Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2020-06-03 , DOI: 10.1007/s00033-020-01320-w Pan Zheng , Jie Xing
This paper deals with a two-dimensional quasilinear chemotaxis-growth system with indirect signal consumption
$$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla v)+\mu (u-u^{2}),&(x,t)\in \Omega \times (0,\infty ), \\&v_t=\Delta v-vw,&(x,t)\in \Omega \times (0,\infty ), \\&w_t=-\delta w+u,&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^{2}\), where \(\delta >0\) and \(\mu >0\), the nonlinear diffusivity D(u) and chemosensitivity S(u) are supposed to satisfy
$$\begin{aligned} D(u)\geqslant u^{m},\quad S(u)\leqslant u^{q}\quad \text {and}\quad D,S>0. \end{aligned}$$When \(m>\max \{2q-3, 1\}\), we study the global boundedness of solutions for this problem. In the case of \(m=0\) and \(0<q<\frac{3}{2}\), we can obtain the boundedness of this system by using different methods. Moreover, under the particular conditions of \(m=0\) and \(q=1\), the global bounded solution (u, v, w) satisfies
$$\begin{aligned} \Vert u(\cdot ,t)-1\Vert _{L^{\infty }(\Omega )}+\Vert v(\cdot ,t)\Vert _{L^{\infty }(\Omega )}+\Vert w(\cdot ,t)-\frac{1}{\delta }\Vert _{L^{\infty }(\Omega )}\rightarrow 0 \end{aligned}$$as \(t\rightarrow \infty \).
中文翻译：
具有间接信号消耗的二维拟线性趋化增长系统的有界性和长时间行为
本文研究具有间接信号消耗的二维拟线性趋化增长系统
$$ \ begin {aligned} \ left \ {\ begin {aligned}＆u_t = \ nabla \ cdot（D（u）\ nabla u）-\ nabla \ cdot（S（u）\ nabla v）+ \ mu（uu ^ {2}），＆（x，t）\ in \ Omega \ times（0，\ infty），\\＆v_t = \ Delta v-vw，＆（x，t）\ in \ Omega \ times（0， \ infty），\\＆w_t =-\ delta w + u，＆（x，t）\ in \ Omega \ times（0，\ infty），\ end {aligned} \ right。\ end {aligned} $$在光滑有界域\（\ Omega \ subset {\ mathbb {R}} ^ {2} \）中的齐次Neumann边界条件下，其中\（\ delta> 0 \）和\（\ mu> 0 \），假定非线性扩散率D（u）和化学敏感性S（u）满足
$$ \ begin {aligned} D（u）\ geqslant u ^ {m}，\ quad S（u）\ leqslant u ^ {q} \ quad \ text {and} \ quad D，S> 0。\ end {aligned} $$当\（m> \ max \ {2q-3，1 \} \）时，我们研究此问题的解的整体有界性。在\（m = 0 \）和\（0 <q <\ frac {3} {2} \）的情况下，我们可以使用不同的方法来获得该系统的有界性。此外，在\（m = 0 \）和\（q = 1 \）的特定条件下，全局有界解（u， v， w）满足
$$ \ begin {aligned} \ Vert u（\ cdot，t）-1 \ Vert _ {L ^ {\ infty}（\ Omega}} + \ Vert v（\ cdot，t）\ Vert _ {L ^ { \ infty}（\ Omega}} + \ Vert w（\ cdot，t）-\ frac {1} {\ delta} \ Vert _ {L ^ {\ infty}（\ Omega}} \ rightarrow 0 \ end {aligned } $$作为\（t \ rightarrow \ infty \）。
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