论文标题
解决方案的全球及时界限是cauchy问题与抛物线抛物线凯勒 - 塞格系统具有物流增长的限制性
Global-in-time Boundedness of solution for Cauchy problem to the Parabolic-Parabolic Keller-Segel system with logistic growth
论文作者
论文摘要
我们研究了带有逻辑术语\ begin \ begin {equation*} \ left的经典凯勒 - 塞格系统中库奇问题的全球及时及时性和解决方案的行为和解决方案。 \对齐 \ 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 $λ\ge 0$.它的灵感来自先前的结果\ cite [m。温克勒，社区。部分。差异。等式，35（2010），1516-1537] {win10}，其中在Smooth \ emph {bounded} convex convex bonains中的上述keller-segel系统的全局界限是针对$ $ $的。但是，他在有限域中的方法停止直接应用于整个空间$ \ rr^d $，然后他们提出了一个有趣的问题，一个有趣的问题是否在凯奇问题上是否仍然如此。在本文中，我们通过制定局部空间估计来回答这个开放问题。更确切地说，我们证明，在假设$μ$很大的情况下，上述凯勒 - 塞格系统对于任何$τ> 0 $都具有独特的全球限制解决方案。我们的证明的关键点在很大程度上依赖于$ l^\ infty（\ rr^d）$ - norm-Norm的“局部效应”引起的解决方案空间的本地化。
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.
