论文标题
趋化系统中的界限，具有消耗的化学吸引剂和产生的化学物质
Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent
论文作者
论文摘要
我们研究了这种零升华的吸引力 - 渗透趋化性模型，具有线性和超线性产生$ g $的化学物质和sublinear速率$ f $ $ f $ $ f $用于趋化剂：\ begin {equination} \ label {cools_absstract} \ tag} （u \ nabla v） +ξ\ nabla \ cdot（u \ nabla w）＆\ text {in}ω\ times（0，t_ {max}），\\ v_t =Δv-f（Δv-f（u） \ text {in}ω\ times（0，t_ {max}）。 ％u（x，0）= u_0（x），\; v（x，0）= v_0（x）＆x \ in \barΩ。 \end{cases} \end{equation} In this problem, $Ω$ is a bounded and smooth domain of $\R^n$, for $n\geq 1$, $χ,ξ,δ>0$, $f(u)$ and $g(u)$ reasonably regular functions generalizing the prototypes $f(u)=K u^α$ and $g(u)=γu^l$, with $ k，γ> 0 $和适当的$α，l> 0 $。一旦表明任何足够光滑的$ u（x，0）= u_0（x）\ geq 0 $ and $ v（x，0）= v_0（x）\ geq 0 $产生独特的经典和非经典解决方案$（u，v，v，v，v，w）$ to \ eqref eqref {cools_abstract}，在$ ne $ n $ n n n n y the $ wement $ wement the $ weshibe the the the the the pectect the the the pection（这样的$（u_0，v_0）$，life span $ \ tm = \ infty $和$ u，v $和$ w $均匀地限制为$ω\ times（0，\ infty）$，（i）$ l = 1 $，$ n \ in \ in \ in \ in \ {1,2 \} $，$α\ in （0，\ frac {1} {2}+\ frac {1} {n}）\ cap（0,1）$和任何$ξ> 0 $，（ii）for $ l = 1 $，$ l = 1 $，$ n \ geq 3 $，$ n \ geq 3 $，$α\ in（in（0，\ frac {1}+frac {1}+frac {1}+frac {1}+\ frac and and}}数量取决于$χ\ lvert v_0 \ rvert_ {l^\ infty（ω）} $，（iii），$ l> 1 $ a> 1 $任何$ξ> 0 $，并且在任何维度设置中。最后，提出了关于逻辑和排斥作用对趋化现象的影响的指示性分析，它是通过将本文生产案例的此处得出的结果与\ cite {lankeitwangconsumptlogistic}中的结果进行比较。
We study this zero-flux attraction-repulsion chemotaxis model, with linear and superlinear production $g$ for the chemorepellent and sublinear rate $f$ for the chemoattractant: \begin{equation}\label{problem_abstract} \tag{$\Diamond$} \begin{cases} u_t= Δu - χ\nabla \cdot (u \nabla v)+ξ\nabla \cdot (u \nabla w) & \text{ in } Ω\times (0,T_{max}),\\ v_t=Δv-f(u)v & \text{ in } Ω\times (0,T_{max}),\\ 0= Δw - δw + g(u)& \text{ in } Ω\times (0,T_{max}). %u(x,0)=u_0(x), \; v(x,0)=v_0(x) & x \in \barΩ. \end{cases} \end{equation} In this problem, $Ω$ is a bounded and smooth domain of $\R^n$, for $n\geq 1$, $χ,ξ,δ>0$, $f(u)$ and $g(u)$ reasonably regular functions generalizing the prototypes $f(u)=K u^α$ and $g(u)=γu^l$, with $K,γ>0$ and proper $ α, l>0$. Once it is indicated that any sufficiently smooth $u(x,0)=u_0(x)\geq 0$ and $v(x,0)=v_0(x)\geq 0$ produce a unique classical and nonnegative solution $(u,v,w)$ to \eqref{problem_abstract}, which is defined in $Ω\times (0,T_{max})$, we establish that for any such $(u_0,v_0)$, the life span $\TM=\infty$ and $u, v$ and $w$ are uniformly bounded in $Ω\times (0,\infty)$, (i) for $l=1$, $n\in \{1,2\}$, $α\in (0,\frac{1}{2}+\frac{1}{n})\cap (0,1)$ and any $ξ>0$, (ii) for $l=1$, $n\geq 3$, $α\in (0,\frac{1}{2}+\frac{1}{n})$ and $ξ$ larger than a quantity depending on $χ\lVert v_0 \rVert_{L^\infty(Ω)}$, (iii) for $l>1$ any $ξ>0$, and in any dimensional settings. Finally, an indicative analysis about the effect by logistic and repulsive actions on chemotactic phenomena is proposed by comparing the results herein derived for the linear production case with those in \cite{LankeitWangConsumptLogistic}.
