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A hybrid method for stiff reaction–diffusion equations
Global bounded and unbounded solutions to a chemotaxis system with indirect signal production
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F–31062 Toulouse Cedex 9, France |
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.
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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