Singular limit of a chemotaxis-growth model

Advances in Differential Equations

Volumn 6, Issue 10, 2001, Pages 1173-1218

  Bonami, A ^a   Hilhorst, D ^b   Logak, E ^c   Mimura, M ^d  

^a UNIVERSITÉ D'ORLÉANS   (France)

^b UNIVERSITÉ PARIS SACLAY   (France)

^c UNIVERSITÉ DE CERGY PONTOISE   (France)

^d HIROSHIMA UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33646553798     PISSN: 10799389     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (34)

1. G. Barles, H. M. Soner, and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control and Opt., 31 (1993), 439-469.
2. A. Bonami, D. Hilhorst, and E. Logak, Modified motion by mean curvature: local existence and uniqueness and qualitative properties, Differential and Integral Equations, 13 (2000), 1371-1392.
3. A. Bonami, D. Hilhorst, E. Logak, and M. Mimura, A free boundary problem arising in a chemotaxis model, "Free Boundary Problems, Theory and Applications," M. Niezgódka and P. Strzelecki, ed.s, Pitman Res. Notes in Math. Series, 363 (1996).
4. L. Bronsard and R.V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau model, J. Diff. Eqns., 90 (1991), 211-237.
5. E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.
6. X. Chen, Generation, and propagation of interfaces for reaction-diffusion equations, J. Diff. Eqns., 96 (1992), 116-141.
7. X. Chen, Generation, and propagation of interfaces for reaction-diffusion systems, Trans. Amer. Math. Soc., 334 (1992), 877-913.
8. X. Chen, D. Hilhorst, and E. Logak, Asymptotic behaviour of an Allen-Cahn equation with a non-local term, Nonlinear Analysis TMA, Vol. 28, No.7 (1997), 1283-1298.
9. Y. G. Chen, Y. Giga, and S. Goto, Uniqueness, and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom., 33 (1991), 749-786.
10. X. Chen, and F. Reitich, Local existence, and uniqueness of solutions of the Stefan problem with surface tension, and kinetic undercooling, J. Math. Anal. Appl., 162 (1992), 350-362.
11. S. Childress, and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1986), 217-237.
12. J. I. Diaz, and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Adv. Math. Sci. Appl., 5 (1995), 659-680.
13. E. Dibenedetto, "Degenerate Parabolic Equations," Springer-Verlag, (1993).
14. L.C. Evans, H.M. Soner, and P.E. Souganidis, Phase transitions, and generalized motion by mean curvature, Communications on Pure, and Applied Mathematics, Vol. 1XLV (1992), 1097-1123.
15. L. C. Evans, and J. Spruck, Motion of level sets by mean curvature, J. Diff. Geom., 33 (1991), 635-681.
16. M. Gage, and R. Hamilton, The shrinking of convex curves by the heat equation, J. Diff. Geom., 23 (1986), 69-96.
17. Y. Giga, S. Goto, and H. Ishii, Global existence of weak solutions for interface equations coupled with diffusion equations, SIAM Journal on Mathematical Analysis, Vol. 23, No. 4 (1992), 821-835.
18. M. Grayson, The heat equation shrinks plane curves to points, J. Diff. Geom., 26 (1987), 283-314.
19. M. A. Herrero, and J. Velazquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), no. 2, 177-194.
20. M. A. Herrero, and J. Velazquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), no. 4, 633-683.
21. D. Hilhorst, E. Logak, and R. Schätzle, Global existence for a nonlocal mean curvature flow as a limit of a parabolic-elliptic phase transition model, Interfaces, and Free Boundaries, 2 (2000), 267-282.
22. G. Huisken, Flow by mean curvature of convex surfaces into spheres, Journal of Differential Geometry, 20 (1984), 237-266.
23. W. Jäger, and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
24. E. F. Keller, and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
25. E. Logak, Singular limit of reaction-diffusion systems, and modified motion by mean curvature , to appear in Proceedings of the Royal Society of Edinburgh.
26. M. Mimura, and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543.
27. M. Mimura, T. Tsujikawa, and A. Yagi, manuscript.
28. M. Mimura, T. Tsujikawa, R. Kobayashi, and D. Ueyama, Dynamics of aggregation patterns in a chemotaxis-diffusion-growth model equation, Forma 8 (1993), 179-195.
29. P. de Mottoni, and M. Schatzman, Development of interfaces in Rn, Proc. Royal Soc. Edin., 116A (1990), 207-220.
30. T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Advances in Math. Sciences, and Appl., Vol. 5, No. 2 (1995), 581-601.
31. R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.
32. P.E. Souganidis, and P. Soravia, Phase field theory for FitzHugh-Nagumo type systems , SIAM Journal on Mathematical Analysis, Vol. 27 (1996), 1341-1359.
33. E. M. Stein, "Singular Integrals, and Differentiable Properties," Princeton University Press, 1993.
34. D. E. Woodward, J. Tyson, M. R. Myerscough, J. D. Murray, E. O. Budrene, and H. C. Berg, Spatio-temporal patterns generated by Salmonella typhymerium, Biophys., 68 (1995), 2181-2189.