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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 and 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), 89-98. doi: 10.3934/cpaa.2013.12.89.

[2]

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

[3]

H. Brézis and W. A. Strauss, Semilinear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565.

[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.

[5]

S. Cantrell, C. Cosner and S. Ruan, Spatial Ecology, Chapman & Hall/CRC, Mathematical and Computational Biology Series, Boca Raton/London/New York, 2010.

[6]

T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publ., Polish Acad. Sci., Warsaw, 81 (2008), 105-117. doi: 10.4064/bc81-0-7.

[7]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009.

[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), 553-580. doi: 10.1017/S0956792511000258.

[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), 317-338. doi: 10.1524/anly.2009.1029.

[10]

A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, New York, 1969.

[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), 831-844.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[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), 633-683.

[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), 15-25. doi: 10.1038/nrmicro2259.

[16]

T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[17]

S. Hittmeir and A. Jüngel, Cross-diffusion preventing blow up in the two-dimensional Keller-Segel model, SIAM J. Math. Anal., 43 (2011), 997-1022. doi: 10.1137/100813191.

[18]

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

[19]

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.

[20]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[21]

F. X. Kelly, K. J. Dapsis and D. A. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition, Microbial Ecology, 16 (1988), 115-131. doi: 10.1007/BF02018908.

[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), 673-730.

[23]

N. Mizoguchi and M. Winkler, Is aggregation a generic phenomenon in the two-dimensional Keller-Segel system?, Preprint.

[24]

J. D. Murray, Mathematical Biology, Second edition. Biomathematics, 19. Springer, Berlin, 1993. doi: 10.1007/b98869.

[25]

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

[26]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433.

[27]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.

[28]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469.

[29]

K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147. doi: 10.1007/s11538-009-9396-8.

[30]

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

[31]

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.

[32]

C. Stinner, J. 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.

[33]

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.

[34]

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.

[35]

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.

[36]

Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225.

[37]

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.

[38]

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.

[39]

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.

[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.

show all references

References:
[1]

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

[2]

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

[3]

H. Brézis and W. A. Strauss, Semilinear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565.

[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.

[5]

S. Cantrell, C. Cosner and S. Ruan, Spatial Ecology, Chapman & Hall/CRC, Mathematical and Computational Biology Series, Boca Raton/London/New York, 2010.

[6]

T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publ., Polish Acad. Sci., Warsaw, 81 (2008), 105-117. doi: 10.4064/bc81-0-7.

[7]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009.

[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), 553-580. doi: 10.1017/S0956792511000258.

[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), 317-338. doi: 10.1524/anly.2009.1029.

[10]

A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, New York, 1969.

[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), 831-844.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[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), 633-683.

[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), 15-25. doi: 10.1038/nrmicro2259.

[16]

T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[17]

S. Hittmeir and A. Jüngel, Cross-diffusion preventing blow up in the two-dimensional Keller-Segel model, SIAM J. Math. Anal., 43 (2011), 997-1022. doi: 10.1137/100813191.

[18]

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

[19]

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.

[20]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[21]

F. X. Kelly, K. J. Dapsis and D. A. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition, Microbial Ecology, 16 (1988), 115-131. doi: 10.1007/BF02018908.

[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), 673-730.

[23]

N. Mizoguchi and M. Winkler, Is aggregation a generic phenomenon in the two-dimensional Keller-Segel system?, Preprint.

[24]

J. D. Murray, Mathematical Biology, Second edition. Biomathematics, 19. Springer, Berlin, 1993. doi: 10.1007/b98869.

[25]

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

[26]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433.

[27]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.

[28]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469.

[29]

K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147. doi: 10.1007/s11538-009-9396-8.

[30]

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

[31]

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.

[32]

C. Stinner, J. 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.

[33]

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.

[34]

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.

[35]

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.

[36]

Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225.

[37]

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.

[38]

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.

[39]

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.

[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.

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