Shock formation in a Chemotaxis model

Mathematical Methods in the Applied Sciences

Volumn 31, Issue 1, 2008, Pages 45-70

  Wang, Zhian ^a   Hillen, Thomas ^a  

^a UNIVERSITY OF ALBERTA   (Canada)

Author keywords

Chemotaxis; Entropy condition; Entropy entropy flux; Shock solutions; Shock structure; Travelling wave

Indexed keywords

ENTROPY; PROBLEM SOLVING; WAVE EFFECTS;

CHEMOTAXIS; ENTROPY CONDITION; SHOCK SOLUTIONS; SHOCK STRUCTURE; TRAVELLING WAVES;

SHOCK TESTING;

EID: 37049029658     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.898     Document Type: Article

References (15)

1. Patlak CS. Random walk with persistence and external bias. Bulletin of Mathematical Biophysics 1953; 15: 388-414.
2. Othmer HG, Stevens A. Aggregation, blowup, and collapse: the ABC's of taxis and reinforced random walks. SIAM Journal on Applied Mathematics 1997; 57:1044-1081.
3. Levine HA, Sleeman BD. A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM Journal of Applied Mathematics 1997; 57:683-730.
4. Yang Y, Chen H, Liu W. On existence of global solutions and blow-up to a system of the reaction-diffusion equations modelling Chemotaxis. SIAM Journal on Mathematical Analysis 2001; 33(4):763-785.
5. Yang Y, Chen H, Liu W, Sleeman BD. The solvability of some Chemotaxis systems. Journal of Differential Equations 2005; 212(2):432-451.
6. Ladyžhenskaya OA, Solonnikov VA, Ural'ceva NN. Linear and Quasilinear Equations of Parabolic Type. AMS: Providence, RI, 1968.
7. Sleeman BD, Ward MJ, Wei JC. The existence and stability of spike patterns in a Chemotaxis model. SIAM Journal on Applied Mathematics 2005; 65(3):790-817.
8. LeFloch PG. Hyperbolic System of Conservation Laws: the Theory of Classical and Nonclassical Shock Waves. Lectures in Mathematics: ETH Zürich. Birkhäuser: Basel, 2002.
9. Bressan A. Hyperbolic System of Conservation Laws: The One-dimensional Cauchy Problem. Oxford University Press: New York, 2001.
10. Smoller J. Shock Waves and Reaction-Diffusion Equations. Springer: Berlin, 1983.
11. LeVeque RJ. Numerical Methods for Conservational Laws. Lectures in Mathematics: ETH Zürich. Birkhäuser: Basel, 1992.
12. Hespanha JP. Uniform stability of switched linear systems: extensions of LaSalle's invariant principle. IEEE Transactions on Automatic Control 2004; 49(4):470-482.
13. LaSalle JP. The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics, vol. 25. SIAM: Philadelphia, PA, 1976.
14. Evans LC. Partial Differential Equations. AMS: Providence, RI, 1997.
15. Rascle W. Sur une eq́uation integro-differentielle non lineaire issue de la biologie. Journal of Differential Equations 1979; 32:420-453.