Nonlinear stability of large amplitude viscous shock waves of a generalized hyperbolicparabolic system arising in chemotaxis

Mathematical Models and Methods in Applied Sciences

Volumn 20, Issue 11, 2010, Pages 1967-1998

  Li, Tong ^a   Wang, Zhi An ^b  

^a UNIVERSITY OF IOWA   (United States)

^b HONG KONG POLYTECHNIC UNIVERSITY   (Hong Kong)

Author keywords

chemotaxis; energy estimates; large amplitude; non diffusive signals; Nonlinear conservation laws; nonlinear kinetics; nonlinear stability; ShizutaKawashima condition; traveling waves

Indexed keywords

CHEMOTAXIS; ENERGY ESTIMATES; LARGE AMPLITUDE; NON-DIFFUSIVE SIGNALS; NON-LINEAR STABILITIES; NONLINEAR CONSERVATION LAW; NONLINEAR KINETICS; SHIZUTAKAWASHIMA CONDITION; TRAVELING WAVE;

BIOCHEMISTRY; PHYSICAL PROPERTIES;

SYSTEM STABILITY;

EID: 78349311551     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202510004830     Document Type: Article

References (38)

1. J. Adler, Chemotaxis in bacteria, Science 153 (1966) 708-716.
2. J. Adler, Chemoreceptors in bacteria, Science 166 (1969) 1588-1597.
3. M. A. J. Chaplain and G. Lolas, Mathematical modeling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med. 1 (2006) 399-439.
4. L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris Ser. I 336 (2003) 141 - 146.
5. L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis system in high space dimensions, Milan J. Math. 72 (2004) 1-28.
6. J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986) 325-344.
7. S. Gueron and N. Liron, A model of herd grazing as a traveling wave: Chemotaxis and stability, J. Math. Biol. 27 (1989) 595-608.
8. D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: I, Jahresber. Deutsch. Math.-Verein 105 (2003) 103-165.
9. D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: II, Jahresber. Deutsch. Math.-Verein 106 (2004) 51-69.
10. D. Horstmann and A. Stevens, A constructive approach to travelling waves in chemotaxis, J. Nonlinear Sci. 14 (2004) 1-25.
11. J. Humpherys and K. Zumbrun, Spectral stability of small-amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems, Z. Angew. Math. Phys. 53 (2002) 20-34.
12. S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Comm. Pure Appl. Math. 47 (1994) 1547-1569.
13. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970) 399-415.
14. E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol. 30 (1971) 225-234.
15. E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol. 30 (1971) 235-248.
16. P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957) 537-566.
17. H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol. 42 (2001) 195-238.
18. H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57 (1997) 683-730.
19. T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal. 40 (2008) 1058-1075.
20. T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math. 70 (2009) 1522-1541.
21. T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 328 (1986) 1 - 108.
22. T. P. Liu and Y. Zeng, Time-asymptotic behavior of wave propagation around a viscous shock profile, Comm. Math. Phys. 290 (2009) 23-82.
23. R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol. 61 (2010) 739-761.
24. A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Jpn. J. Appl. Math. 2 (1985) 17-25.
25. T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol. 30 (1991) 169-184.
26. T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. 78 (1978) 46-53.
27. H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (1997) 1044-1081.
28. C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953) 311-338.
29. B. Perthame, Transport Equations in Biology (Birkhäuser Verlag, 2007).
30. G. Rosen and S. Baloga, On the stability of steadily propagating bands of chemotactic bacteria, Math. Biosco. 24 (1975) 273-279.
31. T. L. Scribner, L. A. Segel and E. H. Rogers, A numerical study of the formation and propagation of traveling bands of chemotactic bacteria, J. Theor. Biol. 46 (1974) 189-219.
32. Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249-275.
33. A. Stevens and J. J. L. Velazquez, Partial differential equations and non-diffusive structures, Nonlinearity 21 (2008) 283-289.
34. A. Szepessy and Z. P. Xin, Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122 (1993) 53-103.
35. M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations, Bull. Math. Biol. 70 (2008) 1570-1607.
36. Z. A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci. 31 (2008) 45-70.
37. S. S. Willard and P. N. Devreotes, Signalling pathways mediating chemotaxis in the social amoeba, dictyostelium discoideum, Euro. J. Cell. Biol. 85 (2006) 897-904.
38. M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc. 135 (2006) 1017-1027.