The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the following parabolic–elliptic chemotaxis system with space and time dependent logistic source,

where \(N\ge 1\) is a positive integer, \(\chi , \lambda \) and \(\mu \) are positive constants, and the functions a(x, t) and b(x, t) are positive and bounded. In the first of the series (Salako and Shen in Math Models Methods Appl Sci 28(11):2237–2273, 2018), we studied the phenomena of pointwise and uniform persistence for solutions with strictly positive initial data, and the asymptotic spreading for solutions with compactly supported or front like initial data. In the second of the series (Salako and Shen in J Math Anal Appl 464(1):883–910, 2018), we investigate the existence, uniqueness and stability of strictly positive entire solutions of (0.1). In particular, in the case of space homogeneous logistic source (i.e. \(a(x,t)\equiv a(t)\) and \(b(x,t)\equiv b(t)\)), we proved in Salako and Shen (J Math Anal Appl 464(1):883–910, 2018) that the unique spatially homogeneous strictly positive entire solution \((u^*(t),v^*(t))\) of (0.1) is uniformly and exponentially stable with respect to strictly positive perturbations when \(0<2\chi \mu <\inf _{t\in {\mathbb R}}b(t)\). In the current part of the series, we discuss the existence of transition front solutions of (0.1) connecting (0, 0) and \((u^*(t),v^*(t))\) in the case of space homogeneous logistic source. We show that for every \(\chi >0\) with \(\chi \mu \big (1+\frac{\sup _{t\in {\mathbb R}}a(t)}{\inf _{t\in {\mathbb R}}a(t)}\big )<\inf _{t\in {\mathbb R}}b(t)\), there is a positive constant \({c}^{*}_\chi \) such that for every \(\underline{c}> {c}^*_{\chi }\) and every unit vector \(\xi \), (0.1) has a transition front solution of the form \((u(x,t),v(x,t))=(U(x\cdot \xi -C(t),t),V(x\cdot \xi -C(t),t))\) satisfying that \(C'(t)=\frac{a(t)+\kappa ^2}{\kappa }\) for some positive number \(\kappa \), \(\liminf _{t-s\rightarrow \infty }\frac{C(t)-C(s)}{t-s}=\underline{c}\), and

Furthermore, we prove that there is no transition front solution \((u(x,t),v(x,t))=(U(x\cdot \xi -C(t),t),V(x\cdot \xi -C(t),t))\) of (0.1) connecting (0, 0) and \((u^*(t),v^*(t))\) with least mean speed less than \(2\sqrt{\underline{a}}\), where \(\underline{a}=\liminf _{t-s\rightarrow \infty }\frac{1}{t-s}\int _{s}^{t}a(\tau )d\tau \).

当前的工作是三篇论文的第三篇，该论文专门研究以下具有空间和时间相关逻辑源的抛物线-椭圆形趋化系统中的渐近动力学，

其中\（N \ ge 1 \）是一个正整数，\（\ chi，\ lambda \）和\（\ mu \）是正常数，函数a（x，  t）和b（x，  t）是肯定的和有界的。在第一个系列中（Salako和Shen在Math Models Methods Appl Sci 28（11）：2237–2273，2018）中，我们研究了具有严格正初始数据的解的逐点和一致持久性现象，以及具有紧凑支持或前端类似初始数据的解决方案。在系列的第二部分（Salako和Shen在J Math Anal Appl 464（1）：883–910，2018年）中，我们研究了（0.1）的严格正整数解的存在性，唯一性和稳定性。特别是，在空间齐次逻辑源（即\（a（x，t）\ equiv a（t）\）和\（b（x，t）\ equiv b（t）\））的情况下，我们证明了在Salako和Shen（J Math Anal Appl 464（1）：883–910，2018）中，独特的空间齐次严格正整体解当（（0 <2 \ chi \ mu <\ inf _ {t ）时，（0.1）的\（（u ^ *（t），v ^ *（t））\）对于严格的正摄动是均匀且指数稳定的\ in {\ mathbb R}} b（t）\）中。在本系列的当前部分中，我们讨论在以下情况下连接（0，0）和\（（u ^ *（t），v ^ *（t））\）的（0.1）过渡前沿解的存在。空间同质后勤来源。我们证明，对于每个\（\ chi> 0 \）与\（\ chi \ mu \ big（1+ \ frac {\ sup _ {t \ in {\ mathbb R}} a（t）} {\ inf _ {t \ in {\ mathbb R}} a（t）} \ big）<\ inf _ {t \ in {\ mathbb R}} b（t）\）中，存在一个正常数\（{c} ^ {*} _ \ chi \），这样对于每个\（\下划线{c}> {c} ^ * _ {\ chi} \）和每个单位向量\（\ xi \），（0.1）具有形式为\（（u（x，t），v（x，t））=（U（x \ cdot \ xi -C（t），t），V（x \ cdot \ xi -C（t），t））\）满足\（C'（t）= \ frac {a（t）+ \ kappa ^ 2} {\ kappa} \）对于某个正数\（ \ kappa \），\（\ liminf _ {ts \ rightarrow \ infty} \ frac {C（t）-C（s）} {ts} = \下划线{c} \）和

此外，我们证明没有过渡前沿解\（（u（x，t），v（x，t））=（U（x \ cdot \ xi -C（t），t），V（x \（0.1）的cdot \ xi -C（t），t））\）以最小平均速度小于（ 0，0）和\（（u ^ *（t），v ^ *（t））\）\（2 \ sqrt {\ underline {a}} \\），其中\（\ underline {a} = \ liminf _ {ts \ rightarrow \ infty} \ frac {1} {ts} \ int _ {s} ^ { t} a（\ tau）d \ tau \）。