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December  2019, 24(12): 6419-6444. doi: 10.3934/dcdsb.2019145

Global bounded and unbounded solutions to a chemotaxis system with indirect signal production

Philippe Laurençot

Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F–31062 Toulouse Cedex 9, France

Received  October 2018 Revised  February 2019 Published  December 2019 Early access  July 2019

The well-posedness of a chemotaxis system with indirect signal production in a two-dimensional domain is shown, all solutions being global unlike for the classical Keller-Segel chemotaxis system. Nevertheless, there is a threshold value $ M_c $ of the mass of the first component which separates two different behaviours: solutions are bounded when the mass is below $ M_c $ while there are unbounded solutions starting from initial conditions having a mass exceeding $ M_c $. This result extends to arbitrary two-dimensional domains a previous result of Tao & Winkler (2017) obtained for radially symmetric solutions to a simplified version of the model in a ball and relies on a different approach involving a Liapunov functional.

Keywords: Chemotaxis system, global existence, boundedness, unbounded global solutions.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35M33, 35K10, 35Q92.
Citation: Philippe Laurençot. Global bounded and unbounded solutions to a chemotaxis system with indirect signal production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6419-6444. doi: 10.3934/dcdsb.2019145
References:
[1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math., 45 (1983), 225-254.  doi: 10.1007/BF02774019.

[2]

H. Amann, Highly degenerate quasilinear parabolic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18 (1991), 135-166. 

[3]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, In Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), volume 133 of Teubner-Texte Math., pages 9–126. Teubner, Stuttgart, 1993. doi: 10.1007/978-3-663-11336-2_1.

[4]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, volume 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. doi: 10.1007/978-3-0348-9221-6.

[5]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223.

[6]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. 

[7]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.

[8]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Math., 66 (1993), 319-334.  doi: 10.4064/cm-66-2-319-334.

[9]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on S2, J. Differential Geom., 27 (1988), 259-296.  doi: 10.4310/jdg/1214441783.

[10]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.

[11]

D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 399-423.  doi: 10.1007/PL00001455.

[12]

D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478.  doi: 10.1007/s002850100134.

[13]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. 

[14]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[15]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[16]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[17]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[18]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[19]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[20]

J. A. Powell, T. McMillen and P. White, Connecting a chemotactic model for mass attack to a rapid integro-difference emulation strategy, SIAM J. Appl. Math., 59 (1999), 547-572.  doi: 10.1137/S0036139996313459.

[21]

T. Senba and T. Suzuki, Blowup behavior of solutions to the rescaled Jäger-Luckhaus system, Adv. Differential Equations, 8 (2003), 787-820. 

[22]

S. Strohm, R. C. Tyson and J. A. Powell, Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data, Bull. Math. Biol., 75 (2013), 1778-1797.  doi: 10.1007/s11538-013-9868-8.

[23]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[24]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc. (JEMS), 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.

[25]

P. White and J. Powell, Spatial invasion of pine beetles into lodgepole forests: A numerical approach, SIAM J. Sci. Comput., 20 (1998), 164-184.  doi: 10.1137/S1064827596297550.

show all references

References:
[1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math., 45 (1983), 225-254.  doi: 10.1007/BF02774019.

[2]

H. Amann, Highly degenerate quasilinear parabolic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18 (1991), 135-166. 

[3]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, In Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), volume 133 of Teubner-Texte Math., pages 9–126. Teubner, Stuttgart, 1993. doi: 10.1007/978-3-663-11336-2_1.

[4]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, volume 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. doi: 10.1007/978-3-0348-9221-6.

[5]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223.

[6]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. 

[7]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5.

[8]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Math., 66 (1993), 319-334.  doi: 10.4064/cm-66-2-319-334.

[9]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on S2, J. Differential Geom., 27 (1988), 259-296.  doi: 10.4310/jdg/1214441783.

[10]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.

[11]

D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 399-423.  doi: 10.1007/PL00001455.

[12]

D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478.  doi: 10.1007/s002850100134.

[13]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. 

[14]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[15]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[16]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[17]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[18]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[19]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[20]

J. A. Powell, T. McMillen and P. White, Connecting a chemotactic model for mass attack to a rapid integro-difference emulation strategy, SIAM J. Appl. Math., 59 (1999), 547-572.  doi: 10.1137/S0036139996313459.

[21]

T. Senba and T. Suzuki, Blowup behavior of solutions to the rescaled Jäger-Luckhaus system, Adv. Differential Equations, 8 (2003), 787-820. 

[22]

S. Strohm, R. C. Tyson and J. A. Powell, Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data, Bull. Math. Biol., 75 (2013), 1778-1797.  doi: 10.1007/s11538-013-9868-8.

[23]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[24]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc. (JEMS), 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.

[25]

P. White and J. Powell, Spatial invasion of pine beetles into lodgepole forests: A numerical approach, SIAM J. Sci. Comput., 20 (1998), 164-184.  doi: 10.1137/S1064827596297550.

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