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-Δn=&-χ\nabla\cdot(n\nabla c)+\la n-μn^2 τ\partial_tc-Δc=&-c+n \endaligned \right\}\quad\text{in}\,\,\,\RR^d\times\RR^+, \end{equation*} where $d\ge 1$, $τ,\, χ,\, μ>0$ and $λ\ge 0$. 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 $μ$. 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 $τ>0$ under the assumption that $μ$ 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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