doi: 10.3934/dcdss.2020009

Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity

1. 

Institute of Mathematics, Polish Academy of Sciences, Warsaw, 00-656, Poland

2. 

Department of Mathematics, Tokyo University of Science, Tokyo, 162-8601, Japan

* Corresponding author: Tomasz Cieślak

Received  May 2017 Revised  January 2018 Published  January 2019

The paper should be viewed as complement of an earlier result in [10]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear diffusion of the form $ 1/u $ all the solutions, independently on the magnitude of initial mass, stay bounded. However, the argument presented in [10] deals with the Jäger-Luckhaus type system. And is very sensitive to this restriction. Namely, the change of variables introduced in [10], being a main step of the method, works only for the Jäger-Luckhaus modification. It does not seem to be applicable in the usual version of the parabolic-elliptic Keller-Segel system. The present paper fulfils this gap and deals with the case of the usual parabolic-elliptic version. To handle it we establish a new Lyapunov-like functional (it is related to what was done in [10]), which leads to global existence of the initial-boundary value problem for any initial mass.

Citation: Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020009
References:
[1]

N. BellomoA. BellouquidY. 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]

B. Bieganowski, T. Cieślak, K. Fujie and T. Senba, Boundedness of solutions to the critical fully parabolic quasilinear one-dimensional Keller-Segel system, Math. Nachr., to appear.

[3]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5.

[4]

A. BlanchetJ. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7.

[5]

H. Brézis and W. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565.

[6]

J. BurczakT. Cieślak and C. Morales-Rodrigo, Global existence vs. blowup in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal., 75 (2012), 5215-5228. doi: 10.1016/j.na.2012.04.038.

[7]

J. Burczak and R. Granero-Belinchon, Critical Keller-Segel meets Burgers on S1:large-time smooth solutions, Nonlinearity, 29 (2016), 3810-3836. doi: 10.1088/0951-7715/29/12/3810.

[8]

T. Cieślak and K. Fujie, No critical nonlinear diffusion in 1D quasilinear fully parabolic chemotaxis system, Proc. Amer. Math. Soc., 146 (2018), 2529-2540. doi: 10.1090/proc/13939.

[9]

T. Cieślak and Ph. Laurençot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system, C. R. Math. Acad. Sci. Paris, 347 (2009), 237-242. doi: 10.1016/j.crma.2009.01.016.

[10]

T. Cieślak and Ph. Laurençot, Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system, Discrete Contin. Dyn. Syst., 26 (2010), 417-430. doi: 10.3934/dcds.2010.26.417.

[11]

T. Cieślak and Ph. Laurençot, Global existence vs. blowup for the one dimensional quasilinear Smoluchowski-Poisson system, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 95-109. doi: 10.1007/978-3-0348-0075-4_6.

[12]

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.

[13]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

[14]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[15]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433.

[16]

E. Nasreddine, Global existence of solutions to a parabolic-elliptic Keller-Segel system with critical degenerate diffusion, J. Math. Anal. Appl., 417 (2014), 144-163. doi: 10.1016/j.jmaa.2014.02.069.

[17]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.

[18]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144.

show all references

References:
[1]

N. BellomoA. BellouquidY. 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]

B. Bieganowski, T. Cieślak, K. Fujie and T. Senba, Boundedness of solutions to the critical fully parabolic quasilinear one-dimensional Keller-Segel system, Math. Nachr., to appear.

[3]

P. BilerW. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5.

[4]

A. BlanchetJ. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7.

[5]

H. Brézis and W. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565.

[6]

J. BurczakT. Cieślak and C. Morales-Rodrigo, Global existence vs. blowup in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal., 75 (2012), 5215-5228. doi: 10.1016/j.na.2012.04.038.

[7]

J. Burczak and R. Granero-Belinchon, Critical Keller-Segel meets Burgers on S1:large-time smooth solutions, Nonlinearity, 29 (2016), 3810-3836. doi: 10.1088/0951-7715/29/12/3810.

[8]

T. Cieślak and K. Fujie, No critical nonlinear diffusion in 1D quasilinear fully parabolic chemotaxis system, Proc. Amer. Math. Soc., 146 (2018), 2529-2540. doi: 10.1090/proc/13939.

[9]

T. Cieślak and Ph. Laurençot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system, C. R. Math. Acad. Sci. Paris, 347 (2009), 237-242. doi: 10.1016/j.crma.2009.01.016.

[10]

T. Cieślak and Ph. Laurençot, Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system, Discrete Contin. Dyn. Syst., 26 (2010), 417-430. doi: 10.3934/dcds.2010.26.417.

[11]

T. Cieślak and Ph. Laurençot, Global existence vs. blowup for the one dimensional quasilinear Smoluchowski-Poisson system, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 95-109. doi: 10.1007/978-3-0348-0075-4_6.

[12]

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.

[13]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

[14]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[15]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433.

[16]

E. Nasreddine, Global existence of solutions to a parabolic-elliptic Keller-Segel system with critical degenerate diffusion, J. Math. Anal. Appl., 417 (2014), 144-163. doi: 10.1016/j.jmaa.2014.02.069.

[17]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.

[18]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144.

[1]

T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125

[2]

Georg Hetzer. Global existence for a functional reaction-diffusion problem from climate modeling. Conference Publications, 2011, 2011 (Special) : 660-671. doi: 10.3934/proc.2011.2011.660

[3]

Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050

[4]

Radek Erban, Hyung Ju Hwang. Global existence results for complex hyperbolic models of bacterial chemotaxis. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1239-1260. doi: 10.3934/dcdsb.2006.6.1239

[5]

Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262

[6]

Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180

[7]

Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220

[8]

Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061

[9]

Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 777-807. doi: 10.3934/krm.2015.8.777

[10]

Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437

[11]

Nikolaos Bournaveas, Vincent Calvez. Global existence for the kinetic chemotaxis model without pointwise memory effects, and including internal variables. Kinetic & Related Models, 2008, 1 (1) : 29-48. doi: 10.3934/krm.2008.1.29

[12]

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

[13]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

[14]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2019064

[15]

Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324

[16]

Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145

[17]

Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463

[18]

Peter Giesl, Sigurdur Hafstein. Existence of piecewise linear Lyapunov functions in arbitrary dimensions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3539-3565. doi: 10.3934/dcds.2012.32.3539

[19]

Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125

[20]

Xin Lai, Xinfu Chen, Mingxin Wang, Cong Qin, Yajing Zhang. Existence, uniqueness, and stability of bubble solutions of a chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 805-832. doi: 10.3934/dcds.2016.36.805

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (13)
  • HTML views (314)
  • Cited by (0)

Other articles
by authors

[Back to Top]