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On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime

Part of: Parabolic equations and systems Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  02 November 2021

NANCY RODRIGUEZ Open the ORCID record for NANCY RODRIGUEZ [Opens in a new window]  and
MICHAEL WINKLER
NANCY RODRIGUEZ
Affiliation:
CU Boulder, Department of Applied Mathematics, Engineering Center, ECOT 225 Boulder, CO 80309-0526, USA email: [email protected]
MICHAEL WINKLER
Affiliation:
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany email: [email protected]
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Abstract

We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system:

\begin{eqnarray*} \left\{ \begin{array}{ll} u_t = u_{xx} - \chi \big(\frac{u}{v} \partial_x v \big)_x - uv +B_1(x,t), \qquad & x\in \Omega, \ t>0, \\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \qquad & x\in \Omega, \ t>0, \end{array} \right. \end{eqnarray*}
which was introduced by Short et al. in [40] with $\chi=2$ to describe the dynamics of urban crime.

In bounded intervals $\Omega\subset\mathbb{R}$ and with prescribed suitably regular non-negative functions $B_1$ and $B_2$ , we first prove the existence of global classical solutions for any choice of $\chi>0$ and all reasonably regular non-negative initial data.

We next address the issue of determining the qualitative behaviour of solutions under appropriate assumptions on the asymptotic properties of $B_1$ and $B_2$ . Indeed, for arbitrary $\chi>0$ , we obtain boundedness of the solutions given strict positivity of the average of $B_2$ over the domain; moreover, it is seen that imposing a mild decay assumption on $B_1$ implies that u must decay to zero in the long-term limit. Our final result, valid for all $\chi\in\left(0,\frac{\sqrt{6\sqrt{3}+9}}{2}\right),$ which contains the relevant value $\chi=2$ , states that under the above decay assumption on $B_1$ , if furthermore $B_2$ appropriately stabilises to a non-trivial function $B_{2,\infty}$ , then (u,v) approaches the limit $(0,v_\infty)$ , where $v_\infty$ denotes the solution of

\begin{eqnarray*} \left\{ \begin{array}{l} -\partial_{xx}v_\infty + v_\infty = B_{2,\infty}, \qquad x\in \Omega, \\[1mm] \partial_x v_{\infty}=0, \qquad x\in\partial\Omega. \end{array} \right. \end{eqnarray*}
We conclude with some numerical simulations exploring possible effects that may arise when considering large values of $\chi$ not covered by our qualitative analysis. We observe that when $\chi$ increases, solutions may grow substantially on short time intervals, whereas only on large timescales diffusion will dominate and enforce equilibration.

Keywords

Urban crimeglobal existencedecay estimateslong-time behaviour

MSC classification

Primary: 35Q91: PDEs in connection with game theory, economics, social and behavioral sciences
Secondary: 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 5 , October 2022 , pp. 919 - 959
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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