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The fast signal diffusion limit in nonlinear chemotaxis systems
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany |
$ n\geq2 $ |
$ \mathit{\Omega }\subset {\mathbb{R}}^n $ |
$ 0\not \equiv u_0\in W^{1, \infty}(\mathit{\Omega }) $ |
$ v_0\in W^{1, \infty}(\mathit{\Omega }) $ |
$ \varepsilon\in(0, 1) $ |
$ \begin{equation*} \begin{cases} u_t = \nabla\cdot((u+1)^{m-1}\nabla u)-\nabla \cdot(u\nabla v) \;\;\; & \text{in} \ \mathit{\Omega }\times\left(0, \infty \right), \\ \varepsilon v_t = \mathit{\Delta } v -v+u & \text{in} \ \mathit{\Omega }\times\left(0, \infty \right), \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 & \text{on} \ \partial\mathit{\Omega }\times\left(0, \infty \right), \\ u(\cdot, 0) = u_0, \ v(\cdot, 0) = v_0 & \text{in} \ \mathit{\Omega } \end{cases} \end{equation*} $ |
$ \varepsilon = 0 $ |
$ v $ |
$ m>1+\frac{n-2}{n} $ |
$ \left\|{{u_0}}\right\|_{L^p(\mathit{\Omega })} $ |
$ p\in[1, \infty] $ |
References:
[1] |
P. Biler,
Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
|
[2] |
M. Freitag,
Blow-up profiles and refined extensibility criteria in quasilinear Keller-Segel systems, Journal of Mathematical Analysis and Applications, 463 (2018), 964-988.
doi: 10.1016/j.jmaa.2018.03.052. |
[3] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[4] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 24 (1997), 633–683. |
[5] |
W. Jäger and S. Luckhaus,
On explosions to solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[6] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. theoret. Biol., 26 (1970), 399–415
doi: 10.1016/0022-5193(70)90092-5. |
[7] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, Amer. math. Soc., Providence RI., 1968. |
[8] |
J. Liu, L. Wang and Z. Zhou,
Positivity-preserving and asymptotic preserving method for 2d Keller-Segel equations, Mathematics of Computation, 87 (2018), 1165-1189.
doi: 10.1090/mcom/3250. |
[9] |
N. Mizoguchi and Ph. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Annales de l'Institut Henri Poincaré, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[10] |
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional parabolic KellerSegel system, J. Math. Pures Appl. (9), 100 (2013), 748–767.
doi: 10.1016/j.matpur.2013.01.020. |
[11] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[12] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
|
[13] |
T. Nagai, T. Senba and T. Suzuki,
Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30 (2000), 463-497.
doi: 10.32917/hmj/1206124609. |
[14] |
Y. Naito and T. Suzuki,
Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34.
doi: 10.4064/cm111-1-2. |
[15] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Eq., 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[16] |
T. Senba,
Type ii blowup of solutions to a simplified Keller-Segel system in two dimensions, Nonlinear Anal., 66 (2007), 1817-1839.
doi: 10.1016/j.na.2006.02.027. |
[17] |
Ph. Souplet and M. Winkler,
Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3, M. Commun. Math. Phys., 367 (2019), 665-681.
doi: 10.1007/s00220-018-3238-1. |
[18] |
T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005.
doi: 10.1007/0-8176-4436-9. |
[19] |
T. Suzuki,
Exclusion of boundary blowup for 2d chemotaxis system provided with Dirichlet boundary condition for the poisson part, J. Math. Pures Appl., 100 (2013), 347-367.
doi: 10.1016/j.matpur.2013.01.004. |
[20] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, Journal of Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[21] |
Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Preprint, 2018, arXiv: 1805.05263. |
[22] |
M. Winkler, Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel system, preprint. |
[23] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[24] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system., Journal de Mathématiques Pures et Appliquees, 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[25] |
J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
doi: 10.1017/CBO9781139171755.![]() ![]() ![]() |
show all references
References:
[1] |
P. Biler,
Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
|
[2] |
M. Freitag,
Blow-up profiles and refined extensibility criteria in quasilinear Keller-Segel systems, Journal of Mathematical Analysis and Applications, 463 (2018), 964-988.
doi: 10.1016/j.jmaa.2018.03.052. |
[3] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[4] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 24 (1997), 633–683. |
[5] |
W. Jäger and S. Luckhaus,
On explosions to solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[6] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. theoret. Biol., 26 (1970), 399–415
doi: 10.1016/0022-5193(70)90092-5. |
[7] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, Amer. math. Soc., Providence RI., 1968. |
[8] |
J. Liu, L. Wang and Z. Zhou,
Positivity-preserving and asymptotic preserving method for 2d Keller-Segel equations, Mathematics of Computation, 87 (2018), 1165-1189.
doi: 10.1090/mcom/3250. |
[9] |
N. Mizoguchi and Ph. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Annales de l'Institut Henri Poincaré, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[10] |
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional parabolic KellerSegel system, J. Math. Pures Appl. (9), 100 (2013), 748–767.
doi: 10.1016/j.matpur.2013.01.020. |
[11] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[12] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
|
[13] |
T. Nagai, T. Senba and T. Suzuki,
Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30 (2000), 463-497.
doi: 10.32917/hmj/1206124609. |
[14] |
Y. Naito and T. Suzuki,
Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34.
doi: 10.4064/cm111-1-2. |
[15] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Eq., 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[16] |
T. Senba,
Type ii blowup of solutions to a simplified Keller-Segel system in two dimensions, Nonlinear Anal., 66 (2007), 1817-1839.
doi: 10.1016/j.na.2006.02.027. |
[17] |
Ph. Souplet and M. Winkler,
Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3, M. Commun. Math. Phys., 367 (2019), 665-681.
doi: 10.1007/s00220-018-3238-1. |
[18] |
T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005.
doi: 10.1007/0-8176-4436-9. |
[19] |
T. Suzuki,
Exclusion of boundary blowup for 2d chemotaxis system provided with Dirichlet boundary condition for the poisson part, J. Math. Pures Appl., 100 (2013), 347-367.
doi: 10.1016/j.matpur.2013.01.004. |
[20] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, Journal of Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[21] |
Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Preprint, 2018, arXiv: 1805.05263. |
[22] |
M. Winkler, Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel system, preprint. |
[23] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[24] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system., Journal de Mathématiques Pures et Appliquees, 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[25] |
J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
doi: 10.1017/CBO9781139171755.![]() ![]() ![]() |
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