American Institute of Mathematical Sciences

Advanced
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

[1]

Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic & Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026
[2]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
[3]

Wolf-Jürgen Beyn, Thorsten Hüls. Continuation and collapse of homoclinic tangles. Journal of Computational Dynamics, 2014, 1 (1) : 71-109. doi: 10.3934/jcd.2014.1.71
[4]

Ming Gao, Jonathan J. Wylie, Qiang Zhang. Inelastic Collapse in a Corner. Communications on Pure & Applied Analysis, 2009, 8 (1) : 275-293. doi: 10.3934/cpaa.2009.8.275
[5]

Abdelkarim Kelleche, Nasser-Eddine Tatar. Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback. Evolution Equations & Control Theory, 2018, 7 (4) : 599-616. doi: 10.3934/eect.2018029
[6]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
[7]

Kota Kumazaki, Adrian Muntean. Local weak solvability of a moving boundary problem describing swelling along a halfline. Networks & Heterogeneous Media, 2019, 14 (3) : 445-469. doi: 10.3934/nhm.2019018
[8]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
[9]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225
[10]

Alina Chertock, Alexander Kurganov, Xuefeng Wang, Yaping Wu. On a chemotaxis model with saturated chemotactic flux. Kinetic & Related Models, 2012, 5 (1) : 51-95. doi: 10.3934/krm.2012.5.51
[11]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[12]

Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669
[13]

Fujun Zhou, Shangbin Cui. Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 929-943. doi: 10.3934/dcds.2008.21.929
[14]

Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185
[15]

Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 573-596. doi: 10.3934/dcdsb.2011.15.573
[16]

Giovanni Russo, Francis Filbet. Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics. Kinetic & Related Models, 2009, 2 (1) : 231-250. doi: 10.3934/krm.2009.2.231
[17]

Piotr Biler, Elio E. Espejo, Ignacio Guerra. Blowup in higher dimensional two species chemotactic systems. Communications on Pure & Applied Analysis, 2013, 12 (1) : 89-98. doi: 10.3934/cpaa.2013.12.89
[18]

Christopher M. Kribs-Zaleta, Christopher Mitchell. Modeling colony collapse disorder in honeybees as a contagion. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1275-1294. doi: 10.3934/mbe.2014.11.1275
[19]

Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809
[20]

A. Chauviere, T. Hillen, L. Preziosi. Modeling cell movement in anisotropic and heterogeneous network tissues. Networks & Heterogeneous Media, 2007, 2 (2) : 333-357. doi: 10.3934/nhm.2007.2.333

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences