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

Youshan Tao; Michael Winkler

doi:10.3934/dcdsb.2015.20.3165

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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

Youshan Tao ^1, and Michael Winkler ^2,

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

Abstract
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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.

Keywords: preventing blow-up., repulsion, attraction, Chemotaxis.

Mathematics Subject Classification: Primary: 35A01, 35B44, 35K57, 35Q92, 92C1.

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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