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Global existence of classical solutions to a system modeling propagation of urban crime with quadratic logistic damping
Abstract
We are concerned with the following partial differential equations arising from urban crime modeling: labelmain ut = Δu − χ∇ · ( u (∇ v)/v ) − uv + B1 + ru − μu2, [4pt] vt = Δv − v + uv + B2, under no-flux boundary conditions in a smoothly bounded domain Ω⊂ ℝn with n ≥ 2, where χ, r, and μ are positive constants. It is shown in this paper that if the nonnegative source terms B1 and B2 are sufficiently regular and μ> 3/n + 1/n( χn + (n(χ(n−1)−2)2)/(4(n−1)) )(n+1)/n · ( ((2n+√n)2)/(2n−1) )1/n + n12n+3, then the system \eqref{main} with suitably regular initial data possesses a global classical solution. Moreover, under the assumption that labelcond.B2 inft>0 ∫ΩB2(·,t) > 0, the solutions are uniformly bounded in time.
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BibTeX
@misc{le2026global,
title = {{Global existence of classical solutions to a system modeling propagation of urban crime with quadratic logistic damping}},
author = {Minh Le},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23373},
eprint = {2609.23373},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23373},
}
RIS
TY - UNPB
TI - Global existence of classical solutions to a system modeling propagation of urban crime with quadratic logistic damping
AU - Minh Le
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23373
UR - https://arxiv.org/abs/2609.23373
AB - We are concerned with the following partial differential equations arising from urban crime modeling: labelmain u_t = Δu − χ∇ · ( u (∇ v)/v ) − uv + B_1 + ru − μu^2, [4pt] v_t = Δv − v + uv + B_2, under no-flux boundary conditions in a smoothly bounded domain Ω⊂ ℝ^n with n ≥ 2, where χ, r, and μ are positive constants. It is shown in this paper that if the nonnegative source terms B_1 and B_2 are sufficiently regular and μ> 3/n + 1/n( χn + (n(χ(n−1)−2)^2)/(4(n−1)) )^(n+1)/n · ( ((2n+√n)^2)/(2n−1) )^1/n + n^12n+3, then the system \eqref{main} with suitably regular initial data possesses a global classical solution. Moreover, under the assumption that labelcond.B2 inf_t>0 ∫_ΩB_2(·,t) > 0, the solutions are uniformly bounded in time.
ER -