MR2177120
Senba, Takasi
Takasi SENBA
Blowup Behavior of Radial Solutions to Jäger-Luckhaus System in High Dimensional Domains
Funkcialaj Ekvacioj. Serio Internacia
48
2005
247--271
http://fe.math.kobe-u.ac.jp/FE/FullPapers/48-2/48_247.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2177120
In this paper, we will consider the blowup and the time-global existence of radial solutions to a simplified system of the so called Keller-Segel model in a domain of three or more dimensional Euclidean space. In the two dimensional case, the blowup solutions have a delta function singularity at each blowup point (see [13]). However, in the three dimensional case, Herrero, Medina and Velázquez [7] show that there exist self-similar blowup solutions and that the solutions have a singularity which is different from a delta function singularity. In this paper, we will consider the blowup criterion and the blowup self-similar solutions in the three or more dimensional cases.
Chemotaxis, Keller-Segel model, Blowup, Self-similar solution.
Primary 35K55, 35K57, 92C17; Secondary 35B35, 25B40.
48-247
2005
Blowup Behavior of Radial Solutions to Jäger-Luckhaus System in High Dimensional Domains
SENBA, Takasi
Takasi SENBA
1
Biler, P.; Hilhorst, D.; Nadzieja, T.
Existence and nonexistence of solutions for a model gravitational of particles, II
Colloq. Math.
67
1994
297-308
MR1305221
2
Childress, S.
Chemotactic collapse in two dimensions
Lecture Notes in Biomath., 55, Springer, Berlin
1984
61-66
MR0813704
3
Childress, S.; Percus, J. K.
Nonlinear aspects of chemotaxis
Math. Biosci.
56
1981
217-237
MR0632161
4
Herrero, M. A.; Velázquez, J. J. L.
Singularity patterns in a chemotaxis model
Math. Ann.
306
1996
583-623
MR1415081
5
Herrero, M. A.; Velázquez, J. J. L.
A blow-up mechanism for a chemotaxis model
Ann. Scoula Norm. Sup. Pisa IV
35
1997
633-683
MR1627338
6
Herrero, M. A.; Medina, E.; Velázquez, J. J. L.
Finite-time aggregation into a single point in a reaction-diffusion system
Nonlinearity
10
1997
1739-1754
MR1483563
7
Herrero, M. A.; Medina, E.; Velázquez, J. J. L.
Self-similar blowup for a reaction-diffusion system
Journal of Computational and Applied Mathematics
97
1998
99-119
MR1651769
8
Jäger, W.; Luckhaus, S.
On explosions of solutions to a system of partial differential equations modelling chemotaxis
Trans. Amer. Math. Soc.
329
1992
819-824
MR1046835
9
Keller, E. F.; Segel, L. A.
Initiation of slime mold aggregation viewed as an instability
J. Theor. Biol.
26
1970
399-415
10
Nagai, T.
Blow-up of radially symmetric solutions to a chemotaxis system
Adv. Math. Sci. Appl.
5
1995
581-601
MR1361006
11
Nagai, T.
Blow-up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains
J. Inequal. Appl.
6
2001
37-55
MR1887324
12
Nanjundiah, V.
Chemotaxis, signal relaying, and aggregation morphology
J. Theor. Biol.
42
1973
63-105
13
Senba, T.; Suzuki, T.
Chemotactic collapse in a parabolic-elliptic system of mathematical biology
Adv. Differential Equations
6
2001
21-50
MR1799679
14
Senba, T.; Suzuki, T.
Parabolic system of chemotaxis: blowup in a finite and the infinite time
Met. Appl. Ana.
8
2001
349-367
MR1904534