论文标题
具有奇异灵敏度和逻辑来源的趋化系统：界限，持久性，吸收集和整个解决方案
Chemotaxis systems with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions
论文作者
论文摘要
本文讨论了以下具有奇异敏感性和逻辑来源的抛物线纤维性趋化系统，\ begin {equination} \ begin {case} u_t =Δu-χ\ nabla \ nabla \ cdot（\ frac {u} {u} 0 =ΔV-μV+νu，＆x \ inω，\ cr \ frac {\ partial u} {\ partial n} = \ frac {\ partial v} {\ partial v} {\ partial n} = 0 \ Mathbb {r}^n $是一个平稳的界面域，$ a（t，x）$和$ b（t，x）$是正平滑功能，$χ$，$μ$和$ν$是正常数。在最近的论文[25]中，我们证明，对于给定的非负初始功能$ 0 \ not \ equiv U_0 \ in C^0（\ barω）$和$ s \ in \ Mathbb {r} $（0.1），（0.1）具有独特的全球定义经典解决方案$（u（t，x; s，x; s，u_0），v（u_0），v（u_0），v（t; $ u（s，x; s，u_0）= u_0（x）$，前提是$ a _ {\ inf} = \ inf_ {t \ in \ mathbb {r}，x \inΩ} a（t，x，x）$相对于$χ$和$ u_0 $不小。在本文中，我们进一步研究了（0.1）全球定义的正溶液的定性属性，假设$ a _ {\ inf} $相对于$χ$，$ u_0 $不小。除其他外，我们还提供了一些$ \int_Ωu^{ - p} $和$ \int_Ωu^q $的具体估计值，对于某些$ p> 0 $和$ q>> \ max \ {2，n \} $，并证明任何全球定义的积极解决方案最终都受到独立于其初始函数独立于某些积极的常数。我们证明存在``矩形''类型的有限不变量集（以$ l^q $）的形式，最终吸引了所有全球定义的积极解决方案。我们还证明（0.1）具有正面的整个经典解决方案$（u^*（t，x），v^*（t，x））$，在$ t $ a（t，x）$（t，x）$和$ b（t，x）$中是$ t $中的周期性的，$ t $ in $ t $，并且独立于$ t $（t，x）$和$ b（t，x）$ b（t，x）$独立于$ t $独立于$ t $。
This paper deals with the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot (\frac{u}{v} \nabla v)+u(a(t,x)-b(t,x) u), & x\in Ω,\cr 0=Δv- μv+ νu, & x\in Ω, \cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $a(t,x)$ and $b(t,x)$ are positive smooth functions, and $χ$, $μ$ and $ν$ are positive constants. In the very recent paper [25], we proved that for given nonnegative initial function $0\not\equiv u_0\in C^0(\bar Ω)$ and $s\in\mathbb{R}$, (0.1) has a unique globally defined classical solution $(u(t,x;s,u_0),v(t,x;s,u_0))$ with $u(s,x;s,u_0)=u_0(x)$, provided that $a_{\inf}=\inf_{t\in\mathbb{R},x\inΩ}a(t,x)$ is large relative to $χ$ and $u_0$ is not small. In this paper, we further investigate qualitative properties of globally defined positive solutions of (0.1) under the assumption that $a_{\inf}$ is large relative to $χ$ and $u_0$ is not small. Among others, we provide some concrete estimates for $\int_Ωu^{-p}$ and $\int_Ωu^q$ for some $p>0$ and $q>\max\{2,N\}$ and prove that any globally defined positive solution is bounded above and below eventually by some positive constants independent of its initial functions. We prove the existence of a ``rectangular'' type bounded invariant set (in $L^q$) which eventually attracts all the globally defined positive solutions. We also prove that (0.1) has a positive entire classical solution $(u^*(t,x),v^*(t,x))$, which is periodic in $t$ if $a(t,x)$ and $b(t,x)$ are periodic in $t$ and is independent of $t$ if $a(t,x)$ and $b(t,x)$ are independent of $t$.
