# American Institute of Mathematical Sciences

December  2013, 18(10): 2627-2646. doi: 10.3934/dcdsb.2013.18.2627

## Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation

 1 College of Liberal Arts and Sciences, Tokyo Medical and Dental University, 2-8-30 Kohnodai, Ichikawa, Chiba 272-0827, Japan 2 Department of Mathematical Sciences, School of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda 669-1337, Japan

Received  December 2012 Revised  July 2013 Published  October 2013

We construct the global bounded solutions and the attractors of a parabolic-parabolic chemotaxis-growth system in two- and three-dimensional smooth bounded domains. We derive new $L_p$ and $H^2$ uniform estimates for these solutions. We then construct the absorbing sets and the global attractors for the dynamical systems generated by the solutions. We also show the existence of exponential attractors by applying the existence theorem of Eden-Foias-Nicolaenko-Temam.
Citation: Etsushi Nakaguchi, Koichi Osaki. Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2627-2646. doi: 10.3934/dcdsb.2013.18.2627
##### References:
 [1] M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system,, J. London Math. Soc. (2), 74 (2006), 453. doi: 10.1112/S0024610706023015. Google Scholar [2] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217. doi: 10.1016/0025-5564(81)90055-9. Google Scholar [3] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5: Evolution Problems I,, With the collaboration of Michel Artola, (1992). doi: 10.1007/978-3-642-58090-1. Google Scholar [4] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Ensembles inertiels pour des équations d'évolution dissipatives, (French) [Inertial sets for dissipative evolution equations],, C. R. Acad. Sci. Paris, 310 (1990), 559. Google Scholar [5] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Research in Applied Mathematics, (1994). Google Scholar [6] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scoula Norm. Sup. Pisa Cl. Sci. IV, 24 (1997), 633. Google Scholar [7] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar [8] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103. Google Scholar [9] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51. Google Scholar [10] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar [11] 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. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [12] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [13] N. Kurata, K. Kuto, K. Osaki, T. Tsujikawa and T. Sakurai, Bifurcation phenomena of pattern solution to Mimura-Tsujikawa model in one dimension,, GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), 265. Google Scholar [14] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model,, Physica D, 241 (2012), 1629. doi: 10.1016/j.physd.2012.06.009. Google Scholar [15] 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 [16] J. D. Murray, Mathematical Biology, II: Spatial Models and Biomedical Applications, 3rd edition,, Springer-Verlag, (2003). Google Scholar [17] E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation,, Nonlinear Anal., 74 (2011), 286. doi: 10.1016/j.na.2010.08.044. Google Scholar [18] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581. Google Scholar [19] T. Nagai, Blow up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domain,, J. Inequal. Appl., 6 (2001), 37. doi: 10.1155/S1025583401000042. Google Scholar [20] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Anal., 51 (2002), 119. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar [21] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar [22] K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $\mathbbR^2$,, Adv. Math. Sci. Appl., 12 (2002), 587. Google Scholar [23] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D: Nonlinear Phenomena, 240 (2011), 363. doi: 10.1016/j.physd.2010.09.011. Google Scholar [24] T. Suzuki, Free Energy and Self-Interacting Particles,, Progress in Nonlinear Differential Equations and their Applications, (2005). doi: 10.1007/0-8176-4436-9. Google Scholar [25] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [26] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition,, Applied Mathematical Sciences, (1997). Google Scholar [27] M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations,, Bull. Math. Biol., 70 (2008), 1570. doi: 10.1007/s11538-008-9322-5. Google Scholar [28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd revised and enlarged edition,, Johann Ambrosius Barth Verlag, (1995). Google Scholar [29] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties,, J. Math. Anal. Appl., 348 (2008), 708. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar [30] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516. doi: 10.1080/03605300903473426. Google Scholar [31] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [32] A. Yagi, Abstract Parabolic Evolution Equations and Their Applications,, Springer Monographs in Mathematics. Springer-Verlag, (2010). doi: 10.1007/978-3-642-04631-5. Google Scholar

show all references

##### References:
 [1] M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system,, J. London Math. Soc. (2), 74 (2006), 453. doi: 10.1112/S0024610706023015. Google Scholar [2] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217. doi: 10.1016/0025-5564(81)90055-9. Google Scholar [3] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5: Evolution Problems I,, With the collaboration of Michel Artola, (1992). doi: 10.1007/978-3-642-58090-1. Google Scholar [4] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Ensembles inertiels pour des équations d'évolution dissipatives, (French) [Inertial sets for dissipative evolution equations],, C. R. Acad. Sci. Paris, 310 (1990), 559. Google Scholar [5] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Research in Applied Mathematics, (1994). Google Scholar [6] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scoula Norm. Sup. Pisa Cl. Sci. IV, 24 (1997), 633. Google Scholar [7] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar [8] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103. Google Scholar [9] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51. Google Scholar [10] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar [11] 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. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [12] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [13] N. Kurata, K. Kuto, K. Osaki, T. Tsujikawa and T. Sakurai, Bifurcation phenomena of pattern solution to Mimura-Tsujikawa model in one dimension,, GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), 265. Google Scholar [14] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model,, Physica D, 241 (2012), 1629. doi: 10.1016/j.physd.2012.06.009. Google Scholar [15] 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 [16] J. D. Murray, Mathematical Biology, II: Spatial Models and Biomedical Applications, 3rd edition,, Springer-Verlag, (2003). Google Scholar [17] E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation,, Nonlinear Anal., 74 (2011), 286. doi: 10.1016/j.na.2010.08.044. Google Scholar [18] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581. Google Scholar [19] T. Nagai, Blow up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domain,, J. Inequal. Appl., 6 (2001), 37. doi: 10.1155/S1025583401000042. Google Scholar [20] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Anal., 51 (2002), 119. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar [21] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar [22] K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $\mathbbR^2$,, Adv. Math. Sci. Appl., 12 (2002), 587. Google Scholar [23] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D: Nonlinear Phenomena, 240 (2011), 363. doi: 10.1016/j.physd.2010.09.011. Google Scholar [24] T. Suzuki, Free Energy and Self-Interacting Particles,, Progress in Nonlinear Differential Equations and their Applications, (2005). doi: 10.1007/0-8176-4436-9. Google Scholar [25] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [26] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition,, Applied Mathematical Sciences, (1997). Google Scholar [27] M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations,, Bull. Math. Biol., 70 (2008), 1570. doi: 10.1007/s11538-008-9322-5. Google Scholar [28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd revised and enlarged edition,, Johann Ambrosius Barth Verlag, (1995). Google Scholar [29] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties,, J. Math. Anal. Appl., 348 (2008), 708. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar [30] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516. doi: 10.1080/03605300903473426. Google Scholar [31] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [32] A. Yagi, Abstract Parabolic Evolution Equations and Their Applications,, Springer Monographs in Mathematics. Springer-Verlag, (2010). doi: 10.1007/978-3-642-04631-5. Google Scholar
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