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.