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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 & Continuous Dynamical Systems, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027
References:
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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] Y. Li and Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. 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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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [23] K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [23] K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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