论文标题
具有密度依赖性运动的反应扩散系统的界限和渐近学
Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility
论文作者
论文摘要
我们考虑具有密度依赖性运动的反应 - 扩散方程系统的初始边界值问题\ begin {qore*} \ label {e1} \ tag {$ \ ast $} \ begin {cases} case} u_t =δ（γ（γ（γ（v）u） v_t =dΔv+u-v，＆x \ inω，~~ t> 0，\\ w_t =Δw-uf（w），＆x \ inω，~~ t> 0，， \ frac {\ partial u} {\partialν} = \ frac {\ partial v} {\partialν} = \ frac {\ partial w} {\ partial w} {\partialν} = 0 （u，v，w）（x，0）=（u_0，v_0，w_0）（x），＆x \inΩ，\ end {cases} \ end {equation {equation*}在有界域$ω\ subset \ subset \ r^2 $的情况下，带有平稳的边界，$α$和$θ$ non-deNot $ denot，$ denot vate ω$。随机运动函数$γ（v）$和功能响应函数$ f（w）$满足以下假设：\ begin {inatize} \ item $γ（v）\ in c^{3}（[0，\ infty），〜0 <0 <γ_{1} {1} \leqγ（v）\leqγ（V） $ v \ geq0 $; \ item $ f（w）\在c^1（[[0，\ infty）），f（0）= 0，f（w）> 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in}〜（0，\ infty）〜\ mathrm {and} 〜f'（w） $γ_1，γ_2$和$η$。基于加权能量估计和MOSER迭代的方法，我们证明问题\ eqref {e1}具有统一的唯一经典全局解决方案。 Furthermore we show that if $θ>0$, the solution $(u,v,w)$ will converge to $(0,0,w_*)$ in $L^\infty$ with some $w_*>0$ as time tends to infinity, while if $θ=0$, the solution $(u,v,w)$ will asymptotically converge to $(u_*,u_*,0)$ in $ l^\ infty $带有$ u _*= \ frac {1} {|ω|}（\ | | u_0 \ | _ {l^1}+α\ | w_0 \ | w_0 \ | ___ {l^1}）$如果$ d> 0 $很大。
We consider the initial-boundary value problem of a system of reaction-diffusion equations with density-dependent motility \begin{equation*}\label{e1}\tag{$\ast$} \begin{cases} u_t=Δ(γ(v)u)+αu F(w) -θu, &x\in Ω, ~~t>0,\\ v_t=DΔv+u-v,& x\in Ω, ~~t>0,\\ w_t=Δw-uF(w),& x\in Ω, ~~t>0, \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}= \frac{\partial w}{\partial ν}=0,&x\in \partialΩ, ~~t>0,\\ (u,v,w)(x,0)=(u_0,v_0,w_0)(x), & x\inΩ, \end{cases} \end{equation*} in a bounded domain $Ω\subset\R^2$ with smooth boundary, $α$ and $θ$ are non-negative constants and $ν$ denotes the outward normal vector of $\partial Ω$. The random motility function $γ(v)$ and functional response function $F(w)$ satisfy the following assumptions: \begin{itemize} \item $γ(v)\in C^{3}([0,\infty)),~0<γ_{1}\leqγ(v)\leq γ_2, \ |γ'(v)|\leq η$ for all $v\geq0$; \item $F(w)\in C^1([0,\infty)), F(0)=0,F(w)>0 \ \mathrm{in}~(0,\infty)~\mathrm{and}~F'(w)>0 \ \mathrm{on}\ \ [0,\infty)$ \end{itemize} for some positive constants $γ_1, γ_2$ and $η$. Based on the method of weighted energy estimates and Moser iteration, we prove that the problem \eqref{e1} has a unique classical global solution uniformly bounded in time. Furthermore we show that if $θ>0$, the solution $(u,v,w)$ will converge to $(0,0,w_*)$ in $L^\infty$ with some $w_*>0$ as time tends to infinity, while if $θ=0$, the solution $(u,v,w)$ will asymptotically converge to $(u_*,u_*,0)$ in $L^\infty$ with $u_*=\frac{1}{|Ω|}(\|u_0\|_{L^1}+α\|w_0\|_{L^1})$ if $D>0$ is suitably large.
