Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global-in-time Boundedness of solution for Cauchy problem to the
  Parabolic-Parabolic Keller-Segel system with logistic growth   We study global-in-time well-posedness and the behaviour and of the solution
to Cauchy problem in the classical Keller-Segel system with logistic term
\begin{equation*} \left. \aligned
  \partial_tn-\Delta n=&-\chi\nabla\cdot(n\nabla c)+\la n-\mu n^2
  \tau\partial_tc-\Delta c=&-c+n \endaligned
\right\}\quad\text{in}\,\,\,\RR^d\times\RR^+, \end{equation*} where $d\ge 1$,
$\tau,\, \chi,\, \mu>0$ and $\lambda\ge 0$. It's inspired by a previous result
\cite[M. Winkler, Commun. Part. Diff. Eq., 35 (2010), 1516-1537]{Win10}, where
the global-in-time boundedness of the above Keller-Segel system in smooth
\emph{bounded }convex domains is established for large $\mu$. However, his
approach in bounded domain ceases to directly apply in the entire space
$\RR^d$, and then they raised an interesting question whether a similar
global-in-time boundedness statement remains true of Cauchy problem. In this
paper, we answer this open problem by developing local-in-space estimates. More
precisely, we prove that the above Keller-Segel system possesses a uniquely
global-in-time bounded solution for any $\tau>0$ under the assumption that
$\mu$ is large. The key point of our proof heavily relies on localization in
space of solution caused by "local effect" of $L^\infty(\RR^d)$-norm.

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The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat