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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 |
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.
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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