Global boundedness and asymptotic stability of the Keller-Segel system with logistic-type source in the whole space

In this paper, we investigate the Cauchy problem of the parabolic-parabolic Keller-Segel system with the logistic-type term $au-b{u}^{\gamma }$ on ${\mathbb{R}}^N,N\ge 2$. We discuss the global boundedness of classical solutions with nonnegative bounded and uniformly continuous initial functions when $\gamma >1$. Moreover, based on the persistence of classical solution we show the large time behavior of the positive constant equilibria with strictly positive initial function in the case of $\gamma \in (1,2)$.