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,\\ {v_t} = \Delta v -\lambda v+\mu u,\quad x\in{{\mathbb R}}^N,\,\,\, \end{cases} \;\;\;\;\;\;\;\;\left( 1 \right)\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\limits_{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. $