Parabolic–Elliptic Chemotaxis Model with Space–Time Dependent Logistic Sources on $\mathbb {R}^N$. III: Transition Fronts

Abstract

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,

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu=\Delta u -\chi \nabla \cdot (u\nabla v)+u(a(x,t)-b(x,t)u),&{}\quad x\in {\mathbb R}^N,\\ 0=\Delta v-\lambda v+\mu u ,&{}\quad x\in {\mathbb R}^N, \end{array}\right. } \end{aligned}$$

(0.1)

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

$$\begin{aligned} \lim _{x\rightarrow -\infty }\sup _{t\in {\mathbb R}}|U(x,t)-u^*(t)|=0 \quad \text {and}\quad \lim _{x\rightarrow \infty }\sup _{t\in {\mathbb R}}|\frac{U(x,t)}{e^{-\kappa x}}-1|=0. \end{aligned}$$

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 $\underline{a}=\liminf _{t-s\rightarrow \infty }\frac{1}{t-s}\int _{s}^{t}a(\tau )d\tau$.