On the global existence and qualitative behavior of one-dimensional solutions to a model for urban crime
arXiv:1903.06331
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} - χ\big(\frac{u}{v} \partial_x u \big)_x - uv +B_1(x,t), \qquad & x\in Ω, \ t>0, \\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \qquad & x\in Ω, \ t>0, \end{array} \right. \end{eqnarray*} which was introduced by Short et al. in [Short2008] with to describe the dynamics of urban crime In bounded intervals and with prescribed suitably regular nonnegative functions and , we first prove the existence of global classical solutions for any choice of and all reasonably regular nonnegative initial data. We next address the issue of determining the qualitative behavior of solutions under appropriate assumptions on the asymptotic properties of and . Indeed, for arbitrary we obtain boundedness of the solutions given strict positivity of the average of over the domain; moreover, it is seen that imposing a mild decay assumption on implies that must decay to zero in the long-term limit. Our final result, valid for all which contains the relevant value , states that under the above decay assumption on , if furthermore appropriately stabilizes to a nontrivial function , then approaches the limit , where denotes the solution of \begin{eqnarray*} \left\{ \begin{array}{l} -\partial_{xx}v_\infty + v_\infty = B_{2,\infty}, \qquad x\in Ω, \\[1mm] \partial_x v_{\infty}=0, \qquad x\in\partialΩ. \end{array} \right. \end{eqnarray*}