Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation

<jats:p>The system</jats:p> <jats:p> <jats:disp-formula> <jats:tex-math>\left\{\begin{matrix} u_{t} = \mathrm{\Delta }u−\chi \mathrm{∇} \cdot \left(\frac{u}{v}\mathrm{∇}v\right)−uv + B_{1}(x,t), \\ v_{t} = \mathrm{\Delta }v + uv−v + B_{2}(x,t), \\ \end{matrix}\right.\:\:( \star )</jats:tex-math> </jats:disp-formula> </jats:p> <jats:p> is considered in a disk <jats:inline-formula> <jats:tex-math>\mathrm{\Omega } \subset \mathbb{R}^{2}</jats:tex-math> </jats:inline-formula> , with a positive parameter <jats:inline-formula> <jats:tex-math>χ</jats:tex-math> </jats:inline-formula> and given nonnegative and suitably regular functions <jats:inline-formula> <jats:tex-math>B_{1}</jats:tex-math> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math>B_{2}</jats:tex-math> </jats:inline-formula> defined on <jats:inline-formula> <jats:tex-math>\mathrm{\Omega } \times (0,\infty )</jats:tex-math> </jats:inline-formula> . In the particular version obtained when <jats:inline-formula> <jats:tex-math>\chi = 2</jats:tex-math> </jats:inline-formula> , ( <jats:inline-formula> <jats:tex-math>\star</jats:tex-math> </jats:inline-formula> ) was proposed in [31] as a model for crime propagation in urban regions. </jats:p> <jats:p> Within a suitable generalized framework, it is shown that under mild assumptions on the parameter functions and the initial data the no-flux initial-boundary value problem for ( <jats:inline-formula> <jats:tex-math>\star</jats:tex-math> </jats:inline-formula> ) possesses at least one global solution in the case when all model ingredients are radially symmetric with respect to the center of <jats:inline-formula> <jats:tex-math>Ω</jats:tex-math> </jats:inline-formula> . Moreover, under an additional hypothesis on stabilization of the given external source terms in both equations, these solutions are shown to approach the solution of an elliptic boundary value problem in an appropriate sense. </jats:p> <jats:p>The analysis is based on deriving a priori estimates for a family of approximate problems, in a first step achieving some spatially global but weak initial regularity information which in a series of spatially localized arguments is thereafter successively improved.</jats:p> <jats:p> To the best of our knowledge, this is the first result on global existence of solutions to the two-dimensional version of the full original system ( <jats:inline-formula> <jats:tex-math>\star</jats:tex-math> </jats:inline-formula> ) for arbitrarily large values of <jats:inline-formula> <jats:tex-math>χ</jats:tex-math> </jats:inline-formula> . </jats:p>