The one-dimensional chemotaxis model: Global existence and asymptotic profile

Mathematical Methods in the Applied Sciences

Volumn 27, Issue 15, 2004, Pages 1783-1801

  Hillen, Thomas ^a   Potapov, Alex ^a  

^a UNIVERSITY OF ALBERTA   (Canada)

Author keywords

Advection diffusion equations; Chemotaxis; Global existence

Indexed keywords

BOUNDARY CONDITIONS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; NUMERICAL ANALYSIS; ONE DIMENSIONAL; PROBLEM SOLVING; THEOREM PROVING;

ADVECTION-DIFFUSION EQUATIONS; CHEMOTAXIS; GLOBAL EXISTENCE;

ASYMPTOTIC STABILITY;

EID: 4744373195     PISSN: 01704214     EISSN: None     Source Type: Journal    
DOI: 10.1002/mma.569     Document Type: Conference Paper

References (20)

1. Patlak CS. Random walk with persistence and external bias. Bulletin of Mathematical Biophysics 1953; 15: 311-338.
2. Keller EF, Segel LA. Model for chemotaxis. Journal of Theoretical Biology 1971; 30:225-234.
3. Jäger W, Luckhaus S. On explosions of solutions to a system of partial differential equations modelling chemotaxis. Transactions of AMS 1992; 329:819-824.
4. Nagai T, Senba T, Yoshida K. Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis. Funkcialaj Ekvacioj 1997; 40(3):411-433.
5. Herrero M, Velázquez J. Chemotactic collapse for the Keller-Segel model. Journal of Mathematical Biology 1996; 35:177-194.
6. Horstmann D, Wang G. Blow-up in a chemotaxis model without symmetry assumptions. European Journal of Applied Mathematics 2001; 12:159-177.
7. Horstmann D. From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I. Jahresberichte der DMV 2004; 105(3):103-165.
8. Horstmann D. From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II. Jahresberichte der DMV 2003, in print.
9. Childress S, Percus JK. Nonlinear aspects of chemotaxis. Mathematical Biosciences 1981; 56:217-237.
10. Nagai T, Nakaki T. Stability of constant steady states and existence of unbounded solutions in time to a reaction-diffusion equation modelling chemotaxis. Preprint, 2004.
11. Levine HA, Sleeman BD. A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM Journal on Applied Mathematics 1997; 57:683-730.
12. Osaki K, Yagi A. Finite dimensional attractor for one-dimensional Keller-Segel equations. Funkcialaj Ekvacioj 2001; 44:441-469.
13. Hillen T, Painter K. Global existence for a parabolic chemotaxis model with prevention of overcrowding. Advances in Applied Mathematics 2001; 26(4):280-301.
14. Yagi A. Norm behavior of solutions to the parabolic system of chemotaxis. Mathematica Japonica 1997; 45:241-265.
15. Osaki K, Tsujikawa T, Yagi A, Mimura M. Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Analysis 2002; 51:119-144.
16. Amann H. Quasilinear evolution equations and parabolic systems. Transactions of AMS 1986; 293:191-227.
17. Taylor ME. Partial Differential Equations, vol. III. Springer: New York, 1996.
18. Ladyžhenskaja OA, Solonnikov VA, Ural'ceva NN. Linear and Quasilinear Equations of Parabolic Type. AMS Providence: Rhode Island, 1968.
19. Othmer HG, Stevens A. Aggregation, blowup and collapse: the ABC's of taxis in reinforced random walks. SIAM Journal on Applied Mathematics 1997; 57:1044-1081.
20. Cao G. Personal communication. University of Alberta, 2003.