&{x \in \Omega ,}\end{array}} \right.\;\;\;\;\left( * \right)$
in a bounded domain
 $\Omega\subset \mathbb{R}^2$
with smooth boundary subject to homogeneous Neumann boundary conditions. By constructing an appropriate Lyapunov functions, we establish the boundedness and asymptotical behavior of solutions to the system (*) with large initial data
 $(u_0,v_0,w_0) \in [W^{1,\infty}(\Omega)]^3$
. Precisely, we show that if the parameters satisfy
 $\frac{\xi\gamma}{\chi\alpha}\geq \max\Big\{\frac{D_1}{D_2},\frac{D_2}{D_1},\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\}$
for all positive parameters
 $D_1,D_2,\chi,\xi,\alpha,\beta,\gamma$
and
 $\delta$
, the system (*) has a unique global classical solution
 $(u,v,w)$
, which converges to the constant steady state
 $(\bar{u}_0,\frac{\alpha}{\beta}\bar{u}_0,\frac{\gamma}{\delta}\bar{u}_0)$
as
 $t\to+\infty$
, where
 $\bar{u}_0 = \frac{1}{|\Omega|}\int_\Omega u_0dx$
. Furthermore, the decay rate is exponential if
 $\frac{\xi\gamma}{\chi\alpha}> \max\Big\{\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\}$
. This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e.
 $D_1\ne D_2$
) in multi-dimensions.
Citation: Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027
##### References:
 [1] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Towards 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. [2] J. P. Bourguignon and H. Brezis, Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5. [3] J. A. Carrillo, A. Juöngle, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.  doi: 10.1007/s006050170032. [4] T. Cieślak, Ph. Laurenct 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] 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. [6] D. Horstemann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Deutsch. Math. Verien., 105 (2003), 103-165. [7] 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. [8] 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. [9] H. Y. Jin and Z. Liu, Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20.  doi: 10.1016/j.aml.2015.03.004. [10] 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. [11] 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. [12] R. Kowalczyk and Z. Szyma$\acute{n}$ska, 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. [13] O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 1968. [14] Y. Li and Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real Word Appl., 30 (2016), 170-183.  doi: 10.1016/j.nonrwa.2015.12.003. [15] K. Lin and C. Mu, Global existence and convergence to steady states for an attraction-repulsion chemotaxis system, Nonlinear Anal. Real Word Appl., 31 (2016), 630-642.  doi: 10.1016/j.nonrwa.2016.03.012. [16] K. Lin, C. 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. [17] K. Lin, C. Mu and D. Zhou, Stabilization in a higher-dimensional attraction-repulsion chemotaxis system if repulsion dominates over attraction, Math. Models Methods Appl. Sci., 28 (2018), 1105-1134.  doi: 10.1142/S021820251850029X. [18] D. Liu and Y. S. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.  doi: 10.1002/mma.3240. [19] P. Liu, J. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597. [20] J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis model in one dimension, J. Biol. Dyn., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722. [21] M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, Microglia, and Alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. [22] 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. [23] K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [24] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [25] R. Shi and W. Wang, Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520.  doi: 10.1016/j.jmaa.2014.10.006. [26] Y. S. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722.  doi: 10.3934/dcdsb.2013.18.2705. [27] Y. S. 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. [28] Y. S. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115. [29] 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. [30] 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.

show all references

##### References:
 [1] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Towards 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. [2] J. P. Bourguignon and H. Brezis, Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5. [3] J. A. Carrillo, A. Juöngle, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.  doi: 10.1007/s006050170032. [4] T. Cieślak, Ph. Laurenct 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] 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. [6] D. Horstemann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Deutsch. Math. Verien., 105 (2003), 103-165. [7] 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. [8] 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. [9] H. Y. Jin and Z. Liu, Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20.  doi: 10.1016/j.aml.2015.03.004. [10] 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. [11] 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. [12] R. Kowalczyk and Z. Szyma$\acute{n}$ska, 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. [13] O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 1968. [14] Y. Li and Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real Word Appl., 30 (2016), 170-183.  doi: 10.1016/j.nonrwa.2015.12.003. [15] K. Lin and C. Mu, Global existence and convergence to steady states for an attraction-repulsion chemotaxis system, Nonlinear Anal. Real Word Appl., 31 (2016), 630-642.  doi: 10.1016/j.nonrwa.2016.03.012. [16] K. Lin, C. 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. [17] K. Lin, C. Mu and D. Zhou, Stabilization in a higher-dimensional attraction-repulsion chemotaxis system if repulsion dominates over attraction, Math. Models Methods Appl. Sci., 28 (2018), 1105-1134.  doi: 10.1142/S021820251850029X. [18] D. Liu and Y. S. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.  doi: 10.1002/mma.3240. [19] P. Liu, J. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597. [20] J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis model in one dimension, J. Biol. Dyn., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722. [21] M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, Microglia, and Alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. [22] 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. [23] K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [24] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [25] R. Shi and W. Wang, Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520.  doi: 10.1016/j.jmaa.2014.10.006. [26] Y. S. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722.  doi: 10.3934/dcdsb.2013.18.2705. [27] Y. S. 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. [28] Y. S. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115. [29] 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. [30] 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.
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