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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 and 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. [2] M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system, Sci. Math. Jpn., 59 (2004), 577-590. [3] M. Aida, M. Efendiev and A. Yagi, Quasilinear abstract parabolic evolution equations and exponential attractors, Osaka J. Math., 42 (2005), 101-132. [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. [5] A. V. Babin and M. I. Vishik, Attractors of evolution equations, "Nauka," Moscow, 1989; English translation, North-Holland, Amsterdam, 1992. [6] A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications," Applied Mathematical Sciences, 151, Springer-Verlag, Berlin, 2002. [7] H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [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. [9] E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 630-633. [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. [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. [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. [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. [14] D. D. Hai and A. Yagi, Numerical computations and pattern formation for chemotaxis-growth model, Sci. Math. Jpn., 70 (2009), 205-211. [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. [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. [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. [18] J. D. Murray, "Mathematical Biology II: Spacial Models and Biomedical Applications," 3$^rd$ edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. [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. [20] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. [21] M. Renardy and C. Rogers, "An Introduction to Partial Differential Equations," Springer, Berlin, 1992. [22] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2$^nd$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. [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. [24] A. Yagi, "Abstract Parabolic Evolution Equations and their Applications," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.

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##### References:
 [1] M. Aida, "Global Behaviour of Solutions and Pattern Formation for Chemotaxis-Growth Equations," (in Japanese), Ph.D thesis, Osaka University, 2003. [2] M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system, Sci. Math. Jpn., 59 (2004), 577-590. [3] M. Aida, M. Efendiev and A. Yagi, Quasilinear abstract parabolic evolution equations and exponential attractors, Osaka J. Math., 42 (2005), 101-132. [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. [5] A. V. Babin and M. I. Vishik, Attractors of evolution equations, "Nauka," Moscow, 1989; English translation, North-Holland, Amsterdam, 1992. [6] A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications," Applied Mathematical Sciences, 151, Springer-Verlag, Berlin, 2002. [7] H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [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. [9] E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 630-633. [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. [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. [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. [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. [14] D. D. Hai and A. Yagi, Numerical computations and pattern formation for chemotaxis-growth model, Sci. Math. Jpn., 70 (2009), 205-211. [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. [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. [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. [18] J. D. Murray, "Mathematical Biology II: Spacial Models and Biomedical Applications," 3$^rd$ edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. [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. [20] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. [21] M. Renardy and C. Rogers, "An Introduction to Partial Differential Equations," Springer, Berlin, 1992. [22] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2$^nd$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. [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. [24] A. Yagi, "Abstract Parabolic Evolution Equations and their Applications," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
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