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Global Solvability and Stabilization in a Three-Dimensional Cross-Diffusion System Modeling Urban Crime Propagation

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Abstract

In this paper, we will focus on the initial-boundary value problem with no-flux boundary conditions for the three-dimensional cross-diffusion system

$$ \left \{ \begin{aligned} &u_{t}=\Delta u-\chi \nabla \cdot (\frac{u}{v}\nabla v)-uv+B_{1}(x,t),& \qquad x\in \Omega ,\,t>0, \\ &v_{t}=\Delta v+uv-v+B_{2}(x,t), &\qquad x\in \Omega ,\,t>0. \end{aligned} \right . $$

Under some basic assumptions on the functions \(B_{1}\) and \(B_{2}\), we will show that for each \(\chi \in (0,\sqrt{3})\), the system has at least one global renormalized solution in case that each ingredient of the system is radially symmetric with regard to the center of \(\Omega \). Moreover, if \(\chi \in (0,\frac{\sqrt{6}}{2})\), the solution will approach the solution of an elliptic boundary value problem as \(t\rightarrow \infty \) under some extra hypotheses.

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References

  1. Ahn, J., Kang, K., Lee, J.: Global well-posedness of logarithmic Keller-Segel type systems. J. Differ. Equ. 287, 185–211 (2021)

    Article  MathSciNet  Google Scholar 

  2. Amann, H.: Dynamic theory of quasilinear parabolic systems III. Global existence. Math. Z. 202, 219–250 (1989)

    Article  MathSciNet  Google Scholar 

  3. Berestycki, H., Nadal, J.: Self-organised critical hot-spots of criminal activity. Eur. J. Appl. Math. 21, 371–399 (2010)

    Article  MathSciNet  Google Scholar 

  4. Berestycki, H., Wei, J., Winter, M.: Existence of symmetric and asymmetric spikes for a crime hotspot model. SIAM J. Math. Anal. 46, 691–719 (2014)

    Article  MathSciNet  Google Scholar 

  5. Biler, P.: Global solutions to some parabolic-elliptic systems of chemotaxis. Adv. Math. Sci. Appl. 9, 347–359 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Cantrell, R.S., Cosner, C., Manásevich, R.: Global bifurcation of solutions for crime modeling equations. SIAM J. Appl. Math. 44, 1340–1358 (2012)

    Article  MathSciNet  Google Scholar 

  7. Cohen, L.E., Felson, M.: Social change and crime rate trends: a routine activity approach. Am. Sociol. Rev. 44, 588–608 (1979)

    Article  Google Scholar 

  8. Espejo, E., Winkler, M.: Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system modeling coral fertilization. Nonlinearity 31, 1227–1259 (2018)

    Article  MathSciNet  Google Scholar 

  9. Felson, M.: Routine activities and crime prevention in the developing metropolis. Criminology 25, 911–932 (1987)

    Article  Google Scholar 

  10. Freitag, M.: Global solutions to a higher-dimensional system related to crime modeling. Math. Methods Appl. Sci. 41, 6326–6335 (2018)

    Article  MathSciNet  Google Scholar 

  11. Fuest, M., Heihoff, F.: Unboundedness phenomenon in a reduced model of urban crime (2021). arXiv preprint. arXiv:2109.01016

  12. Giga, Y., Sohr, H.: Abstract \(L^{p}\) estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)

    Article  MathSciNet  Google Scholar 

  13. Gu, Y., Wang, Q., Guangzeng, Y.: Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect. Eur. J. Appl. Math. 28, 141–178 (2017)

    Article  MathSciNet  Google Scholar 

  14. Hillen, T., Painter, K.J., Winkler, M.: Convergence of a cancer invasion model to a logistic chemotaxis model. Math. Models Methods Appl. Sci. 23, 165–198 (2013)

    Article  MathSciNet  Google Scholar 

  15. Horstmann, D., Winkler, M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107 (2005)

    Article  MathSciNet  Google Scholar 

  16. Johnson, S.D., Bowers, K., Hirschfield, A.: New insights into the spatial and temporal distribution of repeat victimisation. Br. J. Criminol. 37, 224–241 (1997)

    Article  Google Scholar 

  17. Kelling, G.L., Wilson, J.Q.: Broken Windows (1982)

    Google Scholar 

  18. Kolokolnikiv, T., Ward, M.J., Wei, J.: The stability of hot-spot patterns for reaction-diffusion models of urban crime. Discrete Contin. Dyn. Syst., Ser. B 19, 1373–1410 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. Transl. Am. Math. Soc., vol. 23. Am. Math. Soc., Providence (1968)

    Book  Google Scholar 

  20. Pitcher, A.: Adding police to a mathematical model of burglary. Eur. J. Appl. Math. 21, 401–419 (2010)

    Article  MathSciNet  Google Scholar 

  21. Porzio, M.M., Vespri, V.: Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103, 146–178 (1993)

    Article  Google Scholar 

  22. Rodríguez, N.: On the global well-posedness theory for a class of PDE models for criminal activity. Phys. D: Nonlinear Phenom. 260, 191–200 (2013)

    Article  MathSciNet  Google Scholar 

  23. Rodríguez, N., Bertozzi, A.: Local existence and uniqueness of solutions to a PDE model for criminal behavior. Math. Models Methods Appl. Sci. 20, 1425–1457 (2010)

    Article  MathSciNet  Google Scholar 

  24. Rodríguez, N., Winkler, M.: On the global existence and qualitative behavior of one-dimensional solutions to a model for urban crime (2019). arXiv:1903.06331v1

  25. Rodríguez, N., Winkler, M.: Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation. Math. Models Methods Appl. Sci. 30, 2105–2137 (2020)

    Article  MathSciNet  Google Scholar 

  26. Short, M.B., D’Orsogna, M.R., Pasour, V.B., Tita, G.E., Brantingham, P.J., Bertozzi, A.L., Chayes, L.B.: A statistical model of criminal behavior. Math. Models Methods Appl. Sci. 18, 1249–1267 (2008)

    Article  MathSciNet  Google Scholar 

  27. Short, M.B., D’Orsogna, M.R., Brantingham, P.J., Tita, G.E.: Measuring and modeling repeat and near-repeat burglary effects. J. Quant. Criminol. 25, 325–339 (2009)

    Article  Google Scholar 

  28. Short, M.B., Bertozzi, A.L., Brantingham, P.J.: Nonlinear patterns in urban crime: hotspots, bifurcations, and suppression. SIAM J. Appl. Dyn. Syst. 9, 462–483 (2010)

    Article  MathSciNet  Google Scholar 

  29. Tao, Y., Winkler, M.: Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime. Commun. Math. Sci. 19, 829–849 (2021)

    Article  MathSciNet  Google Scholar 

  30. Tse, W.H., Ward, M.J.: Hotspot formation and dynamics for a continuum model of urban crime. Eur. J. Appl. Math. 27, 583–624 (2016)

    Article  MathSciNet  Google Scholar 

  31. Wang, Q., Wang, D., Feng, Y.: Global well-posedness and uniform boundedness of urban crime models: one-dimensional case. J. Differ. Equ. 269, 6216–6235 (2020)

    Article  MathSciNet  Google Scholar 

  32. Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J. Differ. Equ. 248, 2889–2905 (2010)

    Article  MathSciNet  Google Scholar 

  33. Winkler, M.: Global solutions in a fully parabolic chemotaxis system with singular sensitivity. Math. Models Methods Appl. Sci. 34, 176–190 (2011)

    Article  MathSciNet  Google Scholar 

  34. Winkler, M.: Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 36, 1747–1790 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to the referees for their detailed comments and valuable suggestions, which greatly improved the manuscript, and to Professor Zhaoyin Xiang for his helpful guidance. This work is supported by the Applied Fundamental Research Program of Sichuan Province (No. 2020YJ0264).

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  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Yongfeng Jiang & Lan Yang

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  1. Yongfeng Jiang
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Correspondence to Lan Yang.

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Jiang, Y., Yang, L. Global Solvability and Stabilization in a Three-Dimensional Cross-Diffusion System Modeling Urban Crime Propagation. Acta Appl Math 178, 11 (2022). https://doi.org/10.1007/s10440-022-00484-z

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