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March  2012, 11(2): 735-746. doi: 10.3934/cpaa.2012.11.735

The moving boundary problem in a chemotaxis model

Hua Chen 1, and  Shaohua Wu 1,

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received  November 2010 Revised  July 2011 Published  October 2011

In this paper, we prove the local existence and uniqueness of a moving boundary problem modeling chemotactic phenomena. We also get the explicit representative for the moving boundary and show the finite-time blow-up and chemotactic collapse for the solution of the problem.
Keywords: Moving boundary, blow-u, chemosensitive movement, chemotactic collapse..
Mathematics Subject Classification: 35K50, 35M10, 35R35, 92C1.
Citation: Hua Chen, Shaohua Wu. The moving boundary problem in a chemotaxis model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 735-746. doi: 10.3934/cpaa.2012.11.735
References:
[1]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.   Google Scholar

[2]

Miguel A. Herrero and Juan J. L. Velázquez, Singularity patterns in a chemotaxis model,, Math. Ann., 306 (1996), 583.   Google Scholar

[3]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819.   Google Scholar

[4]

V. Nanjundiah, Chemotaxis signal relaying and aggregation morphology,, J. Theor. Biol., 42 (1973), 63.   Google Scholar

[5]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.   Google Scholar

[6]

S. Childress, Chemotactic collapse in two dimension,, Lecture Notes in Biomath., 55 (1984), 61.   Google Scholar

show all references

References:
[1]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.   Google Scholar

[2]

Miguel A. Herrero and Juan J. L. Velázquez, Singularity patterns in a chemotaxis model,, Math. Ann., 306 (1996), 583.   Google Scholar

[3]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819.   Google Scholar

[4]

V. Nanjundiah, Chemotaxis signal relaying and aggregation morphology,, J. Theor. Biol., 42 (1973), 63.   Google Scholar

[5]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.   Google Scholar

[6]

S. Childress, Chemotactic collapse in two dimension,, Lecture Notes in Biomath., 55 (1984), 61.   Google Scholar

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