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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 - 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
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