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

TL;DR

This paper proves the global-in-time boundedness of solutions to the Keller-Segel system with logistic growth in the entire space, extending previous bounded domain results by developing local-in-space estimates.

Contribution

It establishes the global boundedness of solutions in ree space for the first time, using localization techniques to overcome previous limitations.

Findings

Solutions are globally bounded in ree space for large .

The approach relies on local-in-space estimates and the ffect of the ounded norm.

The results extend previous bounded domain findings to the entire space.

Abstract

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.