• Previous Article
    Characterization of turing diffusion-driven instability on evolving domains
  • DCDS Home
  • This Issue
  • Next Article
    On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials
November  2012, 32(11): 3957-3974. doi: 10.3934/dcds.2012.32.3957

Longtime behavior of solutions to chemotaxis-proliferation model with three variables

1. 

Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan

2. 

Department of Applied Physics, Osaka University, Suita, Osaka, 565-0871

Received  May 2011 Revised  August 2011 Published  June 2012

In this paper, we construct a global solution to a mathematical model presented by Murray [18] and investigate longtime behavior of solution. For any initial profile, the solution is proven to tend to a homogeneous stationary solution as $t \rightarrow \infty$. This result is highly congruent with the prediction in [18] which is said that the solution would tend to zero as $t \rightarrow \infty$.
Citation: Doan Duy Hai, Atsushi Yagi. Longtime behavior of solutions to chemotaxis-proliferation model with three variables. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3957-3974. doi: 10.3934/dcds.2012.32.3957
References:
[1]

M. Aida, "Global Behaviour of Solutions and Pattern Formation for Chemotaxis-Growth Equations," (in Japanese),, Ph.D thesis, (2003).   Google Scholar

[2]

M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system,, Sci. Math. Jpn., 59 (2004), 577.   Google Scholar

[3]

M. Aida, M. Efendiev and A. Yagi, Quasilinear abstract parabolic evolution equations and exponential attractors,, Osaka J. Math., 42 (2005), 101.   Google Scholar

[4]

M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of attractor dimension for chemotaxis growth system,, J. London Math. Soc. (2), 74 (2006), 453.  doi: 10.1112/S0024610706023015.  Google Scholar

[5]

A. V. Babin and M. I. Vishik, Attractors of evolution equations,, Nauka, (1989).   Google Scholar

[6]

A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications,", Applied Mathematical Sciences, 151 (2002).   Google Scholar

[7]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Math\'ematiques Appliqu\'ees pour la Ma\^itrise, (1983).   Google Scholar

[8]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630.  doi: 10.1038/349630a0.  Google Scholar

[9]

E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 376 (1995), 630.   Google Scholar

[10]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Physical Origins and Classical Methods,", With the collaboration of Philippe B\'enilan, (1990).   Google Scholar

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations,", RAM: Research in Applied Mathematics, 37 (1994).   Google Scholar

[12]

M. Efendiev and A. Yagi, Continuous dependence on a parameter of exponential attractors for chemotaxis-growth system,, J. Math. Soc. Japan, 57 (2005), 167.  doi: 10.2969/jmsj/1160745820.  Google Scholar

[13]

M. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors for non-autonomous dissipative systems,, J. Math. Soc. Japan, 63 (2011), 647.  doi: 10.2969/jmsj/06320647.  Google Scholar

[14]

D. D. Hai and A. Yagi, Numerical computations and pattern formation for chemotaxis-growth model,, Sci. Math. Jpn., 70 (2009), 205.   Google Scholar

[15]

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

[16]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth,, Physica A, 230 (1996), 499.  doi: 10.1016/0378-4371(96)00051-9.  Google Scholar

[17]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103.   Google Scholar

[18]

J. D. Murray, "Mathematical Biology II: Spacial Models and Biomedical Applications,", 3$^{rd}$ edition, 18 (2003).   Google Scholar

[19]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Analysis, 51 (2002), 119.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[20]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Corrected reprint of the 1967 original, (1967).   Google Scholar

[21]

M. Renardy and C. Rogers, "An Introduction to Partial Differential Equations,", Springer, (1992).   Google Scholar

[22]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, 68 (1997).   Google Scholar

[23]

D. E. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. O. Budrene and H. C. Berg, Spatio-temporal patterns generated by Salmonella typhimurium,, Biophysical J., 68 (1995), 2181.  doi: 10.1016/S0006-3495(95)80400-5.  Google Scholar

[24]

A. Yagi, "Abstract Parabolic Evolution Equations and their Applications,", Springer Monographs in Mathematics, (2010).   Google Scholar

show all references

References:
[1]

M. Aida, "Global Behaviour of Solutions and Pattern Formation for Chemotaxis-Growth Equations," (in Japanese),, Ph.D thesis, (2003).   Google Scholar

[2]

M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system,, Sci. Math. Jpn., 59 (2004), 577.   Google Scholar

[3]

M. Aida, M. Efendiev and A. Yagi, Quasilinear abstract parabolic evolution equations and exponential attractors,, Osaka J. Math., 42 (2005), 101.   Google Scholar

[4]

M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of attractor dimension for chemotaxis growth system,, J. London Math. Soc. (2), 74 (2006), 453.  doi: 10.1112/S0024610706023015.  Google Scholar

[5]

A. V. Babin and M. I. Vishik, Attractors of evolution equations,, Nauka, (1989).   Google Scholar

[6]

A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications,", Applied Mathematical Sciences, 151 (2002).   Google Scholar

[7]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Math\'ematiques Appliqu\'ees pour la Ma\^itrise, (1983).   Google Scholar

[8]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630.  doi: 10.1038/349630a0.  Google Scholar

[9]

E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 376 (1995), 630.   Google Scholar

[10]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Physical Origins and Classical Methods,", With the collaboration of Philippe B\'enilan, (1990).   Google Scholar

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations,", RAM: Research in Applied Mathematics, 37 (1994).   Google Scholar

[12]

M. Efendiev and A. Yagi, Continuous dependence on a parameter of exponential attractors for chemotaxis-growth system,, J. Math. Soc. Japan, 57 (2005), 167.  doi: 10.2969/jmsj/1160745820.  Google Scholar

[13]

M. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors for non-autonomous dissipative systems,, J. Math. Soc. Japan, 63 (2011), 647.  doi: 10.2969/jmsj/06320647.  Google Scholar

[14]

D. D. Hai and A. Yagi, Numerical computations and pattern formation for chemotaxis-growth model,, Sci. Math. Jpn., 70 (2009), 205.   Google Scholar

[15]

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

[16]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth,, Physica A, 230 (1996), 499.  doi: 10.1016/0378-4371(96)00051-9.  Google Scholar

[17]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103.   Google Scholar

[18]

J. D. Murray, "Mathematical Biology II: Spacial Models and Biomedical Applications,", 3$^{rd}$ edition, 18 (2003).   Google Scholar

[19]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Analysis, 51 (2002), 119.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[20]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Corrected reprint of the 1967 original, (1967).   Google Scholar

[21]

M. Renardy and C. Rogers, "An Introduction to Partial Differential Equations,", Springer, (1992).   Google Scholar

[22]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, 68 (1997).   Google Scholar

[23]

D. E. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. O. Budrene and H. C. Berg, Spatio-temporal patterns generated by Salmonella typhimurium,, Biophysical J., 68 (1995), 2181.  doi: 10.1016/S0006-3495(95)80400-5.  Google Scholar

[24]

A. Yagi, "Abstract Parabolic Evolution Equations and their Applications,", Springer Monographs in Mathematics, (2010).   Google Scholar

[1]

Messoud A. Efendiev, Sergey Zelik, Hermann J. Eberl. Existence and longtime behavior of a biofilm model. Communications on Pure & Applied Analysis, 2009, 8 (2) : 509-531. doi: 10.3934/cpaa.2009.8.509

[2]

M. Grasselli, Vittorino Pata. Longtime behavior of a homogenized model in viscoelastodynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 339-358. doi: 10.3934/dcds.1998.4.339

[3]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1

[4]

Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150

[5]

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

[6]

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

[7]

M. Grasselli, Vittorino Pata, Giovanni Prouse. Longtime behavior of a viscoelastic Timoshenko beam. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 337-348. doi: 10.3934/dcds.2004.10.337

[8]

Hua Zhong, Chunlai Mu, Ke Lin. Global weak solution and boundedness in a three-dimensional competing chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3875-3898. doi: 10.3934/dcds.2018168

[9]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145

[10]

Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084

[11]

Yu-Hsien Chang, Guo-Chin Jau. The behavior of the solution for a mathematical model for analysis of the cell cycle. Communications on Pure & Applied Analysis, 2006, 5 (4) : 779-792. doi: 10.3934/cpaa.2006.5.779

[12]

Zhijian Yang, Ke Li. Longtime dynamics for an elastic waveguide model. Conference Publications, 2013, 2013 (special) : 797-806. doi: 10.3934/proc.2013.2013.797

[13]

Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885

[14]

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

[15]

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, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064

[16]

Zhi-An Wang, Kun Zhao. Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3027-3046. doi: 10.3934/cpaa.2013.12.3027

[17]

Youshan Tao. Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2705-2722. doi: 10.3934/dcdsb.2013.18.2705

[18]

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

[19]

Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]