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 \begin{document}$ \mathbb{R}^{N} $\end{document} , \begin{document}$ \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} $\end{document} where \begin{document}$ \chi, \ a,\ b,\ \lambda,\ \mu $\end{document} are positive constants and \begin{document}$ N $\end{document} is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption \begin{document}$ b>\frac{N\chi\mu}{4} $\end{document} , the global existence of a unique classical solution \begin{document}$ (u(x,t;u_0, v_0),v(x,t;u_0, v_0)) $\end{document} of (1) with \begin{document}$ u(x,0;u_0, v_0) = u_0(x) $\end{document} and \begin{document}$ v(x,0;u_0, v_0) = v_0(x) $\end{document} for every nonnegative, bounded, and uniformly continuous function \begin{document}$ u_0(x) $\end{document} , and every nonnegative, bounded, uniformly continuous, and differentiable function \begin{document}$ v_0(x) $\end{document} . Next, under the same assumption \begin{document}$ b>\frac{N\chi\mu}{4} $\end{document} , we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function \begin{document}$ u_0 $\end{document} is bounded below by a positive constant independent of \begin{document}$ (u_0, v_0) $\end{document} when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function \begin{document}$ u_0 $\end{document} . We show that there is \begin{document}$ K = K(a,\lambda,N)>\frac{N}{4} $\end{document} such that if \begin{document}$ b>K \chi\mu $\end{document} and \begin{document}$ \lambda\geq \frac{a}{2} $\end{document} , then for every strictly positive initial function \begin{document}$ u_0(\cdot) $\end{document} , it holds that \begin{document}$ \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. $\end{document}

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