一类具有密度抑制运动的反应扩散方程组的整体经典解
Global classical solutions for a class of reaction-diffusion system with density-suppressed motility
摘要：
研究了一类具有密度抑制运动的反应扩散方程组\begin{equation*} \begin{cases} u_{t}=Δ(γ(v) u)+αu F(w), & x \in Ω, \quad t>0, \\ v_{t}=D Δv+u-v, & x \in Ω, \quad t>0, \\ w_{t}=Δw-u F(w), & x \in Ω, \quad t>0, %\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=\frac{\partial w}{\partial ν}=0, & x \in \partial Ω, \quad t>0, \\ %(u, v, w)(x, 0)=\left(u_{0}, v_{0}, w_{0}\right)(x), & x \in Ω, \end{cases} \end{equation*}在齐次Neumann边界条件下在光滑有界区域$Ω\subset \mathbb{R}^n~(n\leq 2)$中的解，其中$α>0$和$D>0$为常数.随机运动函数$γ$满足\begin{equation*} γ\in C^3((0,+\infty)),\ γ>0,\ γ'<0\,\ \text{on}\,\ (0,+\infty) \ \ \text{and}\ \ \lim_{v\rightarrow+\infty}γ(v)=0. \end{equation*} %和% \begin{equation*} %\lim_{x\rightarrow+\infty}γ(x)=0. %\end{equation*}摄入率函数$F$满足\begin{equation*} F\in C^1([0,+\infty)),\,F(0)=0\,\ \text{and}\ \,F>0\,\ \text{on}\,\ (0,+\infty). \end{equation*}我们证明了上述系统对于所有非负初始数据$$ u_0\in C^0(\overlineΩ),\,v_0\in W^{1,\infty}(Ω),\,w_0\in W^{1,\infty}(Ω). $$都存在唯一的全局经典解。此外，如果存在$k>0$和$\overline{v}>0$使得\begin{equation*} \inf_{v>\overline{v}}v^kγ(v)>0, \end{equation*}，则全局解在时间上一致有界。
英文摘要：
This paper is concerned with a class of reaction-diffusion system with density-suppressed motility \begin{equation*} \begin{cases} u_{t}=Δ(γ(v) u)+αu F(w), & x \in Ω, \quad t>0, \\ v_{t}=D Δv+u-v, & x \in Ω, \quad t>0, \\ w_{t}=Δw-u F(w), & x \in Ω, \quad t>0, %\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=\frac{\partial w}{\partial ν}=0, & x \in \partial Ω, \quad t>0, \\ %(u, v, w)(x, 0)=\left(u_{0}, v_{0}, w_{0}\right)(x), & x \in Ω, \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset \mathbb{R}^n~(n\leq 2)$, where $α>0$ and $D>0$ are constants. The random motility function $γ$ satisfies \begin{equation*} γ\in C^3((0,+\infty)),\ γ>0,\ γ'<0\,\ \text{on}\,\ (0,+\infty) \ \ \text{and}\ \ \lim_{v\rightarrow+\infty}γ(v)=0. \end{equation*} %and %\begin{equation*} %\lim_{x\rightarrow+\infty}γ(x)=0. %\end{equation*} The intake rate function $F$ satisfies \begin{equation*} F\in C^1([0,+\infty)),\,F(0)=0\,\ \text{and}\ \,F>0\,\ \text{on}\,\ (0,+\infty). \end{equation*} We show that the above system admits a unique global classical solution for all non-negative initial data $$ u_0\in C^0(\overlineΩ),\,v_0\in W^{1,\infty}(Ω),\,w_0\in W^{1,\infty}(Ω). $$ Moreover, if there exist $k>0$ and $\overline{v}>0$ such that \begin{equation*} \inf_{v>\overline{v}}v^kγ(v)>0, \end{equation*} then the global solution is bounded uniformly in time.
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