In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on
$ \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
$ \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. $
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