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We study global-in-time well-posedness and the behaviour and of the solutionto 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}, wherethe global-in-time boundedness of the above Keller-Segel system in smooth\emph{bounded }convex domains is established for large $\mu$. However, hisapproach in bounded domain ceases to directly apply in the entire space$\RR^d$, and then they raised an interesting question whether a similarglobal-in-time boundedness statement remains true of Cauchy problem. In thispaper, we answer this open problem by developing local-in-space estimates. Moreprecisely, we prove that the above Keller-Segel system possesses a uniquelyglobal-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 inspace of solution caused by "local effect" of $L^\infty(\RR^d)$-norm.

Global-in-time Boundedness of solution for Cauchy problem to the Parabolic-Parabolic Keller-Segel system with logistic growth