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On a fully parabolic singular chemotaxis-(growth) system with indirect signal production or consumption
Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2021-04-30 , DOI: 10.1007/s00033-021-01534-6
Jie Xing , Pan Zheng , Yuting Xiang , Hui Wang

This paper deals with a fully parabolic singular chemotaxis-(growth) system with indirect signal production or consumption

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot \left( \frac{u}{v}\nabla v\right) +f(u),&(x,t)\in \Omega \times (0,\infty ), \\&v_t=\Delta v+h(v,w),&(x,t)\in \Omega \times (0,\infty ), \\&w_t=\Delta w- w+u,&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth convex bounded domain \(\Omega \subset {\mathbb {R}}^{2}\), where \(\chi >0\). When \(f(u)=0\), \(h(v,w)=-v+w\), we prove that the solution (uvw) of this model is globally uniformly bounded in time and exponentially converges to the steady state \((\overline{u_{0}},\overline{u_{0}},\overline{u_{0}})\) in the norm of \(L^{\infty }(\Omega )\) provided that \(\chi <\chi _{0}\) with \(\chi _{0}>0\), where \(\overline{u_{0}}:=\frac{1}{|\Omega |}\int _{\Omega }u_{0}(x)\mathrm{d}x\). Moreover, in the case of \(f(u)=ru-\mu u^{2}\), \(h(v,w)=-v+w\), where \(r\in {\mathbb {R}}\), and \(\mu >0\), we obtain the global existence of solutions for this system. Furthermore, under the conditions of \(f(u)=ru-\mu u^{2}\), \(h(v,w)=-vw\) and \(\mu >0\), \(r>0\), the solution (uvw) is also global in time.



中文翻译:

在具有间接信号产生或消耗的完全抛物线奇异趋化(生长)系统上

本文研究了具有间接信号产生或消耗的全抛物奇异趋化(增长)系统

$$ \ begin {aligned} \ left \ {\ begin {aligned}&u_t = \ Delta u- \ chi \ nabla \ cdot \ left(\ frac {u} {v} \ nabla v \ right)+ f(u) ,&(x,t)\ in \ Omega \ times(0,\ infty),\\&v_t = \ Delta v + h(v,w),&(x,t)\ in \ Omega \ times(0, \ infty),\\&w_t = \ Delta w- w + u,&(x,t)\ in \ Omega \ times(0,\ infty),\ end {aligned} \ right。\ end {aligned} $$

在光滑凸有界域\(\ Omega \ subset {\ mathbb {R}} ^ {2} \)中的齐次Neumann边界条件下,其中\(\ chi> 0 \)。当\(f(u)= 0 \)\(h(v,w)=-v + w \)时,我们证明了该模型的解(u,  v,  w)在时间上全局一致且以\(L ^ {\ infty}的范数指数收敛到稳态\((\ overline {u_ {0}},\ overline {u_ {0}},\ overline {u_ {0}})\)(\ Omega)\)提供\(\ chi <\ chi _ {0} \)\(\ chi _ {0}> 0 \),其中\(\ overline {u_ {0}}:= \ frac {1} {| \ Omega |} \ int _ {\ Omega} u_ {0}(x)\ mathrm {d} x \)。此外,在\(f(u)= ru- \ mu u ^ {2} \)的情况下\(h(v,w)=-v + w \),其中\(r \ in {\ mathbb {R}} \)\(\ mu> 0 \),我们获得了该系统解的全局存在性。此外,在\(f(u)= ru- \ mu u ^ {2} \)的情况下\(h(v,w)=-vw \)\(\ mu> 0 \)\( r> 0 \),解(u,  v,  w)在时间上也是全局的。

更新日期:2021-04-30
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