Previous |  Up |  Next

Article

Keywords:
chemotaxis; angiogenesis; degenerate parabolic equations; kinetic equations; global weak solutions; blow-up
Summary:

References:
[1] W.  Alt: Biased random walk models for chemotaxis and related diffusion approximations. J.  Math. Biol. 9 (1980), 147–177. MR 0661424 | Zbl 0434.92001
[2] W.  Alt, G.  Hoffmann: Biological motion. Proceedings of a workshop held in Königswinter, Germany, March 16–19, 1989. Lecture Notes in Biomathematics, 89. Springer-Verlag, Berlin, 1990.
[3] A. R. A.  Anderson, M. A. J.  Chaplain: A mathematical model for capillary network formation in the absence of endothelial cell proliferation. Appl. Math. Lett. 11 (1998), 109–114.
[4] C.  Bardos, R.  Santos, and R.  Sentis: Diffusion approximation and computation of the critical size. Trans. Amer. Math. Soc. 284 (1984), 617–649. MR 0743736
[5] N.  Bellomo, L.  Preziosi: Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Modelling 32 (2000), 413–452. MR 1775113
[6] M. D.  Betterton, M. P.  Brenner: Collapsing bacterial cylinders. Phys. Rev.  E 64 (2001).
[7] P.  Biler: Global solutions to some parabolic-elliptic systems of chemotaxis. Adv. Math. Sci. Appl. 9 (1999), 347–359. MR 1690388 | Zbl 0941.35009
[8] P.  Biler, T.  Nadzieja: A class of nonlocal parabolic problems occurring in statistical mechanics. Colloq. Math. 66 (1993), 131–145. MR 1242651
[9] P.  Biler, T.  Nadzieja: Global and exploding solutions in a model of self-gravitating systems. Rep. Math. Phys. 52 (2003), 205–225. MR 2016216
[10] M. P.  Brenner, P. Constantin, L. P.  Kadanoff, A. Schenkel, and S. C. Venkataramani: Diffusion, attraction and collapse. Nonlinearity 12 (1999), 1071–1098. MR 1709861
[11] M. P.  Brenner, L.  Levitov, and E. O.  Budrene: Physical mechanisms for chemotactic pattern formation by bacteria. Biophysical Journal 74 (1995), 1677–1693.
[12] C.  Cercignani, R.  Illner, and M.  Pulvirenti: The Mathematical Theory of Dilute Gases. Applied Math. Sciences Vol. 106, Springer-Verlag, New York, 1994. MR 1307620
[13] F.  Chalub, P.  Markowich, B.  Perthame, and C.  Schmeiser: Kinetic models for chemotaxis and their drift-diffusion limits. Monatsh. Math. 142 (2004), 123–141. MR 2065025
[14] M. A. J.  Chaplain: Avascular growth, angiogenesis and vascular growth in solid tumors: the mathematical modelling of the stages of tumor development. Math. Comput. Modelling 23 (1996), 47–87.
[15] M. A. J.  Chaplain, L.  Preziosi: Macroscopic modelling of the growth and developement of tumor masses. Preprint No.  27, Politecnico di Torino, 2000.
[16] L. Corrias, B.  Perthame, and H.  Zaag: A chemotaxis model motivated by angiogenesis. C. R. Acad. Sci.  Paris, Ser.  I 336 (2003), 141–146. MR 1969568
[17] L. Corrias, B. Perthame, and H.  Zaag: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milano J.  Math. 72 (2004), 1–29. MR 2099126
[18] F. A.  Davidson, A. R. A.  Anderson, and M. A. J.  Chaplain: Steady-state solutions of a generic model for the formation of capillary networks. Appl. Math. Lett. 13 (2000), 127–132. MR 1760274
[19] P.  Degond, T.  Goudon, and F.  Poupaud: Diffusion limit for nonhomogeneous and non-micro-reversible processes. Indiana Univ. Math.  J. 49 (2000), 1175–1198. MR 1803225
[20] Y.  Dolak, T.  Hillen: Cattaneo models for chemotaxis, numerical solution and pattern formation. J.  Math. Biol. 46 (2003), 461–478. MR 1963070
[21] J.  Dolbeault, B.  Perthame: Optimal critical mass in the two dimensional Keller-Segel model in $\mathbb{R}^2$. C. R. Acad. Sci. (2004) (to appear). MR 2103197
[22] Y.  Dolak, C.  Schmeiser: Kinetic Models for Chemotaxis. ANUM preprint. (2003). MR 2093271
[23] L. C.  Evans: Partial Differential Equations. Amer. Math. Soc., Providence, 1998. Zbl 0902.35002
[24] F.  Filbet, P.  Laurençot, and B.  Perthame: Derivation of hyperbolic models for chemosensitive movement. Preprint. Ecole Normale Supérieure, 2003. MR 2120548
[25] M. A.  Fontelos, A. Friedman, and B.  Hu: Mathematical analysis of a model for the initiation of angiogenesis. SIAM J.  Math. Anal. 33 (2002), 1330–1355. MR 1920634
[26] A.  Friedman, I.  Tello: Stability of solutions of chemotaxis equations in reinforced random walks. J.  Math. Anal. Appl. 272 (2002), 138–163. MR 1930708
[27] H.  Gajewski, K.  Zacharias: Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 195 (1998), 77–114. MR 1654677
[28] A.  Gamba, D.  Ambrosi, A.  Coniglio, A.  de Candia, S.  Di  Talia, E.  Giraudo, G.  Serini, L.  Preziosi, and F.  Bussolino: Percolation, morphogenesis, and Burgers dynamics in blood vessels formation. Phys. Rev. Lett. 90 (2003), .
[29] I. Gasser, P.-E. Jabin, and B. Perthame: Regularity and propagation of moments in some nonlinear Vlasov systems. Proc. Roy. Soc. Edinburgh Sect.  A 130 (2000), 1259–1273. MR 1809103
[30] R. T.  Glassey: The Cauchy Problem in Kinetic Theory. SIAM, Philadelphia, 1996. MR 1379589 | Zbl 0858.76001
[31] M. A.  Herrero, J. J. L.  Velázquez: Singularity patterns in a chemotaxis model. Math. Ann. 306 (1996), 583–623. MR 1415081
[32] M. A.  Herrero, E. Medina, and J. J. L. Velázquez: Finite-time aggregation into a single point in a reaction-diffusion system. Nonlinearity 10 (1997), 1739–1754. MR 1483563
[33] T.  Hillen, H.  Othmer: The diffusion limit of transport equations derived from velocity-jump processes. SIAM  J.  Appl. Math. 61 (2000), 751–775. MR 1788017
[34] D.  Horstmann: Lyapunov functions and $L^p$  estimates for a class of reaction-diffusion systems. Colloq. Math. 87 (2001), 113–127. MR 1812147
[35] D.  Horstmann: From  1970 until present: the Keller-Segel model in chemotaxis and its consequences. Jahresber. Dtsch. Math.-Ver. Vol. 105, 2003, pp. 103–165. MR 2013508 | Zbl 1071.35001
[36] H. J.  Hwang, K.  Kang, and A.  Stevens: Global solutions of nonlinear transport equations for chemosensitive movement. SIAM J. Math. Anal (to appear). MR 2139206
[37] W.  Jäger, S.  Luckhaus: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329 (1992), 819–824. MR 1046835
[38] E. F.  Keller: Assessing the Keller-Segel model: How has it fared? Biological growth and spread. Proc. Conf., Heidelberg,  1979. Lecture Notes in Biomath. Vol. 38, Springer-Verlag, Berlin-New York, 1980, pp. 379–387. MR 0609374
[39] E. F.  Keller, L. A.  Segel: Initiation of slime mold aggregation viewed as an instability. J.  Theoret. Biol. 26 (1970), 399–415.
[40] E. F.  Keller, L. A.  Segel: Model for chemotaxis. J.  Theoret. Biol. 30 (1971), 225–234.
[41] E. F.  Keller, L. A.  Segel: Travelling bands of chemotactic bacteria: a theoretical analysis. J.  Theoret. Biol. 30 (1971), 235–248.
[42] H. A.  Levine, 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. MR 1450846
[43] H. A.  Levine, B. D.  Sleeman: Partial differential equations of chemotaxis and angiogenesis. Math. Methods Appl. Sci. 24 (2001), 405–426. MR 1821934
[44] H. A.  Levine, M.  Nilsen-Hamilton, and B. D. Sleeman: Mathematical modelling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42 (2001), 195–238. MR 1828815
[45] P. K.  Maini: Applications of mathematical modelling to biological pattern formation. Coherent Structures in Complex Systems (Sitges, 2000). Lecture Notes in Phys. Vol.  567, Springer-Verlag, Berlin, 2001, pp. 205–217. MR 1995108
[46] D.  Manoussaki: Modeling and simulation of the formation of vascular networks. ESAIM Proc. 12 (2002 (electronic)), 108–114.
[47] A.  Marrocco: 2D simulation of chemotactic bacteria agreggation. ESAIM: Math. Model. Numer. Anal. 37 (2003), 617–630. MR 2018433
[48] P.  Michel, S.  Mischler, and B.  Perthame: General entropy equations for structured population models and scattering. C.  R.  Acad. Sci. Paris (to appear). MR 2065377
[49] J. D.  Murray: Mathematical Biology, Vol.  2, third revised edition. Spatial Models and Biomedical Applications. Springer-Verlag, , 2003. MR 1952568
[50] T.  Nagai: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5 (1995), 581–601. MR 1361006 | Zbl 0843.92007
[51] T.  Nagai, T.  Senba: Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis. Adv. Math. Sci. Appl. 8 (1998), 145–156. MR 1623326
[52] J. Nieto, F. Poupaud, and J.  Soler: High field limit for the Vlasov-Poisson-Fokker-Planck system. Arch. Rational. Mech. Anal. 158 (2001), 29–59. MR 1834113
[53] H. G.  Othmer, A. Stevens: Aggregation, blowup and collapse: the ABC’s of taxis in reinforced random walks. SIAM J.  Appl. Math. 57 (1997), 1044–1081. MR 1462051
[54] H. G. Othmer, S. R.  Dunbar, and W.  Alt: Models of dispersal in biological systems. J.  Math. Biol. 26 (1988), 263–298. MR 0949094
[55] C. S.  Patlak: Random walk with persistence and external bias. Bull. Math. Biophys. 15 (1953), 311–338. MR 0081586
[56] B.  Perthame: Mathematical tools for kinetic equations. Bull.  Amer. Math. Soc. (NS) 41 (2004), 205–244. MR 2043752 | Zbl 1151.82351
[57] M.  Rascle: On a system of non-linear strongly coupled partial differential equations arising in biology. Proc. Conf. on  Ordinary and Partial Differential Equation. Lectures Notes in Math. Vol. 846, Everitt and Sleeman (eds.), Springer-Verlag, New-York, 1981, pp. 290–298.
[58] M.  Rascle, C.  Ziti: Finite time blow-up in some models of chemotaxis. J.  Math. Biol. 33 (1995), 388–414. MR 1320430
[59] G.  Serini, D.  Ambrosi, E.  Giraudo, A.  Gamba, L.  Preziosi, and F.  Bussolino: Modeling the early stages of vascular network assembly. The EMBO Journal 22 (2003), 1771–1779.
[60] T.  Sanba and T.  Suzuki: Weak solutions to a parabolic-elliptic system of chemotaxis. J.  Functional. Analysis 47 (2001), 17–51. MR 1909263
[61] H. R. Schwetlick: Travelling fronts for multidimensional nonlinear transport equations. Ann. Inst. H.  Poincaré. Anal. non Linéaire 17 (2000), 523–550. MR 1782743
[62] A.  Stevens: The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J.  Appl. Math. 61 (2000), 183–212. MR 1776393 | Zbl 0963.60093
[63] A.  Stevens, M.  Schwelick: Work in preparation.
[64] M. I.  Weinstein: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1983), 567–576. MR 0691044 | Zbl 0527.35023
[65] Y.  Yang, H.  Chen, and W.  Liu: On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis. SIAM J.  Math. Anal. 33 (2001), 763–785. MR 1884721
Partner of
EuDML logo