Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Relaxation by nonlinear diffusion enhancement in a two-dimensional
  cross-diffusion model for urban crime propagation   We consider a class of macroscopic models for the spatio-temporal evolution
of urban crime, as originally going back to Short et al. (Math. Mod. Meth.
Appl. Sci. 18, 2008). The focus here is on the question how far a certain
nonlinear enhancement in the random diffusion of criminal agents may exert
visible relaxation effects. Specifically, in the context of the system
\begin{eqnarray*}
  \left\{ \begin{array}{l}
  u_t = \nabla \cdot (u^{m-1} \nabla u) - \chi \nabla \cdot \Big(\frac{u}{v}
\nabla v \Big) - uv + B_1(x,t), \\[1mm]
  v_t = \Delta v +uv - v + B_2(x,t),
  \end{array} \right. \end{eqnarray*}
  it is shown that whenever $\chi>0$ and the given nonnegative source terms
$B_1$ and $B_2$ are sufficiently regular, the assumption \begin{eqnarray*}
m>\frac{3}{2} \end{eqnarray*} is sufficient to ensure that a corresponding
Neumann-type initial-boundary value problem, posed in a smoothly bounded planar
convex domain, admits locally bounded solutions for a wide class of arbitrary
initial data. Furthermore, this solution is seen to be globally bounded if both
$B_1$ and $B_2$ are bounded and $\liminf_{t\to\infty} \int_\Omega B_2(\cdot,t)$
is positive. This is supplemented by numerical evidence which, besides
illustrating associated smoothing effects in particular situations of sharply
structured initial data in the presence of such porous medium type diffusion
mechanisms, indicates a significant tendency toward support of singular
structures in the linear diffusion case $m=1$.

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The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys