  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 u \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 [Short2008] with $\chi=2$ to describe
the dynamics of urban crime
  In bounded intervals $\Omega\subset\mathbb{R}$ and with prescribed suitably
regular nonnegative
  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 nonnegative initial data. We next address the issue of determining
the qualitative behavior 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 stabilizes to a
  nontrivial 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*}
