论文标题
具有信号依赖性动力的完全抛物线凯勒 - 塞格系统的全球界限
Global boundedness of the fully parabolic Keller-Segel system with signal-dependent motilities
论文作者
论文摘要
本文建立了解决方案对以下凯勒 - 安排系统的全局均匀界限，并具有信号依赖性扩散和趋化性\ begin {equation} \ left \ left \ {\ oken {array} {ll} {ll} u_t = \ nabla \ nabla \ nabla \ cdot \ cd（v） ω，t> 0，\\ v_t =dΔv-v+u，\ quad＆x \ inω，t> 0 \ end {arnay} \ right。 $ ϕ（v）$分别表示扩散和趋化系数。该模型最初是由Keller和Segel在\ cite {Keller-1}中提出的，以描述Dictyostelium discoideum细胞的聚集阶段，其中两个运动函数满足成比例关系$χ（v）=（α-1）γ'（v）$ ting $α> 0 $ n $ tos y tose的尺寸（I.分析的主要技术困难是扩散的可能退化。在这项工作中，我们表明，如果$γ（v）> 0 $和$ ϕ（v）> 0 $在$ [0，\ infty）$上平滑，并且满足$$ \ inf_ {v \ geq0} \ frac {dγ（v）}在凯勒 - 塞格系统的上方受诺伊曼边界条件约束的情况下，经典的解决方案在时间上均匀边界。证明我们的结果的主要思想是对加权功能的$ \int_ΩU^{p} v^{ - q} dx $ for $ p> \ frac {n} {2} $的估计，通过选择一个不知名的$ v $，我们可以根据我们能够取决于符合的$ l^$ l^$ l^$ l^$ l^$ l^$ l^$ l^$ l-- $ p $ and and and and and and and of und^$ l^$ - 退化。
This paper establishes the global uniform-in-time boundedness of solutions to the following Keller-Setel system with signal-dependent diffusion and chemotaxis \begin{equation}\left\{ \begin{array}{ll} u_t=\nabla\cdot(γ(v)\nabla u - uϕ(v)\nabla v),\quad & x\in Ω, t>0,\\ v_t = dΔv- v+u,\quad & x\in Ω, t>0 \end{array}\right.\end{equation} in a bounded domain $Ω\subset\mathbb{R}^N(N\leq4)$ with smooth boundary, where the density-dependent motility functions $γ(v)$ and $ϕ(v)$ denote the diffusive and chemotactic coefficients, respectively. The model was originally proposed by Keller and Segel in \cite{Keller-1} to describe the aggregation phase of Dictyostelium discoideum cells, where the two motility functions satisfy a proportional relation $χ(v)=(α-1)γ'(v)$ with $α>0$ denoting the ratio of effective body length (i.e. distance between receptors) to the step size. The major technical difficulty in the analysis is the possible degeneracy of diffusion. In this work, we show that if $γ(v)>0$ and $ϕ(v)>0$ are smooth on $[0,\infty)$ and satisfy $$\inf_{v\geq0} \frac{dγ(v)}{vϕ(v)(vϕ(v)+d-γ(v))_+}>\frac{N}{2},$$ then the above Keller-Segel system subject to Neumann boundary conditions admits classical solutions uniformly bounded in time. The main idea of proving our results is the estimates of a weighted functional $\int_Ωu^{p}v^{-q}dx$ for $p>\frac{N}{2}$ by choosing a suitable exponent $p$ depending on the unknown $v$, by which we are able to derive a uniform $L^\infty$-norm of $v$ and hence rule out the diffusion degeneracy.
