Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity

Kentarou Fujie; Tomomi Yokota

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 4, page 639-647
  • ISSN: 0862-7959

Abstract

top
This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function χ ( v ) and the growth term f ( u ) under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that 0 < χ ( v ) ≤ χ 0 / v k ( k ≥ 1 , χ 0 > 0 ) and λ 1 - μ 1 u ≤ f ( u ) ≤ λ 2 - μ 2 u ( λ 1 , λ 2 , μ 1 , μ 2 > 0 ) . It is shown that if χ 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.

How to cite

top

Fujie, Kentarou, and Yokota, Tomomi. "Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity." Mathematica Bohemica 139.4 (2014): 639-647. <http://eudml.org/doc/269838>.

@article{Fujie2014,
abstract = {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)\le \{\{\chi \}_0\}/\{v^k\}$$(k\ge 1$, $\{\chi \}_0>0)$ and $\lambda _1-\mu _1 u \le f(u)\le \lambda _2-\mu _2 u$$(\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.},
author = {Fujie, Kentarou, Yokota, Tomomi},
journal = {Mathematica Bohemica},
keywords = {chemotaxis; global existence; boundedness; chemotaxis; global existence; boundedness},
language = {eng},
number = {4},
pages = {639-647},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity},
url = {http://eudml.org/doc/269838},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Fujie, Kentarou
AU - Yokota, Tomomi
TI - Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 4
SP - 639
EP - 647
AB - 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)\le {{\chi }_0}/{v^k}$$(k\ge 1$, ${\chi }_0>0)$ and $\lambda _1-\mu _1 u \le f(u)\le \lambda _2-\mu _2 u$$(\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.
LA - eng
KW - chemotaxis; global existence; boundedness; chemotaxis; global existence; boundedness
UR - http://eudml.org/doc/269838
ER -

References

top
  1. Biler, P., Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl. 9 (1999), 347-359. (1999) Zbl0941.35009MR1690388
  2. Biler, P., Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715-743. (1998) Zbl0913.35021MR1657160
  3. Fujie, K., Winkler, M., Yokota, T., 10.1002/mma.3149, (to appear) in Math. Methods Appl. Sci. DOI:10.1002/mma.3149. DOI10.1002/mma.3149
  4. Hillen, T., Painter, K. J., 10.1007/s00285-008-0201-3, J. Math. Biol. 58 (2009), 183-217. (2009) Zbl1161.92003MR2448428DOI10.1007/s00285-008-0201-3
  5. Horstmann, D., Winkler, M., 10.1016/j.jde.2004.10.022, J. Differ. Equations 215 (2005), 52-107. (2005) Zbl1085.35065MR2146345DOI10.1016/j.jde.2004.10.022
  6. Keller, E. F., Segel, L. A., 10.1016/0022-5193(71)90051-8, J. Theor. Biol. 30 (1971), 235-248. (1971) Zbl1170.92308DOI10.1016/0022-5193(71)90051-8
  7. Keller, E. F., Segel, L. A., 10.1016/0022-5193(70)90092-5, J. Theor. Biol. 26 (1970), 399-415. (1970) Zbl1170.92306DOI10.1016/0022-5193(70)90092-5
  8. Manásevich, R., Phan, Q. H., Souplet, P., 10.1017/S095679251200040X, Eur. J. Appl. Math. 24 (2013), 273-296. (2013) Zbl1284.35445MR3031780DOI10.1017/S095679251200040X
  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. (2013) Zbl1261.35072MR3004526
  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. (1998) Zbl0902.35010MR1623326
  11. Negreanu, M., Tello, J. I., 10.1016/j.na.2012.12.004, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 80 (2013), 1-13. (2013) Zbl1260.35238MR3010749DOI10.1016/j.na.2012.12.004
  12. Osaki, K., Tsujikawa, T., Yagi, A., Mimura, M., 10.1016/S0362-546X(01)00815-X, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 51 (2002), 119-144. (2002) Zbl1005.35023MR1915744DOI10.1016/S0362-546X(01)00815-X
  13. Osaki, K., Yagi, A., Global existence for a chemotaxis-growth system in ℝ 2 , Adv. Math. Sci. Appl. 12 (2002), 587-606. (2002) MR1943982
  14. Othmer, H. G., Stevens, A., 10.1137/S0036139995288976, SIAM J. Appl. Math. 57 (1997), 1044-1081. (1997) Zbl0990.35128MR1462051DOI10.1137/S0036139995288976
  15. Sleeman, B. D., Levine, H. A., 10.1002/mma.212, Applied mathematical analysis in the last century Math. Methods Appl. Sci. 24 (2001), 405-426. (2001) Zbl0990.35014MR1821934DOI10.1002/mma.212
  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. (2011) Zbl1268.35072MR2833007
  17. Winkler, M., 10.1002/mma.1346, Math. Methods Appl. Sci. 34 (2011), 176-190. (2011) Zbl1291.92018MR2778870DOI10.1002/mma.1346
  18. Winkler, M., 10.1002/mana.200810838, Math. Nachr. 283 (2010), 1664-1673. (2010) Zbl1205.35037MR2759803DOI10.1002/mana.200810838
  19. Winkler, M., 10.1080/03605300903473426, Commun. Partial Differ. Equations 35 (2010), 1516-1537. (2010) Zbl1290.35139MR2754053DOI10.1080/03605300903473426

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.