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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, 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, Osaka University, 2003. Google Scholar

[2]

M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system, Sci. Math. Jpn., 59 (2004), 577-590.  Google Scholar

[3]

M. Aida, M. Efendiev and A. Yagi, Quasilinear abstract parabolic evolution equations and exponential attractors, Osaka J. Math., 42 (2005), 101-132.  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), 453-474. doi: 10.1112/S0024610706023015.  Google Scholar

[5]

A. V. Babin and M. I. Vishik, Attractors of evolution equations, "Nauka," Moscow, 1989; English translation, North-Holland, Amsterdam, 1992.  Google Scholar

[6]

A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications," Applied Mathematical Sciences, 151, Springer-Verlag, Berlin, 2002.  Google Scholar

[7]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[8]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633. 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), 630-633. 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 Bénilan, Michel Cessenat, André Gervat, Alain Kavenoky and Hélène Lanchon, Translated from the French by Ian N. Sneddon, With a preface by Jean Teillac, Springer-Verlag, Berlin, 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, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 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), 167-181. 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), 647-673. 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), 205-211.  Google Scholar

[15]

E. F. Keller and L. A. Segel, Initiation of slime mould aggregation viewed as instability, J. Theor. Biol., 26 (1970), 399-415. 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), 499-543. 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 "Handbook of Differential Equations: Evolutionary Equations," Vol. 4 (eds. C. M. Dafermos and M. Pokorný), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 103-200.  Google Scholar

[18]

J. D. Murray, "Mathematical Biology II: Spacial Models and Biomedical Applications," 3$^rd$ edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 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), 119-144. 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, Springer-Verlag, New York, 1984.  Google Scholar

[21]

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

[22]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2$^nd$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 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), 2181-2189. doi: 10.1016/S0006-3495(95)80400-5.  Google Scholar

[24]

A. Yagi, "Abstract Parabolic Evolution Equations and their Applications," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 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, Osaka University, 2003. Google Scholar

[2]

M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system, Sci. Math. Jpn., 59 (2004), 577-590.  Google Scholar

[3]

M. Aida, M. Efendiev and A. Yagi, Quasilinear abstract parabolic evolution equations and exponential attractors, Osaka J. Math., 42 (2005), 101-132.  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), 453-474. doi: 10.1112/S0024610706023015.  Google Scholar

[5]

A. V. Babin and M. I. Vishik, Attractors of evolution equations, "Nauka," Moscow, 1989; English translation, North-Holland, Amsterdam, 1992.  Google Scholar

[6]

A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications," Applied Mathematical Sciences, 151, Springer-Verlag, Berlin, 2002.  Google Scholar

[7]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[8]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633. 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), 630-633. 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 Bénilan, Michel Cessenat, André Gervat, Alain Kavenoky and Hélène Lanchon, Translated from the French by Ian N. Sneddon, With a preface by Jean Teillac, Springer-Verlag, Berlin, 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, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 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), 167-181. 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), 647-673. 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), 205-211.  Google Scholar

[15]

E. F. Keller and L. A. Segel, Initiation of slime mould aggregation viewed as instability, J. Theor. Biol., 26 (1970), 399-415. 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), 499-543. 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 "Handbook of Differential Equations: Evolutionary Equations," Vol. 4 (eds. C. M. Dafermos and M. Pokorný), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 103-200.  Google Scholar

[18]

J. D. Murray, "Mathematical Biology II: Spacial Models and Biomedical Applications," 3$^rd$ edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 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), 119-144. 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, Springer-Verlag, New York, 1984.  Google Scholar

[21]

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

[22]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2$^nd$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 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), 2181-2189. doi: 10.1016/S0006-3495(95)80400-5.  Google Scholar

[24]

A. Yagi, "Abstract Parabolic Evolution Equations and their Applications," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.  Google Scholar

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