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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

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. 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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