We study a logarithmic Keller-Segel system proposed by Rodríguez for crime modeling as follows:
$ \begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot\left(u\nabla\ln v\right)- \kappa uv+ h_1,\\ &v_t = \Delta v- v+ u+h_2, \end{split} \right. \end{equation*} $
in a bounded and smooth spatial domain $ \Omega\subset \mathbb R^n $ with $ n\geq3 $, with the parameters $ \chi > 0 $ and $ \kappa > 0 $, and with the nonnegative functions $ h_1 $ and $ h_2 $. For the case that $ \kappa = 0 $, $ h_1\equiv0 $ and $ h_2\equiv0 $, recent results showed that the corresponding initial-boundary value problem admits a global generalized solution provided that $ \chi < \chi_0 $ with some $ \chi_0 > 0 $.
In the present work, our first result shows that for the case of $ \kappa > 0 $ such problem possesses global generalized solutions provided that $ \chi < \chi_1 $ with some $ \chi_1 > \chi_0 $, which seems to confirm that the mixed-type damping $ -\kappa uv $ has a regularization effect on solutions. Besides the existence result for generalized solutions, a statement on the large-time behavior of such solutions is derived as well.
Citation: Bin Li, Zhi Wang, Li Xie. Regularization effect of the mixed-type damping in a higher-dimensional logarithmic Keller-Segel system related to crime modeling[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4532-4559. doi: 10.3934/mbe.2023210
We study a logarithmic Keller-Segel system proposed by Rodríguez for crime modeling as follows:
$ \begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot\left(u\nabla\ln v\right)- \kappa uv+ h_1,\\ &v_t = \Delta v- v+ u+h_2, \end{split} \right. \end{equation*} $
in a bounded and smooth spatial domain $ \Omega\subset \mathbb R^n $ with $ n\geq3 $, with the parameters $ \chi > 0 $ and $ \kappa > 0 $, and with the nonnegative functions $ h_1 $ and $ h_2 $. For the case that $ \kappa = 0 $, $ h_1\equiv0 $ and $ h_2\equiv0 $, recent results showed that the corresponding initial-boundary value problem admits a global generalized solution provided that $ \chi < \chi_0 $ with some $ \chi_0 > 0 $.
In the present work, our first result shows that for the case of $ \kappa > 0 $ such problem possesses global generalized solutions provided that $ \chi < \chi_1 $ with some $ \chi_1 > \chi_0 $, which seems to confirm that the mixed-type damping $ -\kappa uv $ has a regularization effect on solutions. Besides the existence result for generalized solutions, a statement on the large-time behavior of such solutions is derived as well.
| [1] |
N. Rodríguez, On the global well-posedness theory for a class of PDE models for criminal activity, Phys. D Nonlinear Phenom., 260 (2013), 191–200. https://doi.org/10.1016/j.physd.2012.08.003 doi: 10.1016/j.physd.2012.08.003
|
| [2] |
M. Short, M. D'Orsogna, V. Pasour, G. Tita, P. Brantingham, A. Bertozzi, et al., A statistical model of criminal behavior, Math. Mod. Meth. Appl. Sci., 18 (2008), 1249–1267. https://doi.org/10.1142/S0218202508003029 doi: 10.1142/S0218202508003029
|
| [3] |
M. Short, A. Bertozzi, P. Brantingham, G. Tita, Dissipation and displacement of hotspots in reaction-diffusion model of crime, Proc. Natl. Acad. Sci. USA, 107 (2010), 3961–3965. https://doi.org/0.1073/pnas.0910921107 doi: 10.1073/pnas.0910921107
|
| [4] |
H. Berestycki, J. Wei, M. Winter, Existence of symmetric and asymmetric spikes for a crime hotspot model, SIAM J. Math. Anal., 46 (2014), 691–719. https://doi.org/10.1137/130922744 doi: 10.1137/130922744
|
| [5] |
R. Cantrell, C. Cosner, R. Manásevich, Global bifurcation of solutions for crime modeling equations, SIAM J. Math. Anal., 44 (2012), 1340–1358. https://doi.org/10.1137/110843356 doi: 10.1137/110843356
|
| [6] |
Y. Gu, Q. Wang, G. Yi, Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect, Eur. J. Appl. Math., 28 (2017), 141–178. https://doi.org/10.1017/S0956792516000206 doi: 10.1017/S0956792516000206
|
| [7] |
T. Kolokolnikov, M. Ward, J. Wei, The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1373–1410. https://doi.org/10.3934/dcdsb.2014.19.1373 doi: 10.3934/dcdsb.2014.19.1373
|
| [8] |
D. Lloyd, H. O'Farrell, On localised hotspots of an urban crime model, Phys. D Nonlinear Phenom., 253 (2013), 23–39. https://doi.org/10.1016/j.physd.2013.02.005 doi: 10.1016/j.physd.2013.02.005
|
| [9] |
D. Lloyd, N. Santitissadeekorn, M. Short, Exploring data assimilation and forecasting issues for an urban crime model, Eur. J. Appl. Math., 27 (2016), 451–478. https://doi.org/10.1017/S0956792515000625 doi: 10.1017/S0956792515000625
|
| [10] |
L. Mei, J. Wei, The existence and stability of spike solutions for a chemotax is system modeling crime pattern formation, Math. Models Methods Appl. Sci., 30 (2020), 1727–1764. https://doi.org/10.1142/S0218202520500359 doi: 10.1142/S0218202520500359
|
| [11] |
M. Short, A. Bertozzi, P. Brantingham, Nonlinear patterns in urban crime: hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst., 9 (2010), 462–483. https://doi.org/10.1137/090759069 doi: 10.1137/090759069
|
| [12] |
W. Tse, M. Ward, Hotspot formation and dynamics for a continuum model of urban crime, Eur. J. Appl. Math., 27 (2016), 583–624. https://doi.org/10.1017/S0956792515000376 doi: 10.1017/S0956792515000376
|
| [13] |
N. Rodríguez, A. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior, Math. Models Methods Appl. Sci., 20 (2010), 1425–1457. https://doi.org/10.1142/S0218202510004696 doi: 10.1142/S0218202510004696
|
| [14] |
N. Rodríguez, M. Winkler, On the global existence and qualitative behavior of one-dimensional solutions to a model for urban crime, Eur. J. Appl. Math., 33 (2022), 919–959. https://doi.org/10.1017/S0956792521000279 doi: 10.1017/S0956792521000279
|
| [15] |
Q. Wang, D. Wang, Y. Feng, Global well-posedness and uniform boundedness of urban crime models: One-dimensional case, J. Differ. Equations, 269 (2020), 6216–6235. https://doi.org/10.1016/j.jde.2020.04.035 doi: 10.1016/j.jde.2020.04.035
|
| [16] |
M. Freitag, Global solutions to a higher-dimensional system related to crime modeling, Math. Meth. Appl. Sci., 41 (2018), 6326–6335. https://doi.org/10.1002/mma.5141 doi: 10.1002/mma.5141
|
| [17] |
J. Shen, B. Li, Mathematical analysis of a continuous version of statistical models for criminal behavior, Math. Meth. Appl. Sci., 43 (2020), 409–426. https://doi.org/10.1002/mma.5898 doi: 10.1002/mma.5898
|
| [18] |
J. Ahn, K. Kang, J. Lee, Global well-posedness of logarithmic Keller-Segel type systems, J. Differ. Equations, 287 (2021), 185–211. https://doi.org/10.1016/j.jde.2021.03.053 doi: 10.1016/j.jde.2021.03.053
|
| [19] |
Y. Tao, M. Winkler, Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime, Commun. Math. Sci., 19 (2021), 829–849. https://doi.org/10.4310/CMS.2021.v19.n3.a12 doi: 10.4310/CMS.2021.v19.n3.a12
|
| [20] |
M. Winkler, Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 36 (2019), 1747–1790. https://doi.org/10.1016/j.anihpc.2019.02.004 doi: 10.1016/j.anihpc.2019.02.004
|
| [21] |
Y. Jiang, L. Yang, Global solvability and stabilization in a three-dimensional cross-diffusion system modeling urban crime propagation, Acta Appl. Math., 178 (2022). https://doi.org/10.1007/s10440-022-00484-z doi: 10.1007/s10440-022-00484-z
|
| [22] |
N. Rodríguez, M. Winkler, Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation, Math. Models Methods Appl. Sci., 30 (2020), 2105–2137. https://doi.org/10.1142/S0218202520500396 doi: 10.1142/S0218202520500396
|
| [23] |
F. Heihoff, Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source term, Z. Für Angew. Math. Phys., 71 (2020). https://doi.org/10.1007/s00033-020-01304-w doi: 10.1007/s00033-020-01304-w
|
| [24] | B. Li, L. Xie, Generalized solution to a 2D parabolic-parabolic chemotaxis system for urban crime: Global existence and large time behavior, submitted for publication, 2022. |
| [25] |
P. Jones, P. Brantingham, L. Chayes, Statistical models of criminal behavior: The effects of law enforcement actions, Math. Models Methods Appl. Sci., 20 (2010), 1397–1423. https://doi.org/10.1142/S0218202510004647 doi: 10.1142/S0218202510004647
|
| [26] |
A. Pitcher, Adding police to a mathematical model of burglary, Eur. J. Appl. Math., 21 (2010), 401–419. https://doi.org/10.1017/S0956792510000112 doi: 10.1017/S0956792510000112
|
| [27] |
J. Zipkin, M. Short, A. Bertozzi, Cops on the dots in a mathematical model of urban crime and police response, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), no. 5, 1479–1506. https://doi.org/10.3934/dcdsb.2014.19.1479 doi: 10.3934/dcdsb.2014.19.1479
|
| [28] |
W. Tse, M. Ward, Asynchronous instabilities of crime hotspots for a 1-D reaction-diffusion model of urban crime with focused police patrol, SIAM J. Appl. Dyn. Syst., 17 (2018), 2018–2075. https://doi.org/10.1137/17M1162585 doi: 10.1137/17M1162585
|
| [29] |
A. Buttenschoen, T. Kolokolnikov, M. Ward, J. Wei, Cops-on-the-dots: the linear stability of crime hotspots for a 1-D reaction-diffusion model of urban crime, Eur. J. Appl. Math., 31 (2020), 871–917. https://doi.org/10.1017/S0956792519000305 doi: 10.1017/S0956792519000305
|
| [30] |
B. Li, L. Xie, Global large-data generalized solutions to a two-dimensional chemotaxis system stemming from crime modelling, Discrete Contin. Dyn. Syst. Ser. B, 2022 (2022). https://doi.org/10.3934/dcdsb.2022167 doi: 10.3934/dcdsb.2022167
|
| [31] |
N. Rodríguez, Q. Wang, L. Zhang, Understanding the effects of on- and off-hotspot policing: Evidence of hotspot, oscillating, and chaotic activities, SIAM J. Appl. Dyn. Syst., 20 (2021), 1882–1916. https://doi.org/10.1137/20M1359572 doi: 10.1137/20M1359572
|
| [32] |
E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399–415. https://doi.org/10.1016/0022-5193(70)90092-5 doi: 10.1016/0022-5193(70)90092-5
|
| [33] |
K. Fujie, Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675–684. https://doi.org/10.1016/j.jmaa.2014.11.045 doi: 10.1016/j.jmaa.2014.11.045
|
| [34] |
M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176–190. https://doi.org/10.1002/mma.1346 doi: 10.1002/mma.1346
|
| [35] |
J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Meth. Appl. Sci., 39 (2016), 394–404. https://doi.org/10.1002/mma.3489 doi: 10.1002/mma.3489
|
| [36] |
C. Stinner, M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonliear Anal. Real Word Appl., 12 (2011), 3727–3740. https://doi.org/10.1016/j.nonrwa.2011.07.006 doi: 10.1016/j.nonrwa.2011.07.006
|
| [37] |
J. Lankeit, M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: global solvability for large nonradial data, NoDEA-Nonlinear Differ. Equ. Appl., 24 (2017). https://doi.org/10.1007/s00030-017-0472-8 doi: 10.1007/s00030-017-0472-8
|
| [38] |
A. Zhigun, Generalised supersolutions with mass control for the Keller-Segel system with logarithmic sensitivity, J. Math. Anal. Appl., 467 (2018), 1270–1286. https://doi.org/10.1016/j.jmaa.2018.08.001 doi: 10.1016/j.jmaa.2018.08.001
|
| [39] |
M. Winkler, Unlimited growth in logarithmic Keller-Segel systems, J. Differ. Equations, 309 (2022), 74–97. https://doi.org/10.1016/j.jde.2021.11.026 doi: 10.1016/j.jde.2021.11.026
|
| [40] |
M. Winkler, T. Yokota, Stabilization in the logarithmic Keller-Segel system, Nonlinear Anal., 170 (2018), 123–141. https://doi.org/10.1016/j.na.2018.01.002 doi: 10.1016/j.na.2018.01.002
|
| [41] |
J. Ahn, Global well-posedness and asymptotic stabilization for chemotaxis system with signal-dependent sensitivity, J. Differ. Equations, 266 (2019), 6866–6904. https://doi.org/10.1016/j.jde.2018.11.015 doi: 10.1016/j.jde.2018.11.015
|
| [42] |
Q. Hou, Z. Wang, K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differ. Equations, 261 (2016), 5035–5070. https://doi.org/10.1016/j.jde.2016.07.018 doi: 10.1016/j.jde.2016.07.018
|
| [43] |
H. Jin, J. Li, Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differ. Equations, 255 (2013), 193–219. doilinkhttps://doi.org/10.1016/j.jde.2013.04.002 doi: 10.1016/j.jde.2013.04.002
|
| [44] |
H. Li, K. Zhao, Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differ. Equations, 258 (2015), 302–308. https://doi.org/10.1016/j.jde.2014.09.014 doi: 10.1016/j.jde.2014.09.014
|
| [45] |
J. Li, T. Li, Z. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819–2849. https://doi.org/10.1142/S0218202514500389 doi: 10.1142/S0218202514500389
|
| [46] |
Y. Tao, L. Wang, Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 821–845. https://doi.org/10.3934/dcdsb.2013.18.821 doi: 10.3934/dcdsb.2013.18.821
|
| [47] |
Z. Wang, Z. Xiang, P. Yu, Asymptotic dynamics in a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differ. Equations, 260 (2016), 2225–2258. https://doi.org/10.1016/j.jde.2015.09.063 doi: 10.1016/j.jde.2015.09.063
|
| [48] |
M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption, J. Differ. Equations, 264 (2018), 2310–2350. https://doi.org/10.1016/j.jde.2017.10.029 doi: 10.1016/j.jde.2017.10.029
|
| [49] |
M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987–1024. https://doi.org/10.1142/S0218202516500238 doi: 10.1142/S0218202516500238
|
| [50] | B. Li, L. Xie, Generalized solution and its eventual smoothness to a logarithmic Keller-Segel system for criminal activities, submitted for publication, 2022. |
| [51] |
M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092–3115. https://doi.org/dx.doi.org/10.1137/140979708 doi: 10.1137/140979708
|
| [52] |
M. Aida, K. Osaka, T. Tsujikawa, M. Mimura, Chemotaxis and growth system with sigular sensitivity function, Nonliear Anal. Real Word Appl., 6 (2005), 323–336. https://doi.org/10.1016/j.nonrwa.2004.08.011 doi: 10.1016/j.nonrwa.2004.08.011
|
| [53] |
X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891–1904. https://doi.org/10.3934/dcds.2015.35.1891 doi: 10.3934/dcds.2015.35.1891
|
| [54] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equations, 248 (2010), 2889–2905. https://doi.org/10.1016/j.jde.2010.02.008 doi: 10.1016/j.jde.2010.02.008
|
| [55] |
J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1996), 65–96. https://doi.org/10.1007/BF01762360 doi: 10.1007/BF01762360
|
| [56] | O. Ladyzhenskaya, N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. |
Bin Li, Zhi Wang, Li Xie. Regularization effect of the mixed-type damping in a higher-dimensional logarithmic Keller-Segel system related to crime modeling[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4532-4559. doi: 10.3934/mbe.2023210
DownLoad: