arXiv 1406.01042014-06math.AP

次临界质量Patlak-Keller-Segel模型胞元密度向平衡态的一致收敛性的指数速度

Exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a Patlak-Keller-Segel model

Alexandre Montaru

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摘要：

本文研究了细胞的趋化聚集模型，更精确地说，研究了一个抛物-椭圆型半线性Patlak-Keller-Segel系统在一个$\mathbb{R}^N$球中对$N\geq 2$的趋化聚集.对于$N=2$，该系统以其临界质量$8π$而闻名。在\cite{Montaru2}中已经证明，对于$N\geq 3$，它也表现出临界质量现象。本文的主要结果是指数速度的一致收敛性的径向解决方案的唯一的稳定状态的亚临界情况下$N\geq 2$。我们强调，这涵盖了特别是经典的凯勒-西格尔系统与$N=2$，结果改善了已知的结果，即使是这个最研究的问题。一个关键的工具是一个相关的一维退化抛物问题$(PDE_m)$，其中$m$是成比例的细胞的总质量。证明利用其形式梯度流结构$u_t=-\nabla \mathcal{F}[u(t)]$上的“无限维黎曼流形“。特别地，我们证明了一个新的Hardy型不等式，等价于$\mathcal{F}$在亚临界质量的任何稳态下的严格凸性，这解释了指数收敛速度。

英文摘要：

This paper is concerned with a chemotaxis aggregation model for cells, more precisely with a parabolic-elliptic semilinear Patlak-Keller-Segel system in a ball of $\mathbb{R}^N$ for $N\geq 2$. For $N=2$, this system is well known for its critical mass $8π$. It has been proved in \cite{Montaru2} that it also exhibits a critical mass phenomenon for $N\geq 3$. The main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case for $N\geq 2$. We stress that this covers in particular the classical Keller-Segel system with $N=2$, and that the result improves on the known results even for this most studied problem. A key tool is an associated one-dimensional degenerate parabolic problem $(PDE_m)$ where $m$ is proportional to the total mass of cells. The proof exploits its formal gradient flow structure $u_t=-\nabla \mathcal{F}[u(t)]$ on an "infinite dimensional Riemannian manifold". In particular, we show a new Hardy type inequality, equivalent to the strict convexity of $\mathcal{F}$ at any steady state of subcritical mass, which heuristically explains the exponential speed of convergence.

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