{"intvolume":" 33","date_created":"2025-12-18T19:23:28Z","date_updated":"2025-12-18T20:08:49Z","volume":33,"type":"journal_article","publication_identifier":{"issn":["0956-7925","1469-4425"]},"status":"public","author":[{"full_name":"RODRIGUEZ, NANCY","last_name":"RODRIGUEZ","first_name":"NANCY"},{"first_name":"Michael","id":"31496","full_name":"Winkler, Michael","last_name":"Winkler"}],"abstract":[{"text":"We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system:\r\n\\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*}\r\nwhich was introduced by Short et al. in [40] with \r\n$\\chi=2$\r\n to describe the dynamics of urban crime.In bounded intervals \r\n$\\Omega\\subset\\mathbb{R}$\r\n and with prescribed suitably regular non-negative functions \r\n$B_1$\r\n and \r\n$B_2$\r\n, we first prove the existence of global classical solutions for any choice of \r\n$\\chi>0$\r\n 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 \r\n$B_1$\r\n and \r\n$B_2$\r\n. Indeed, for arbitrary \r\n$\\chi>0$\r\n, we obtain boundedness of the solutions given strict positivity of the average of \r\n$B_2$\r\n over the domain; moreover, it is seen that imposing a mild decay assumption on \r\n$B_1$\r\n implies that u must decay to zero in the long-term limit. Our final result, valid for all \r\n$\\chi\\in\\left(0,\\frac{\\sqrt{6\\sqrt{3}+9}}{2}\\right),$\r\n which contains the relevant value \r\n$\\chi=2$\r\n, states that under the above decay assumption on \r\n$B_1$\r\n, if furthermore \r\n$B_2$\r\n appropriately stabilises to a non-trivial function \r\n$B_{2,\\infty}$\r\n, then (u,v) approaches the limit \r\n$(0,v_\\infty)$\r\n, where \r\n$v_\\infty$\r\n denotes the solution of \r\n\\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*}\r\nWe conclude with some numerical simulations exploring possible effects that may arise when considering large values of \r\n$\\chi$\r\n not covered by our qualitative analysis. We observe that when \r\n$\\chi$\r\n increases, solutions may grow substantially on short time intervals, whereas only on large timescales diffusion will dominate and enforce equilibration.","lang":"eng"}],"language":[{"iso":"eng"}],"publication_status":"published","doi":"10.1017/s0956792521000279","issue":"5","publisher":"Cambridge University Press (CUP)","publication":"European Journal of Applied Mathematics","user_id":"31496","_id":"63297","title":"On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime","citation":{"ieee":"N. RODRIGUEZ and M. Winkler, “On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime,” European Journal of Applied Mathematics, vol. 33, no. 5, pp. 919–959, 2021, doi: 10.1017/s0956792521000279.","mla":"RODRIGUEZ, NANCY, and Michael Winkler. “On the Global Existence and Qualitative Behaviour of One-Dimensional Solutions to a Model for Urban Crime.” European Journal of Applied Mathematics, vol. 33, no. 5, Cambridge University Press (CUP), 2021, pp. 919–59, doi:10.1017/s0956792521000279.","bibtex":"@article{RODRIGUEZ_Winkler_2021, title={On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime}, volume={33}, DOI={10.1017/s0956792521000279}, number={5}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={RODRIGUEZ, NANCY and Winkler, Michael}, year={2021}, pages={919–959} }","ama":"RODRIGUEZ N, Winkler M. On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime. European Journal of Applied Mathematics. 2021;33(5):919-959. doi:10.1017/s0956792521000279","short":"N. RODRIGUEZ, M. Winkler, European Journal of Applied Mathematics 33 (2021) 919–959.","apa":"RODRIGUEZ, N., & Winkler, M. (2021). On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime. European Journal of Applied Mathematics, 33(5), 919–959. https://doi.org/10.1017/s0956792521000279","chicago":"RODRIGUEZ, NANCY, and Michael Winkler. “On the Global Existence and Qualitative Behaviour of One-Dimensional Solutions to a Model for Urban Crime.” European Journal of Applied Mathematics 33, no. 5 (2021): 919–59. https://doi.org/10.1017/s0956792521000279."},"year":"2021","page":"919-959"}