论文标题
具有奇异灵敏度的两个物种趋化效果系统：全球存在，有限和持久性
Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence
论文作者
论文摘要
本文涉及以下具有奇异灵敏度和Lotka-Volterra竞争动力学的抛物线抛物性 - 羟基递质趋化性系统， \ begin {equination} \ begin {case} u_t =ΔU-χ_1\ nabla \ cdot（\ frac {\ frac {u} {w} {w} \ nabla w）+u（a_1-b_1u-c_1v） （\ frac {v} {w} \ nabla w）+v（a_2-b_2v-c_2u），\ quad＆x \ inω\ cr 0 =Δw-μw+νu+νu+νu+λv，\ quad＆x \ quad＆x \ inm. v} {\ partial n} = \ frac {\ partial w} {\ partial n} = 0，\ quad＆x \ in \partialΩ，\ end {cases} \ end {case} \ end {equication} $ b_i $，$ c_i $（$ i = 1,2 $）和$μ，\，ν，\，λ$是正常数。这是具有奇异灵敏度和Lotka-volterra竞争动力学的两种趋化性系统上的第一项作品。除其他外，我们证明，对于任何给定的非负初始数据$ u_0，v_0 \ in c^0（\barΩ）$，带有$ u_0+v_0+v_0 \ not \ equiv 0 $，（0.1）具有独特的全球定义经典解决方案$（u（t，x; x; x; u_0，v_0，v_0，v_0，v_0，v_0，v（t，t，t，x; u_0，v_0）使用$ u（0，x; u_0，v_0）= u_0（x）$和$ v（0，x; u_0，v_0）= v_0（x）$，规定$ \ min \ {a_1，a_2 \} $相对于$χ_1，χ_2$ andχ_2$和$ _2 $和$ u_0+v_0 $不是小。此外，在相同的条件下，我们证明\ begin {equation*} \ limsup_ {t \ to \ infty} \ | u（t，\ cdot; u_0，v_0）+v（t，\ cdot; u_0，v_0，v_0）\ | _ \ | _ \ infty \ infty \ le m^*， \ liminf_ {t \ to \ infty} \ inf_ {x \inΩ}（u（t，t，x，x，u_0，v_0，v_0）+v（t，x; u_0，v_0，v_0））\ ge m^*，\ ge m^*，\ end End {equation {equation*}对于某些积极的常数$ m^*，m^*，m^*，m^*，m^*$ us of $ u _0 $ u_0 $ u_0，commentive of $ u_0 $ u_0，commentive of $ u_0 $ u_0 $ ud $ u_0，commention commention，持久性。
This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begin{equation} \begin{cases} u_t=Δu-χ_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) ,\quad &x\in Ω\cr v_t=Δv-χ_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u),\quad &x\in Ω\cr 0=Δw-μw +νu+ λv,\quad &x\in Ω\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, and $χ_i$, $a_i$, $b_i$, $ c_i$ ($i=1,2$) and $μ,\, ν, \, λ$ are positive constants. This is the first work on two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra competitive kinetics. Among others, we prove that for any given nonnegative initial data $u_0,v_0\in C^0(\barΩ)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution $(u(t,x;u_0,v_0),v(t,x;u_0,v_0),w(t,x;u_0,v_0))$ with $u(0,x;u_0,v_0)=u_0(x)$ and $v(0,x;u_0,v_0)=v_0(x)$ provided that $\min\{a_1,a_2\}$ is large relative to $χ_1,χ_2$ and $u_0+v_0$ is not small. Moreover, under the same condition, we prove that \begin{equation*} \limsup_{t\to\infty} \|u(t,\cdot;u_0,v_0)+v(t,\cdot;u_0,v_0)\|_\infty\le M^*, \end{equation*} and \begin{equation*} \liminf_{t\to\infty} \inf_{x\inΩ}(u(t,x,u_0,v_0)+v(t,x;u_0,v_0))\ge m^*, \end{equation*} for some positive constants $M^*,m^*$ independent of $u_0,v_0$, the latter is referred to as combined pointwise persistence.
