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The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system
A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant
1. | School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China |
2. | School of Mathematics and Statistics, Beijing, Institute of Technology, Beijing 100081, China |
$\left\{ \begin{array}{l}{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (u\nabla v),\;\;\;\;x \in \Omega ,t < 0,\\{v_t} = \Delta v - uv,\;\;\;\;\;x \in \Omega ,t < 0,\end{array} \right. \tag{KS}\label{KS} $ |
$Ω\subset \mathbb{R}^N(N≥2)$ |
$D(u)≥ C_D(u+1)^{m-1}~~ \mbox{for all}~~ u≥0~~\mbox{with some}~~ m > 1~~\mbox{and}~~ C_D>0.$ |
$m >\frac{3N}{2N+2}$ |
$m>2-\frac{6}{N+4}$ |
$N≥3$ |
$N=2,m>1$ |
$N= 3$ |
$m > \frac{8}{7}$ |
References:
[1] |
N. Bellomo, A. Belloquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
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M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Continuous Dynam. Systems, 33 (2013), 2271-2297.
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M. Chae, K. Kang and J. Lee,
Global Existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Part. Diff. Eqns., 39 (2014), 1205-1235.
doi: 10.1080/03605302.2013.852224. |
[4] |
R. Dal Passo, H. Garcke and G. Grün,
A fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.
doi: 10.1137/S0036141096306170. |
[5] |
R. Duan, A. Lorz and P. A. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Comm. Part. Diff. Eqns., 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[6] |
M. Di Francesco, A. Lorz and P. Markowich,
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.
doi: 10.3934/dcds.2010.28.1437. |
[7] |
J. Lankeit,
Eventual smoothness and asymptotics in a threedimensional chemotaxis system with logistic source, J. Diff. Eqns., 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[8] |
J. Liu and A. Lorz,
A coupled chemotaxis-fluid model: global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.
doi: 10.1016/j.anihpc.2011.04.005. |
[9] |
T. Li, A. Suen, C. Xue and M. Winkler,
Global small-data solutions of a two-dimensional chemotaxis system with rotational ux term, Math. Models Methods Appl. Sci., 25 (2015), 721-746.
doi: 10.1142/S0218202515500177. |
[10] |
A. Lorz,
Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[11] |
K. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
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G. Rosen,
Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 641-674.
doi: 10.1016/S0092-8240(78)80025-1. |
[13] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[14] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[15] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[16] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Diff. Eqns., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[17] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[18] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[19] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl.,vol.2, North-Holland, Amsterdam, 1977. |
[20] |
I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler and R. Goldstein,
Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.
|
[21] |
G. Viglialoro,
Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.
doi: 10.1016/j.jmaa.2016.02.069. |
[22] |
L. Wang, Y. Li and C. Mu,
Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 789-802.
doi: 10.3934/dcds.2014.34.789. |
[23] |
L. Wang, C. Mu and S. Zhou,
Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion, Z. Angew. Math. Phys., 65 (2014), 1137-1152.
doi: 10.1007/s00033-013-0375-4. |
[24] |
L. Wang, C. Mu, K. Lin and J. Zhao,
Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Z. Angew. Math. Phys., 66 (2015), 1633-1648.
doi: 10.1007/s00033-014-0491-9. |
[25] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 3159-3179.
doi: 10.1007/s00033-015-0557-3. |
[26] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[27] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Diff. Eqns., 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[28] |
M. Winkler,
Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.
doi: 10.1007/s00205-013-0678-9. |
[29] |
M. Winkler,
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.
doi: 10.1016/j.anihpc.2015.05.002. |
[30] |
M. Winkler,
Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calculus of Variations and Partial Diff. Eqns., 54 (2015), 3789-3828.
doi: 10.1007/s00526-015-0922-2. |
[31] |
Q. Zhang and X. Zheng,
Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.
doi: 10.1137/130936920. |
[32] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
show all references
References:
[1] |
N. Bellomo, A. Belloquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[2] |
M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Continuous Dynam. Systems, 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[3] |
M. Chae, K. Kang and J. Lee,
Global Existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Part. Diff. Eqns., 39 (2014), 1205-1235.
doi: 10.1080/03605302.2013.852224. |
[4] |
R. Dal Passo, H. Garcke and G. Grün,
A fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.
doi: 10.1137/S0036141096306170. |
[5] |
R. Duan, A. Lorz and P. A. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Comm. Part. Diff. Eqns., 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[6] |
M. Di Francesco, A. Lorz and P. Markowich,
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.
doi: 10.3934/dcds.2010.28.1437. |
[7] |
J. Lankeit,
Eventual smoothness and asymptotics in a threedimensional chemotaxis system with logistic source, J. Diff. Eqns., 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[8] |
J. Liu and A. Lorz,
A coupled chemotaxis-fluid model: global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.
doi: 10.1016/j.anihpc.2011.04.005. |
[9] |
T. Li, A. Suen, C. Xue and M. Winkler,
Global small-data solutions of a two-dimensional chemotaxis system with rotational ux term, Math. Models Methods Appl. Sci., 25 (2015), 721-746.
doi: 10.1142/S0218202515500177. |
[10] |
A. Lorz,
Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[11] |
K. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
[12] |
G. Rosen,
Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 641-674.
doi: 10.1016/S0092-8240(78)80025-1. |
[13] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[14] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[15] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[16] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Diff. Eqns., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[17] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[18] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[19] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl.,vol.2, North-Holland, Amsterdam, 1977. |
[20] |
I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler and R. Goldstein,
Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.
|
[21] |
G. Viglialoro,
Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.
doi: 10.1016/j.jmaa.2016.02.069. |
[22] |
L. Wang, Y. Li and C. Mu,
Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 789-802.
doi: 10.3934/dcds.2014.34.789. |
[23] |
L. Wang, C. Mu and S. Zhou,
Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion, Z. Angew. Math. Phys., 65 (2014), 1137-1152.
doi: 10.1007/s00033-013-0375-4. |
[24] |
L. Wang, C. Mu, K. Lin and J. Zhao,
Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Z. Angew. Math. Phys., 66 (2015), 1633-1648.
doi: 10.1007/s00033-014-0491-9. |
[25] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 3159-3179.
doi: 10.1007/s00033-015-0557-3. |
[26] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[27] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Diff. Eqns., 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[28] |
M. Winkler,
Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.
doi: 10.1007/s00205-013-0678-9. |
[29] |
M. Winkler,
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.
doi: 10.1016/j.anihpc.2015.05.002. |
[30] |
M. Winkler,
Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calculus of Variations and Partial Diff. Eqns., 54 (2015), 3789-3828.
doi: 10.1007/s00526-015-0922-2. |
[31] |
Q. Zhang and X. Zheng,
Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.
doi: 10.1137/130936920. |
[32] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
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