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

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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 and 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.
[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.

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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