Acta Applicandae Mathematicae ( IF 1.6 ) Pub Date : 2021-02-08 , DOI: 10.1007/s10440-021-00392-8 Zhi-An Wang , Jiashan Zheng
This paper establishes the global uniform-in-time boundedness of solutions to the following Keller-Segel system with signal-dependent diffusion and chemotaxis
$$ \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\nabla \cdot (\gamma (v)\nabla u-u\phi (v)\nabla v),\quad & x \in \Omega , t>0, \\ \displaystyle { v_{t}=d\Delta v- v+u},\quad & x\in \Omega , t>0 \\ \end{array}\displaystyle \right . $$in a bounded domain \(\Omega \subset \mathbb{R}^{N}(N\leq 4)\) with smooth boundary, where the density-dependent motility functions \(\gamma (v)\) and \(\phi (v)\) denote the diffusive and chemotactic coefficients, respectively. The model was originally proposed by Keller and Segel in (J. Theor. Biol. 30:225–234, 1970) to describe the aggregation phase of Dictyostelium discoideum cells, where the two motility functions satisfy a proportional relation \(\phi (v)=(\alpha -1)\gamma '(v)\) with \(\alpha >0\) 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 \(\gamma (v)>0\) and \(\phi (v)>0\) are smooth on \([0,\infty )\) and satisfy
$$ \inf _{v\geq 0} \frac{d\gamma (v)}{v\phi (v)(v\phi (v)+d-\gamma (v))_{+}}>\frac{N}{2}, $$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 \(\int _{\Omega }u^{p}v^{-q}dx\) 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.
中文翻译：
具有信号依赖性的完全抛物线Keller-Segel系统的全局有界性
本文建立了具有信号依赖扩散和趋化作用的以下Keller-Segel系统的整体时间一致有界性
$$ \ left \ {\ textstyle \ begin {array} {l @ {\ quad} l} u_ {t} = \ nabla \ cdot（\ gamma（v）\ nabla uu \ phi（v）\ nabla v）， \ quad＆x \ in \ Omega，t> 0，\\ \ displaystyle {v_ {t} = d \ Delta v- v + u}，\ quad＆x \ in \ Omega，t> 0 \\ \ end {数组} \ displaystyle \ right。$$在具有平滑边界的有界域\（\ Omega \ subset \ mathbb {R} ^ {N}（N \ leq 4）\）中，其中依赖于密度的运动函数\ {\ gamma（v）\）和\（ \ phi（v）\）分别表示扩散系数和趋化系数。该模型最初是由Keller和Segel在（J. Theor。Biol。30：225–234，1970）中提出的，用于描述Dictyostelium discoideum细胞的聚集阶段，其中两个运动功能满足比例关系\（\ phi（v ）=（\\ alpha -1）\ gamma'{v} \）与\（\ alpha> 0 \）表示有效体长（即受体之间的距离）与步长之比。分析中的主要技术困难是扩散的简并性。在这项工作中，我们证明如果\（\ gamma（v）> 0 \）和\（\ phi（v）> 0 \）在\（[0，\ infty} \）上是光滑的，并且满足
$$ \ inf _ {v \ geq 0} \ frac {d \ gamma（v）} {v \ phi（v）（v \ phi（v）+ d- \ gamma（v））_ {+}}> \ frac {N} {2}，$$则上述的受Neumann边界条件约束的Keller-Segel系统接受了时间均匀有界的经典解。证明我们的结果的主要思想是对\（p> \ frac {N} {2}的加权函数\（\ int _ {\ Omega} u ^ {p} v ^ {-q} dx \）的估计\）通过选择合适的指数\（p \）取决于未知\（v \） ，使我们能够得到一个均匀\（L ^ {\ infty} \）范数的\（v \）和因此排除了扩散退化。
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