Boundedness of solutions of a haptotaxis model

Mathematical Models and Methods in Applied Sciences

Volumn 20, Issue 3, 2010, Pages 449-476

  Marciniak Czochra, Anna ^a   Ptashnyk, Mariya ^a  

^a UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

Boundedness of solutions; Chemotaxis model; Haptotaxis

Indexed keywords

BOUNDEDNESS; BOUNDEDNESS OF SOLUTION; CANCER INVASION; CARRYING CAPACITY; CELL DENSITY; CHEMOTAXIS MODEL; DIFFUSION COEFFICIENTS; DIFFUSION MATRICES; DIVERGENCE FORM; EXISTENCE OF GLOBAL SOLUTION; GROWTH PROPERTIES; HAPTOTAXIS MODEL; INITIAL CONDITIONS; LOGISTIC EQUATIONS; LOGISTIC GROWTH; REACTION DIFFUSION; REACTION DIFFUSION EQUATIONS; SENSITIVITY FUNCTIONS;

BIOCHEMISTRY; DIFFUSION IN LIQUIDS;

GROWTH KINETICS;

EID: 77951597072     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202510004301     Document Type: Article

References (34)

1. A. R. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, J. Theor. Med. 2 (2000) 129-154.
2. H. M. Byrne, M. A. J. Chaplain, G. J. Pettet and D. L. S. McElwain, A mathematical model of trophoblast invasion, J. Theor. Med. 1 (1998) 275-286.
3. V. Calvez and R. H. Khonsari, Mathematical description of concentric demyelination in the human brain: Self-organization models, from Liesegang rings to chemotaxis, Math. Comput. Model. 47 (2008) 726-742, doi:10.1016/j.mcm.2007. 06.01. (Pubitemid 351335340)
4. M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. Sci. 15 (2005) 1685-1734, doi:10.1142/S0218202505000947. (Pubitemid 41551043)
5. M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med. 1 (2006) 399-439.
6. K. Chueh, C. Conley and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Ind. Univ. Math. J. 26 (1977) 373-392.
7. L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angio-genesis systems in high space dimensions, Milan J. Math. 72 (2004) 1-28, doi:10.1007/S00032-003-0026-X.
8. ∞ a priori estimates for some chemotaxis models and applications to the Cauchy problem, The Mechanism of the Spatio-Temporal Pattern Arising in Reaction Diffusion System, Kyoto (2004), preprint available at http://www.math.univ-paris13.fr/zaag/publications/CPZkyoto04.pdf.
9. L. C. Evans, Partial Differential Equations (Amer. Math. Soc, 1998).
10. F. R. Guarguaglini and R. Natalini, Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology, Comm. Pure Appl. Anal. 6 (2007) 287-309.
11. T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math. 26 (2001) 280-301, doi:10.1006/aama.2001.0721. (Pubitemid 33668982)
12. T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009) 183-217, doi:10.1007/S00285-008-0201-3.
13. D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2004) 103-165;
14. II. Jahresber. Deutsch. Math.-Verein. 106 (2004) 51-69.
15. D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns. 215 (2005) 52-107, doi:10.1016/j.jde.2004.10.022. (Pubitemid 40621900)
16. W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations, Trans. AMS 329 (1992) 819-824, doi:10.2307/2153966.
17. A. Kettemann and M. Neuss-Radu, Derivation and analysis of a system modeling the chemotactic movement of hematopietic stem cells, J. Math. Biol. 56 (2008) 579-610, doi:10.1007/S00285-007-0132-4. (Pubitemid 351177977)
18. R. H. Khonsari and V. Calvez, The origins of concentric demyelination: Self-organization in the human brain, PLoS ONE 2 (2007) 1-7, doi:10.1371/journal.pone.0000150.
19. M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumours, Math. Mod. Meth. Appl. Sci. 15 (2005) 1667-1683, doi:10.1142/S0218202505000935. (Pubitemid 41551042)
20. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasi-Linear Equations of Parabolic Type (Amer. Math. Soc., 1968).
21. G. M. Lieberman, Second Order Parabolic Differential Equations (World Scientific, 1996).
22. J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (Dunod, 1969).
23. B. P. Marchant, J. Norbury and A. J. Perumpanani, Travelling shock waves arising in a model of malignant invasion, SIAM J. Appl. Math. 2 (2000) 463-476.
24. C. Morales-Rodrigo, Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Model. 47 (2008) 604-613, doi:10.1016/j.mcm.2007.02.0310. (Pubitemid 351199983)
25. G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, preprint (2008).
26. T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995) 581-601.
27. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, 1983).
28. A. J. Perumpanani, J. A. Sherratt, J. Norbury and H. M. Byrne, A two-parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cell invasion, Phys. D 126 (1999) 145-159, doi:10.1016/S0167-2789 (98) 00272-3. (Pubitemid 129496521)
29. A. J. Perumpanani, D. L. Simmons, A. J. H. Gearing, K. M. Miller, G. Ward, J. Norbury, M. Schneemann and J. A. Sherratt, Extracellular matrix-mediated chemotaxis can impede cell migration, Proc. Roy. Soc. London, Ser. B 265 (1998) 2347-2352, doi:10.1098/rspb.1998.0582. (Pubitemid 29015439)
30. W. Schen and S. Zheng, On the coupled Cahn-Hilliard equations, Comm. Part. Diff. Eqns. 18 (1993) 701-727, doi:10.1080/03605309308820946.
31. J. A. Sherratt, Chemotaxis and chemokinesis in eukaryotic cells: The Keller-Segel equations as an approximation to a detailed model, Bull. Math. Biol. 56 (1994) 129-146, doi:10.1016/S0092-8240 (05) 80208-3. (Pubitemid 2019778)
32. J. Smoller, Shock-Waves and Reaction-Diffusion Equations (Springer-Verlag, 1994).
33. Z. Szymańska, J. Urbański and A. Marciniak-Czochra, Mathematical modelling of the influence of heat shock proteins on cancer invasion of tissue, J. Math. Biol. 58 (2009) 819-840, doi:10.1007/S00285-008- 0220-0.
34. Z. Szymańska, C. Morales-Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Model. Meth. Appl. Sci. 19 (2009) 257-281, doi:10.1142/S0218202509003425.