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- W4381943298 abstract "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 nablacdotleft ( unabla v right )+uleft(a-b uright)&xin mathbb{R}^{N},, t>0, &tau v_{t} = Delta v+lambda u-mu v&xin 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)dsin 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." @default.
- W4381943298 created "2023-06-26" @default.
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- W4381943298 date "2023-01-01" @default.
- W4381943298 modified "2023-09-28" @default.
- W4381943298 title "Asymptotic stability and the hair-trigger effect in Cauchy problem of the parabolic-parabolic Keller-Segel system with logistic source" @default.
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- W4381943298 doi "https://doi.org/10.3934/dcdss.2023092" @default.
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