Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^{N}$

In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^{N}$, \begin{equation} \begin{cases} u_t=\Delta u-\chi\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in\mathbb{R}^{N}\,\,\, t>0\cr {v_t}=\Delta v -\lambda v+\mu u,\quad x\in \mathbb{R}^{N}\,\,\, t>0 \end{cases}(1) \end{equation} where $\chi, \ a,\ b,\ \lambda,\ \mu$ are positive constants and $N$ is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption $b>\frac{N\chi\mu}{4}$, the global existence of a unique classical solution $(u(x,t;u_0, v_0),v(x,t;u_0, v_0))$ of (1) with $u(x,0;u_0, v_0)=u_0(x)$ and $v(x,0;u_0, v_0)=v_0(x)$ for every nonnegative, bounded, and uniformly continuous function $u_0(x)$, and every nonnegative, bounded, uniformly continuous, and differentiable function $v_0(x)$. Next, under the same assumption $b>\frac{N\chi\mu}{4}$, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function $u_0$ is bounded below by a positive constant independent of $(u_0, v_0)$ when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function $u_0$. We show that there is $K=K(a,\lambda,N)>\frac{N}{4}$ such that if $b>K \chi\mu$ and $\lambda\geq \frac{a}{2}$, then for every strictly positive initial function $u_0(\cdot)$, it holds that $$\lim_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\frac{\mu}{\lambda}\frac{a}{b}\|_{\infty}\big]=0.$$