Local existence and uniqueness of solutions to a PDE model for criminal behavior

Mathematical Models and Methods in Applied Sciences

Volumn 20, Issue SUPPL. 1, 2010, Pages 1425-1457

  Rodriguez, Nancy ^a   Bertozzi, Andrea ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

chemotaxis; Crime models; local existence; reaction diffsion equation

Indexed keywords

AGENT-BASED STOCHASTIC MODEL; BLOW-UP; CHEMOTAXIS; COUPLED SYSTEMS; CRIME MODELS; CRIMINAL ACTIVITIES; FINITE TIME; GLOBAL EXISTENCE; HOTSPOTS; KELLER-SEGEL MODEL; LOCAL EXISTENCE; MATHEMATICAL TOOLS; PDE MODEL; QUALITATIVE BEHAVIOR; REACTION DIFFSION EQUATION; SPATIO-TEMPORAL;

AIRCRAFT ENGINES; BIOCHEMISTRY; CRIME; PARTIAL DIFFERENTIAL EQUATIONS;

STOCHASTIC MODELS;

EID: 77955924345     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202510004696     Document Type: Article

References (40)

1. p bounds of solutions of reaction-diffusion equations, Comm. Partial Diff. Eqns. 4 (1979) 827-868.
2. L. Anselin, J. Cohen, D. Cook, W. Gorr and G. Tita, Spatial Analysis of Crime, Criminal Justice 2000, Vol. 4 (1998), pp. 213-262.
3. A. L. Bertozzi and J. B. Greer, Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes, Comm. Pure Appl. Math. 57 (2004) 764-790.
4. P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl. 9 (1999) 347-359.
5. H. M. Byrne and M. R. Owen, A new interpretation of the Keller-Segel model based on multiphase modeling, J. Math. Biol. 49 (2004) 604-626.
6. 2, Comm. Math. Sci. 6 (2008) 417-447.
7. F. Chalub, Y. Dolak-Struss, P. Markowich, D. Oelz, C. Schmeiser and A. Soreff, Model hierarchies for cell aggregation by chemotaxis, Math. Models Methods Appl. Sci. 16 (2006) 1173-1197.
8. S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981) 217-237.
9. L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math. 72 (2004) 1-28.
10. M. del Pino and J. Wei, Collapsing steady states of the Keller-Segel system, Nonlinearity 19 (2006) 661-684.
11. M. Di Francesco and J. Rosado, Fully parabolic Keller-Segel model for chemotaxis with prevention of overcrowding, Nonlinearity 21 (2008) 2715-2730.
12. Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math. 66 (2005) 286-308 (electronic).
13. C. Escudero, The fractional Keller-Segel model, Nonlinearity 19 (2006) 2909-2918.
14. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19 (Amer. Math. Soc, 1998).
15. F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol. 50 (2005) 189-207.
16. A. Friedman, Partial Differential Equations of Parabolic Type (Prentice-Hall, 1964).
17. A. Friedman, Partial Differential Equations (Holt, Rinehart and Winston, Inc., 1969).
18. J. B Greer, Fourth-order diffusions for image processing, Ph.D. thesis, Duke Univeristy, 2003.
19. P. Hartman, Ordinary Differential Equations, 2nd edn. (Birkhäuser, 1982).
20. M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35 (1996) 177-194.
21. M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996) 583-623.
22. T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math. 26 (2001) 280-301.
23. T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009) 183-217.
24. D. Hortsmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresberichte DMV 105 (2003) 103-165.
25. S. Johnson and K. Bowers, The burglary as clue to the future: The beginnings of prospective hot-spotting, Eur. J. Criminol. 1 (2004) 237-255.
26. S. Johnson and K. Bowers, Domestic burglary repeats and space-time clusters: The dimensions of risk, Eur. J. Criminol. 2 (2005) 67-92.
27. S. Johnson, K. Bowers and A. Hirschfield, New insights into the spatial and temporal distribution of repeat victimisation, Brit. J. Criminol. 37 (1997) 224-244.
28. E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol. 30 (1971) 225-234.
29. R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl. 305 (2005) 566-588.
30. I. R. Lapidus and M. Levandowsky, Modeling chemosensory responses of swimming eukaryotes, in Biological Growth and Spread, Proc. Conf., Heidelberg, 1979, Lecture Notes in Biomath., Vol. 38 (Springer, 1980), pp. 388-396.
31. A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Vol. 27 (Cambridge Univ. Press, 2002).
32. K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. 51 (2002) 119-144.
33. R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc. 292 (1985) 531-556.
34. M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249-1267.
35. 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 (electronic);
36. Erratum-ibid. 61 (2001) 2200 (electronic).
37. M. E. Taylor, Partial Differential Equations. III, Applied Mathematical Sciences, Vol. 117 (Springer-Verlag, 1997) [Nonlinear Equations, corrected reprint of the 1996 original].
38. J. J. L. Velázquez, Well-posedness of a model of point dynamics for a limit of the Keller-Segel system, J. Diff. Eqns. 206 (2004) 315-352.
39. J. Q. Wilson and G. L Kelling, Broken windows and police and neighborhood safety, Atlantic Mon. 249 (1998) 29-38.
40. W. H. Young, On the multiplication of successions of Fourier constants, Proc. Roy. Soc. Lond. 87 (1912) 331-339.