American Institute of Mathematical Sciences

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March  2017, 22(2): 669-686. doi: 10.3934/dcdsb.2017032

A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

Jiashan Zheng 1,, and  Yifu Wang 2,

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

* Corresponding author

Received  February 2016 Revised  August 2016 Published  December 2016

The Neumann boundary value problem for the chemotaxis system generalizing the prototype
$\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} $
is considered in a smooth bounded convex domain
$Ω\subset \mathbb{R}^N(N≥2)$
, where
$D(u)≥ C_D(u+1)^{m-1}~~ \mbox{for all}~~ u≥0~~\mbox{with some}~~ m > 1~~\mbox{and}~~ C_D>0.$
If
$m >\frac{3N}{2N+2}$
 and suitable regularity assumptions on the initial data are given, the corresponding initial-boundary problem possesses a global classical solution. Our paper extends the results of Wang et al. ([24]), who showed the global existence of solutions in the cases
$m>2-\frac{6}{N+4}$
 (
$N≥3$
). If the flow of fluid is ignored, our result is consistent with and improves the result of Tao, Winkler ([15]) and Tao, Winkler ([17]), who proved the possibility of global boundedness, in the case that
$N=2,m>1$
 and
$N= 3$
,
$m > \frac{8}{7}$
, respectively.
Keywords: Chemotaxis system, global existence, chemoattractant.
Mathematics Subject Classification: 35K55, 35Q92, 35Q35, 92C1.
Citation: Jiashan Zheng, Yifu Wang. A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 669-686. doi: 10.3934/dcdsb.2017032
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Lorz, Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507.  Google Scholar

[11]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl.,vol.2, North-Holland, Amsterdam, 1977. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Lorz, Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507.  Google Scholar

[11]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl.,vol.2, North-Holland, Amsterdam, 1977. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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