| Title:
|
Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity (English) |
| Author:
|
Fujie, Kentarou |
| Author:
|
Yokota, Tomomi |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
639-647 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function $\chi (v)$ and the growth term $f(u)$ under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that $0< \chi (v)\leq {{\chi }_0}/{v^k}$ $(k\geq 1$, ${\chi }_0>0)$ and $\lambda _1-\mu _1 u \leq f(u)\leq \lambda _2-\mu _2 u$ $(\lambda _1,\lambda _2,\mu _1,\mu _2>0)$. It is shown that if $\chi _0$ is sufficiently small, then the system has a unique global-in-time classical solution that is uniformly bounded. This boundedness result is a generalization of a recent result by K. Fujie, M. Winkler, T. Yokota. (English) |
| Keyword:
|
chemotaxis |
| Keyword:
|
global existence |
| Keyword:
|
boundedness |
| MSC:
|
35A01 |
| MSC:
|
35B40 |
| MSC:
|
35B45 |
| MSC:
|
35K60 |
| MSC:
|
35M33 |
| MSC:
|
92C17 |
| idZBL:
|
Zbl 06433687 |
| idMR:
|
MR3306853 |
| DOI:
|
10.21136/MB.2014.144140 |
| . |
| Date available:
|
2015-02-04T09:21:56Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144140 |
| . |
| Reference:
|
[1] Biler, P.: Global solutions to some parabolic-elliptic systems of chemotaxis.Adv. Math. Sci. Appl. 9 (1999), 347-359. Zbl 0941.35009, MR 1690388 |
| Reference:
|
[2] Biler, P.: Local and global solvability of some parabolic systems modelling chemotaxis.Adv. Math. Sci. Appl. 8 (1998), 715-743. Zbl 0913.35021, MR 1657160 |
| Reference:
|
[3] Fujie, K., Winkler, M., Yokota, T.: Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity.(to appear) in Math. Methods Appl. Sci. DOI:10.1002/mma.3149. 10.1002/mma.3149 |
| Reference:
|
[4] Hillen, T., Painter, K. J.: A user's guide to PDE models for chemotaxis.J. Math. Biol. 58 (2009), 183-217. Zbl 1161.92003, MR 2448428, 10.1007/s00285-008-0201-3 |
| Reference:
|
[5] Horstmann, D., Winkler, M.: Boundedness vs. blow-up in a chemotaxis system.J. Differ. Equations 215 (2005), 52-107. Zbl 1085.35065, MR 2146345, 10.1016/j.jde.2004.10.022 |
| Reference:
|
[6] Keller, E. F., Segel, L. A.: Traveling bands of chemotactic bacteria: A theoretical analysis.J. Theor. Biol. 30 (1971), 235-248. Zbl 1170.92308, 10.1016/0022-5193(71)90051-8 |
| Reference:
|
[7] Keller, E. F., Segel, L. A.: Initiation of slime mold aggregation viewed as an instability.J. Theor. Biol. 26 (1970), 399-415. Zbl 1170.92306, 10.1016/0022-5193(70)90092-5 |
| Reference:
|
[8] Manásevich, R., Phan, Q. H., Souplet, P.: Global existence of solutions for a chemotaxis-type system arising in crime modelling.Eur. J. Appl. Math. 24 (2013), 273-296. Zbl 1284.35445, MR 3031780, 10.1017/S095679251200040X |
| Reference:
|
[9] Mu, C., Wang, L., Zheng, P., Zhang, Q.: Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system.Nonlinear Anal., Real World Appl. 14 (2013), 1634-1642. Zbl 1261.35072, MR 3004526 |
| Reference:
|
[10] Nagai, T., Senba, T.: Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis.Adv. Math. Sci. Appl. 8 (1998), 145-156. Zbl 0902.35010, MR 1623326 |
| Reference:
|
[11] Negreanu, M., Tello, J. I.: On a parabolic-elliptic chemotactic system with non-constant chemotactic sensitivity.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 80 (2013), 1-13. Zbl 1260.35238, MR 3010749, 10.1016/j.na.2012.12.004 |
| Reference:
|
[12] Osaki, K., Tsujikawa, T., Yagi, A., Mimura, M.: Exponential attractor for a chemotaxis- growth system of equations.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 51 (2002), 119-144. Zbl 1005.35023, MR 1915744, 10.1016/S0362-546X(01)00815-X |
| Reference:
|
[13] Osaki, K., Yagi, A.: Global existence for a chemotaxis-growth system in $\mathbb R^2$.Adv. Math. Sci. Appl. 12 (2002), 587-606. MR 1943982 |
| Reference:
|
[14] Othmer, H. G., Stevens, A.: Aggregation, blowup, and collapse: The ABC's of taxis in reinforced random walks.SIAM J. Appl. Math. 57 (1997), 1044-1081. Zbl 0990.35128, MR 1462051, 10.1137/S0036139995288976 |
| Reference:
|
[15] Sleeman, B. D., Levine, H. A.: Partial differential equations of chemotaxis and angiogenesis.Applied mathematical analysis in the last century Math. Methods Appl. Sci. 24 (2001), 405-426. Zbl 0990.35014, MR 1821934, 10.1002/mma.212 |
| Reference:
|
[16] Stinner, C., Winkler, M.: Global weak solutions in a chemotaxis system with large singular sensitivity.Nonlinear Anal., Real World Appl. 12 (2011), 3727-3740. Zbl 1268.35072, MR 2833007 |
| Reference:
|
[17] Winkler, M.: Global solutions in a fully parabolic chemotaxis system with singular sensitivity.Math. Methods Appl. Sci. 34 (2011), 176-190. Zbl 1291.92018, MR 2778870, 10.1002/mma.1346 |
| Reference:
|
[18] Winkler, M.: Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity.Math. Nachr. 283 (2010), 1664-1673. Zbl 1205.35037, MR 2759803, 10.1002/mana.200810838 |
| Reference:
|
[19] Winkler, M.: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source.Commun. Partial Differ. Equations 35 (2010), 1516-1537. Zbl 1290.35139, MR 2754053, 10.1080/03605300903473426 |
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