Asymptotic stability and the hair-trigger effect in Cauchy problem of the parabolic-parabolic Keller-Segel system with logistic source

In this paper, we study the asymptotic stability and the hair-trigger effect for Cauchy problem of the following parabolic-parabolic Keller-Segel system with logistic term$ \begin{equation} \left\{ \begin{aligned} &u_{t} = \Delta u-\chi \nabla\cdot\left ( u\nabla v \right )+u\left(a-b u\right)&x\in \mathbb{R}^{N},\, t>0,\\ &\tau v_{t} = \Delta v+\lambda u-\mu v&x\in \mathbb{R}^{N},\, t>0, \end{aligned} \right. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)\end{equation} $where $ \chi $, $ a $, $ b $, $ \lambda $, $ \mu $ and $ \tau $ are positive constants and $ N $ is a positive integer. To this end, for small $ \chi $, we firstly obtain the global boundedness of solution by loop-argument based on $ L^p-L^q $ estimates of heat semigroup, with which we can further obtain the asymptotic stability of the positive constant equilibria in $ L^\infty(\mathbb R^N) $ for any initial data with positive lower bound. Moreover, for the special case $ \tau = 1 $, if $ \int_{B(x,\delta)}\ln u_0(s)ds\in L^\infty(\mathbb R^N) $ for some $ \delta>0 $, by constructing localized Lyapunov type functional, the solutions are shown to converge to the positive constant equilibria uniformly on any compact subset of $ \mathbb R^N $, which is known as the hair-trigger effect. Our contribution lies in the generalization of the results on asymptotic stability from the special case $ \tau = 1 $([31]) to any $ \tau>0 $, and the generalization of classical results on hair-trigger effect for Fisher-KPP equation to Keller-Segel system.