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November  2015, 20(9): 3165-3183. doi: 10.3934/dcdsb.2015.20.3165

## Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals

 1 Department of Applied Mathematics, Dong Hua University, Shanghai 200051 2 Institut für Mathematik, Universität Paderborn, 33098 Paderborn

Received  August 2014 Revised  April 2015 Published  September 2015

We consider a model for two species interacting through chemotaxis in such a way that each species produces a signal which directs the respective motion of the other. Specifically, we shall be concerned with nonnegative solutions of the Neumann problem, posed in bounded domains $\Omega\subset \mathbb{R}^n$ with smooth boundary, for the system $$\begin{cases} u_t= \Delta u - \chi \nabla \cdot (u\nabla v), & x\in \Omega, \, t>0, \\ 0=\Delta v-v+w, & x\in \Omega, \, t>0, \qquad (\star)\\ w_t= \Delta w - \xi \nabla \cdot (w\nabla z), & x\in \Omega, \, t>0, \\ 0=\Delta z-z+u, & x\in \Omega, \, t>0, \end{cases}$$
with parameters $\chi \in \{\pm 1\}$ and $\xi\in \{\pm 1\}$, thus allowing the interaction of either attraction-repulsion, or attraction-attraction, or repulsion-repulsion type.
It is shown that
$\bullet$ in the attraction-repulsion case $\chi=1$ and $\xi=-1$, if $n\le 3$ then for any nonnegative initial data $u_0\in C^0(\bar{\Omega})$ and $w_0\in C^0 (\bar{\Omega})$, there exists a unique global classical solution which is bounded;
$\bullet$ in the doubly repulsive case when $\chi=\xi=-1$, the same holds true;
$\bullet$ in the attraction-attraction case $\chi=\xi=1$,
$-$ if either $n=2$ and $\int_\Omega u_0 + \int_\Omega w_0$ lies below some threshold, or $n\ge 3$ and $\|u_0\|_{L^\infty(\Omega)}$ and $\|w_0\|_{L^\infty(\Omega)}$ are sufficiently small, then solutions exist globally and remain bounded, whereas
$-$ if either $n=2$ and $m$ is suitably large, or $n\ge 3$ and $m>0$ is arbitrary, then there exist smooth initial data $u_0$ and $w_0$ such that $\int_\Omega u_0 + \int_\Omega w_0=m$ and such that the corresponding solution blows up in finite time.
In particular, these results demonstrate that the circular chemotaxis mechanism underlying ($\star$) goes along with essentially the same destabilizing features as known for the classical Keller-Segel system in the doubly attractive case, but totally suppresses any blow-up phenomenon when only one, or both, taxis directions are repulsive.
Citation: Youshan Tao, Michael Winkler. Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3165-3183. doi: 10.3934/dcdsb.2015.20.3165
##### References:
 [1] P. Biler, E. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems,, Commun. Pure Appl. Anal., 12 (2013), 89.  doi: 10.3934/cpaa.2013.12.89.  Google Scholar [2] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions,, Nonlinear Analysis, 23 (1994), 1189.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar [3] H. Brézis and W. A. Strauss, Semilinear second-order elliptic equations in $L^{1}$,, J. Math. Soc. Japan, 25 (1973), 565.  doi: 10.2969/jmsj/02540565.  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.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar [5] S. Cantrell, C. Cosner and S. Ruan, Spatial Ecology,, Chapman & Hall/CRC, (2010).   Google Scholar [6] T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system,, Banach Center Publ., 81 (2008), 105.  doi: 10.4064/bc81-0-7.  Google Scholar [7] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar [8] C. Conca, E. E. Espejo and K. Vilches, Global existence and blow-up for a two species Keller-Segel model for chemotaxis,, European J. Appl. Math., 22 (2011), 553.  doi: 10.1017/S0956792511000258.  Google Scholar [9] E. E. Espejo, A. Stevens and J. L. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis,, Analysis (Munich), 29 (2009), 317.  doi: 10.1524/anly.2009.1029.  Google Scholar [10] A. Friedman, Partial Differential Equations,, Holt, (1969).   Google Scholar [11] M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Saptially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-sriatal circuit,, Euro. J. Neuroscicen, 19 (2004), 831.   Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Grundlehren der Mathematischen Wissenschaften, (1977).   Google Scholar [13] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).   Google Scholar [14] M. A. Herrero and J. L. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633.   Google Scholar [15] M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle,, Nature Reviews Microbiology, 8 (2010), 15.  doi: 10.1038/nrmicro2259.  Google Scholar [16] T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar [17] S. Hittmeir and A. Jüngel, Cross-diffusion preventing blow up in the two-dimensional Keller-Segel model,, SIAM J. Math. Anal., 43 (2011), 997.  doi: 10.1137/100813191.  Google Scholar [18] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.- Verien, 105 (2003), 103.   Google Scholar [19] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar [20] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [21] F. X. Kelly, K. J. Dapsis and D. A. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition,, Microbial Ecology, 16 (1988), 115.  doi: 10.1007/BF02018908.  Google Scholar [22] 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.   Google Scholar [23] N. Mizoguchi and M. Winkler, Is aggregation a generic phenomenon in the two-dimensional Keller-Segel system?,, Preprint., ().   Google Scholar [24] J. D. Murray, Mathematical Biology,, Second edition. Biomathematics, (1993).  doi: 10.1007/b98869.  Google Scholar [25] T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains,, J. of Inequal. & Appl., 6 (2001), 37.  doi: 10.1155/S1025583401000042.  Google Scholar [26] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkc. Ekvacioj, 40 (1997), 411.   Google Scholar [27] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Analysis, 51 (2002), 119.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar [28] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcialaj Ekvacioj, 44 (2001), 441.   Google Scholar [29] K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis,, Bull. Math. Biol., 71 (2009), 1117.  doi: 10.1007/s11538-009-9396-8.  Google Scholar [30] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Quart., 10 (2002), 501.   Google Scholar [31] K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations,, Journal of Theoretical Biology, 225 (2003), 327.  doi: 10.1016/S0022-5193(03)00258-3.  Google Scholar [32] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model,, J. Math. Biology, 68 (2014), 1607.  doi: 10.1007/s00285-013-0681-7.  Google Scholar [33] Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity,, Discr. Cont. Dyn. Syst. B, 18 (2013), 2705.  doi: 10.3934/dcdsb.2013.18.2705.  Google Scholar [34] Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1.  doi: 10.1142/S0218202512500443.  Google Scholar [35] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar [36] Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model,, Nonlinearity, 27 (2014), 1225.  doi: 10.1088/0951-7715/27/6/1225.  Google Scholar [37] J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source,, Nonlinearity, 25 (2012), 1413.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar [38] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Commun. Partial Differential Equations, 35 (2010), 1516.  doi: 10.1080/03605300903473426.  Google Scholar [39] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  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.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

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##### References:
 [1] P. Biler, E. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems,, Commun. Pure Appl. Anal., 12 (2013), 89.  doi: 10.3934/cpaa.2013.12.89.  Google Scholar [2] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions,, Nonlinear Analysis, 23 (1994), 1189.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar [3] H. Brézis and W. A. Strauss, Semilinear second-order elliptic equations in $L^{1}$,, J. Math. Soc. Japan, 25 (1973), 565.  doi: 10.2969/jmsj/02540565.  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.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar [5] S. Cantrell, C. Cosner and S. Ruan, Spatial Ecology,, Chapman & Hall/CRC, (2010).   Google Scholar [6] T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system,, Banach Center Publ., 81 (2008), 105.  doi: 10.4064/bc81-0-7.  Google Scholar [7] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar [8] C. Conca, E. E. Espejo and K. Vilches, Global existence and blow-up for a two species Keller-Segel model for chemotaxis,, European J. Appl. Math., 22 (2011), 553.  doi: 10.1017/S0956792511000258.  Google Scholar [9] E. E. Espejo, A. Stevens and J. L. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis,, Analysis (Munich), 29 (2009), 317.  doi: 10.1524/anly.2009.1029.  Google Scholar [10] A. Friedman, Partial Differential Equations,, Holt, (1969).   Google Scholar [11] M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Saptially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-sriatal circuit,, Euro. J. Neuroscicen, 19 (2004), 831.   Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Grundlehren der Mathematischen Wissenschaften, (1977).   Google Scholar [13] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).   Google Scholar [14] M. A. Herrero and J. L. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633.   Google Scholar [15] M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle,, Nature Reviews Microbiology, 8 (2010), 15.  doi: 10.1038/nrmicro2259.  Google Scholar [16] T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar [17] S. Hittmeir and A. Jüngel, Cross-diffusion preventing blow up in the two-dimensional Keller-Segel model,, SIAM J. Math. Anal., 43 (2011), 997.  doi: 10.1137/100813191.  Google Scholar [18] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.- Verien, 105 (2003), 103.   Google Scholar [19] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar [20] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [21] F. X. Kelly, K. J. Dapsis and D. A. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition,, Microbial Ecology, 16 (1988), 115.  doi: 10.1007/BF02018908.  Google Scholar [22] 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.   Google Scholar [23] N. Mizoguchi and M. Winkler, Is aggregation a generic phenomenon in the two-dimensional Keller-Segel system?,, Preprint., ().   Google Scholar [24] J. D. Murray, Mathematical Biology,, Second edition. Biomathematics, (1993).  doi: 10.1007/b98869.  Google Scholar [25] T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains,, J. of Inequal. & Appl., 6 (2001), 37.  doi: 10.1155/S1025583401000042.  Google Scholar [26] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkc. Ekvacioj, 40 (1997), 411.   Google Scholar [27] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Analysis, 51 (2002), 119.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar [28] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcialaj Ekvacioj, 44 (2001), 441.   Google Scholar [29] K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis,, Bull. Math. Biol., 71 (2009), 1117.  doi: 10.1007/s11538-009-9396-8.  Google Scholar [30] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Quart., 10 (2002), 501.   Google Scholar [31] K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations,, Journal of Theoretical Biology, 225 (2003), 327.  doi: 10.1016/S0022-5193(03)00258-3.  Google Scholar [32] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model,, J. Math. Biology, 68 (2014), 1607.  doi: 10.1007/s00285-013-0681-7.  Google Scholar [33] Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity,, Discr. Cont. Dyn. Syst. B, 18 (2013), 2705.  doi: 10.3934/dcdsb.2013.18.2705.  Google Scholar [34] Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1.  doi: 10.1142/S0218202512500443.  Google Scholar [35] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar [36] Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model,, Nonlinearity, 27 (2014), 1225.  doi: 10.1088/0951-7715/27/6/1225.  Google Scholar [37] J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source,, Nonlinearity, 25 (2012), 1413.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar [38] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Commun. Partial Differential Equations, 35 (2010), 1516.  doi: 10.1080/03605300903473426.  Google Scholar [39] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  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.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar
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