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

<jats:p>We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system:<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0956792521000279_eqnU1.png"/><jats:tex-math> \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 &amp; x\in \Omega, \ t&gt;0, \\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \qquad &amp; x\in \Omega, \ t&gt;0, \end{array} \right. \end{eqnarray*} </jats:tex-math></jats:alternatives></jats:disp-formula>which was introduced by Short <jats:italic>et al.</jats:italic> in [40] with <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline1.png"/><jats:tex-math> $\chi=2$ </jats:tex-math></jats:alternatives></jats:inline-formula> to describe the dynamics of urban crime.</jats:p><jats:p>In bounded intervals <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline2.png"/><jats:tex-math> $\Omega\subset\mathbb{R}$ </jats:tex-math></jats:alternatives></jats:inline-formula> and with prescribed suitably regular non-negative functions <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline3.png"/><jats:tex-math> $B_1$ </jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline4.png"/><jats:tex-math> $B_2$ </jats:tex-math></jats:alternatives></jats:inline-formula>, we first prove the existence of global classical solutions for any choice of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline5.png"/><jats:tex-math> $\chi&gt;0$ </jats:tex-math></jats:alternatives></jats:inline-formula> and all reasonably regular non-negative initial data.</jats:p><jats:p>We next address the issue of determining the qualitative behaviour of solutions under appropriate assumptions on the asymptotic properties of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline6.png"/><jats:tex-math> $B_1$ </jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline7.png"/><jats:tex-math> $B_2$ </jats:tex-math></jats:alternatives></jats:inline-formula>. Indeed, for arbitrary <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline8.png"/><jats:tex-math> $\chi&gt;0$ </jats:tex-math></jats:alternatives></jats:inline-formula>, we obtain boundedness of the solutions given strict positivity of the average of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline9.png"/><jats:tex-math> $B_2$ </jats:tex-math></jats:alternatives></jats:inline-formula> over the domain; moreover, it is seen that imposing a mild decay assumption on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline10.png"/><jats:tex-math> $B_1$ </jats:tex-math></jats:alternatives></jats:inline-formula> implies that <jats:italic>u</jats:italic> must decay to zero in the long-term limit. Our final result, valid for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline11.png"/><jats:tex-math> $\chi\in\left(0,\frac{\sqrt{6\sqrt{3}+9}}{2}\right),$ </jats:tex-math></jats:alternatives></jats:inline-formula> which contains the relevant value <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline12.png"/><jats:tex-math> $\chi=2$ </jats:tex-math></jats:alternatives></jats:inline-formula>, states that under the above decay assumption on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline13.png"/><jats:tex-math> $B_1$ </jats:tex-math></jats:alternatives></jats:inline-formula>, if furthermore <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline14.png"/><jats:tex-math> $B_2$ </jats:tex-math></jats:alternatives></jats:inline-formula> appropriately stabilises to a non-trivial function <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline15.png"/><jats:tex-math> $B_{2,\infty}$ </jats:tex-math></jats:alternatives></jats:inline-formula>, then (<jats:italic>u</jats:italic>,<jats:italic>v</jats:italic>) approaches the limit <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline16.png"/><jats:tex-math> $(0,v_\infty)$ </jats:tex-math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline17.png"/><jats:tex-math> $v_\infty$ </jats:tex-math></jats:alternatives></jats:inline-formula> denotes the solution of <jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0956792521000279_eqnU2.png"/><jats:tex-math> \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*} </jats:tex-math></jats:alternatives></jats:disp-formula>We conclude with some numerical simulations exploring possible effects that may arise when considering large values of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline18.png"/><jats:tex-math> $\chi$ </jats:tex-math></jats:alternatives></jats:inline-formula> not covered by our qualitative analysis. We observe that when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792521000279_inline19.png"/><jats:tex-math> $\chi$ </jats:tex-math></jats:alternatives></jats:inline-formula> increases, solutions may grow substantially on short time intervals, whereas only on large timescales diffusion will dominate and enforce equilibration.</jats:p>