| Abstract: |
| \begin{center}
\large{\textbf{Global solutions to a higher-dimensional chemotaxis system related to crime modelling}}
\end{center}
\vspace{1cm}
We are concerned with a problem that differs from the ususal setting both with respect to the system as well as the modelled process. Given a bounded domain $\Omega\subset\mathds{R}^n$ for $n\geq2$, a positive constant $\chi$ and two nonnegative functions $B_1$ and $B_2$ in $L^\infty(\Omega)$ (as well as Neumann boundary conditions and suitable initial data), we consider the set of coupled PDEs
\begin{equation*}
\begin{cases}
u_t=\laplacian u-\chi\nabla\cdot\braces{\frac{u}{v}\nabla v}-uv+B_1 \ &\text{in}\ \Omega\times(0,\infty),\
v_t=\laplacian v-v+uv+B_2\ &\text{in}\ \Omega\times(0,\infty)
\end{cases}
\end{equation*}
which several works ([1],[2]) have used to explain an aspect of the decision-making process of burglars. After Rodr\`{i}guez and Winkler have proven the existence and uniqueness of global solutions for $n=1$ in [3], it is our aim to add correspondent results for higher dimensions. The crucial question herein is how large we can choose $\chi$ while retaining our ability to detect the existence of global solutions.\vspace{2cm}\
{[1]Manasevich, R., Phan, Q. H., Souplet, P., Global existence of solutions for a chemotaxis-type system arising in crime modeling, European Journal of Applied mathematics 24(2), 2013}\
{[2] Short, M. B., D`Orsogna, M. R., Pasour, V. B., Tita, P. J., Brantingham, A. L., Bertozzi, A. L., Chayes, L. B., A statistical modelof criminal behavior, Math. Models Methods Appl. Sci. 18, 2008}\
{[3] Rodr\`{i}guez, N., Winkler, M., On the global existence and qualitative behavior of one-dimensional solutionsto a model for urban crime, preprint, 2017} |
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