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