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Strong solutions to compressible barotropic viscoelastic flow with vacuum
Global existence and steady states of a two competing species Keller--Segel chemotaxis model
1. | Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China, China, China |
References:
[1] |
J. Adler and W. Tso, Decision making in bacteria: Chemotactic response of Escherichia coli to conflic stimuli, Science, 184 (1974), 1292-1294.
doi: 10.1126/science.184.4143.1292. |
[2] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[3] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. |
[4] |
________, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[5] |
P. Biler, 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. |
[6] |
C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math, 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[7] |
_______, Sharp Condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbb R^ 2$, European J. Appl. Math, 24 (2013), 297-313.
doi: 10.1017/S0956792512000411. |
[8] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[9] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[10] |
________, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. |
[11] |
E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[12] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. |
[13] |
D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci, 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[14] |
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. |
[15] |
T. Kato, Functional Analysis, Springer Classics in Mathematics, 1995. |
[16] |
F. Kelly, K. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition, Microbial Ecology, 16 (1988), 115-131.
doi: 10.1007/BF02018908. |
[17] |
K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains, J. Differential Equations, 58 (1985), 15-21.
doi: 10.1016/0022-0396(85)90020-8. |
[18] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968, 648 pages. |
[19] |
D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microbial Ecology, 22 (1991), 175-185.
doi: 10.1007/BF02540222. |
[20] |
D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys. J., 40 (1982), 209-219.
doi: 10.1016/S0006-3495(82)84476-7. |
[21] |
D. Lauffenburger and P. Calcagno, Competition between two microbial populations in a nonmixed environment: Effect of cell random motility, Biotechnol Bioeng., 25 (1983), 2103-2125.
doi: 10.1002/bit.260250902. |
[22] |
P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[23] |
M. Ma, C. Ou and Z. A. Wang, Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math, 72 (2012), 740-766.
doi: 10.1137/110843964. |
[24] |
P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[25] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[26] |
G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. |
[27] |
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. |
[28] |
N. Tsang, R. Macnab and J. Koshland, Common mechanism for repellents and attractants in bacterial chemotaxis, Science, 181 (1973), 60-63.
doi: 10.1126/science.181.4094.60. |
[29] |
F. Verhagen and H. Laanbroek, Competition for ammonium between nitrifying and heterotrophic bacteria in dual energy-limited chemostats, Appl. and Enviro. Microbiology, 57 (1991), 3255-3263. |
[30] |
Q. Wang, C. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.
doi: 10.3934/dcds.2015.35.1239. |
[31] |
Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth,, preprint, ().
|
[32] |
X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math, 60 (2002), 505-531. |
[33] |
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. |
show all references
References:
[1] |
J. Adler and W. Tso, Decision making in bacteria: Chemotactic response of Escherichia coli to conflic stimuli, Science, 184 (1974), 1292-1294.
doi: 10.1126/science.184.4143.1292. |
[2] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[3] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. |
[4] |
________, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[5] |
P. Biler, 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. |
[6] |
C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math, 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[7] |
_______, Sharp Condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbb R^ 2$, European J. Appl. Math, 24 (2013), 297-313.
doi: 10.1017/S0956792512000411. |
[8] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[9] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[10] |
________, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. |
[11] |
E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[12] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. |
[13] |
D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci, 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[14] |
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. |
[15] |
T. Kato, Functional Analysis, Springer Classics in Mathematics, 1995. |
[16] |
F. Kelly, K. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition, Microbial Ecology, 16 (1988), 115-131.
doi: 10.1007/BF02018908. |
[17] |
K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains, J. Differential Equations, 58 (1985), 15-21.
doi: 10.1016/0022-0396(85)90020-8. |
[18] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968, 648 pages. |
[19] |
D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microbial Ecology, 22 (1991), 175-185.
doi: 10.1007/BF02540222. |
[20] |
D. Lauffenburger, R. Aris and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys. J., 40 (1982), 209-219.
doi: 10.1016/S0006-3495(82)84476-7. |
[21] |
D. Lauffenburger and P. Calcagno, Competition between two microbial populations in a nonmixed environment: Effect of cell random motility, Biotechnol Bioeng., 25 (1983), 2103-2125.
doi: 10.1002/bit.260250902. |
[22] |
P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[23] |
M. Ma, C. Ou and Z. A. Wang, Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math, 72 (2012), 740-766.
doi: 10.1137/110843964. |
[24] |
P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[25] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[26] |
G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. |
[27] |
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. |
[28] |
N. Tsang, R. Macnab and J. Koshland, Common mechanism for repellents and attractants in bacterial chemotaxis, Science, 181 (1973), 60-63.
doi: 10.1126/science.181.4094.60. |
[29] |
F. Verhagen and H. Laanbroek, Competition for ammonium between nitrifying and heterotrophic bacteria in dual energy-limited chemostats, Appl. and Enviro. Microbiology, 57 (1991), 3255-3263. |
[30] |
Q. Wang, C. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.
doi: 10.3934/dcds.2015.35.1239. |
[31] |
Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth,, preprint, ().
|
[32] |
X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math, 60 (2002), 505-531. |
[33] |
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. |
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