Boundedness in a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source

Ji Liu; Jia-Shan Zheng

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 1117-1136
  • ISSN: 0011-4642

Abstract

top
We study a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. By establishing proper a priori estimates we prove that, with both the diffusion function and the chemotaxis sensitivity function being positive, the corresponding initial boundary value problem admits a unique global classical solution which is uniformly bounded. The result of this paper is a generalization of that of Cao (2014).

How to cite

top

Liu, Ji, and Zheng, Jia-Shan. "Boundedness in a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source." Czechoslovak Mathematical Journal 65.4 (2015): 1117-1136. <http://eudml.org/doc/276089>.

@article{Liu2015,
abstract = {We study a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. By establishing proper a priori estimates we prove that, with both the diffusion function and the chemotaxis sensitivity function being positive, the corresponding initial boundary value problem admits a unique global classical solution which is uniformly bounded. The result of this paper is a generalization of that of Cao (2014).},
author = {Liu, Ji, Zheng, Jia-Shan},
journal = {Czechoslovak Mathematical Journal},
keywords = {boundedness; chemotaxis; nonlinear logistic source},
language = {eng},
number = {4},
pages = {1117-1136},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Boundedness in a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source},
url = {http://eudml.org/doc/276089},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Liu, Ji
AU - Zheng, Jia-Shan
TI - Boundedness in a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 1117
EP - 1136
AB - We study a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. By establishing proper a priori estimates we prove that, with both the diffusion function and the chemotaxis sensitivity function being positive, the corresponding initial boundary value problem admits a unique global classical solution which is uniformly bounded. The result of this paper is a generalization of that of Cao (2014).
LA - eng
KW - boundedness; chemotaxis; nonlinear logistic source
UR - http://eudml.org/doc/276089
ER -

References

top
  1. Cao, X., 10.1016/j.jmaa.2013.10.061, J. Math. Anal. Appl. 412 (2014), 181-188. (2014) MR3145792DOI10.1016/j.jmaa.2013.10.061
  2. Cieślak, T., Stinner, C., 10.1007/s10440-013-9832-5, Acta Appl. Math. 129 (2014), 135-146. (2014) Zbl1295.35123MR3152080DOI10.1007/s10440-013-9832-5
  3. Cieślak, T., Stinner, C., 10.1016/j.jde.2012.01.045, J. Differ. Equations 252 (2012), 5832-5851. (2012) Zbl1252.35087MR2902137DOI10.1016/j.jde.2012.01.045
  4. Herrero, M. A., Velázquez, J. J. L., A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci. 4. 24 (1997), 633-683. (1997) Zbl0904.35037MR1627338
  5. Horstmann, D., Wang, G., 10.1017/S0956792501004363, Eur. J. Appl. Math. 12 (2001), 159-177. (2001) Zbl1017.92006MR1931303DOI10.1017/S0956792501004363
  6. 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
  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. Nagai, T., Senba, T., Yoshida, K., Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj. Ser. Int. 40 (1997), 411-433. (1997) Zbl0901.35104MR1610709
  9. Osaki, K., Yagi, A., Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkc. Ekvacioj. Ser. Int. 44 (2001), 441-469. (2001) Zbl1145.37337MR1893940
  10. Painter, K. J., Hillen, T., Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q. 10 (2002), 501-543. (2002) Zbl1057.92013MR2052525
  11. Tao, Y., Winkler, M., 10.1016/j.jde.2011.08.019, J. Differ. Equations 252 (2012), 692-715. (2012) MR2852223DOI10.1016/j.jde.2011.08.019
  12. Winkler, M., 10.1016/j.matpur.2013.01.020, J. Math. Pures Appl. 100 (2013), 748-767. (2013) Zbl1326.35053MR3115832DOI10.1016/j.matpur.2013.01.020
  13. Winkler, M., 10.1016/j.jde.2010.02.008, J. Differ. Equations 248 (2010), 2889-2905. (2010) Zbl1190.92004MR2644137DOI10.1016/j.jde.2010.02.008
  14. 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.