具有弱奇异敏感性和非局部项的趋化系统在任意维数下的全局有界性
Global boundedness of the chemotaxis system with weakly singular sensitivity and nonlocal term in any dimension
- Dalian University of Technology（大连理工大学）
- China West Normal University（西华师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究具有弱奇异敏感性和非局部源的抛物-椭圆趋化系统，在任意维数下给出了经典解全局有界的充分条件，并利用Moser迭代在无需先验正下界的情况下获得一致估计。
AI中文摘要：
本文研究涉及弱奇异敏感性和非局部源的抛物-椭圆趋化系统：$u_t=\Delta u-\chi\nabla\cdot\left(\frac{u}{v^k}\nabla v\right) +u^\alpha\left(r-\mu\int_\Omega u^\beta\\,dx\right)$ 和 $0=\Delta v-v+u^\gamma$，在光滑有界域 $\Omega\subset\mathbb{R}^N$（$N\ge1$）中满足齐次Neumann边界条件，其中 $\chi,r,\mu,\gamma>0$，$k\in(0,1)$，$\alpha,\beta\ge1$。鉴于奇异敏感性引起的增强聚集以及非局部阻尼对过度种群增长的抑制，我们确定了解保持全局有界的充分条件。我们证明经典解在以下情况下全局有界：\par\smallskip \begin{center} \small \setlength{\tabcolsep}{5pt} \setlength{\arrayrulewidth}{0.3pt} \renewcommand{\arraystretch}{1.25} \begin{tabular}{c|c|c|c} \multirow{2}{*}{$\beta>1$} & \multirow{2}{*}{$\gamma=1$} & $N=1$ & $1\leq\alpha<1+2\beta$ \\\\ \cline{3-4} & & $N\geq2$ & $\begin{gathered} 1\leq\alpha<2,\quad \alpha+\beta>2+\tfrac{N}{2},\\\\[-0.5mm] \text{or}\quad 2\leq\alpha<1+\tfrac{2\beta}{N} \end{gathered}$ \\\\ \hline \multirow{2}{*}{$\beta=1$} & $\gamma=1$ & $N=1$ & \multirow{2}{*}{ $\begin{array}{l@{\quad}l@{\quad}l} \text{Case 1:} & 1\leq\alpha<1+\tfrac{2}{N} & \text{if}\quad m_0<\tfrac{r}{\mu},\\\\[4pt] \text{Case 2:} & \alpha\geq1 & \text{if}\quad m_0\geq\tfrac{r}{\mu} \end{array}$ } \\\\ \cline{2-3} & $0<\gamma<\tfrac{2}{N}$ & $N\geq2$ & \\\\ \end{tabular} \end{center} \par \smallskip \noindent 这里 $m_0:=\int_\Omega u_0$ 表示初始总质量。值得注意的是，一致时间 $L^p$ 估计和通过Moser迭代获得的 $L^\infty$ 界都是在没有首先建立 $v$ 的一致正下界的情况下推导出来的。
英文摘要：
This paper is concerned with the parabolic-elliptic chemotaxis system involving weakly singular sensitivity and a nonlocal source: $u_t=Δu-χ\nabla\cdot\left(\frac{u}{v^k}\nabla v\right) +u^α\left(r-μ\int_Ωu^β\,dx\right)$ and $0=Δv-v+u^γ$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(Ω\subset\mathbb{R}^N\) with \(N\ge1\), where \(χ,r,μ,γ>0\), \(k\in(0,1)\) and \(α,β\ge1\). In view of the enhanced aggregation induced by singular sensitivity and the suppression of excessive population growth by the nonlocal damping, we identify sufficient conditions under which solutions remain globally bounded. We show that classical solutions are globally bounded in the following cases: \par\smallskip \begin{center} \small \setlength{\tabcolsep}{5pt} \setlength{\arrayrulewidth}{0.3pt} \renewcommand{\arraystretch}{1.25} \begin{tabular}{c|c|c|c} \multirow{2}{*}{$β>1$} & \multirow{2}{*}{$γ=1$} & $N=1$ & $1\leqα<1+2β$ \\ \cline{3-4} & & $N\geq2$ & $\begin{gathered} 1\leqα<2,\quad α+β>2+\tfrac{N}{2},\\[-0.5mm] \text{or}\quad 2\leqα<1+\tfrac{2β}{N} \end{gathered}$ \\ \hline \multirow{2}{*}{$β=1$} & $γ=1$ & $N=1$ & \multirow{2}{*}{ $\begin{array}{l@{\quad}l@{\quad}l} \text{Case 1:} & 1\leqα<1+\tfrac{2}{N} & \text{if}\quad m_0<\tfrac{r}μ,\\[4pt] \text{Case 2:} & α\geq1 & \text{if}\quad m_0\geq\tfrac{r}μ \end{array}$ } \\ \cline{2-3} & $0<γ<\tfrac{2}{N}$ & $N\geq2$ & \\ \end{tabular} \end{center} \par \smallskip \noindent Here \(m_0:=\int_Ωu_0\) denotes the initial total mass. It is worth noting that both the uniform-in-time \(L^p\)-estimate and the \(L^\infty\)-bound obtained by Moser iteration are derived without first establishing a uniform positive lower bound for \(v\).
