ASYMPTOTIC STABILITY AND THE HAIR-TRIGGER EFFECT IN CAUCHY PROBLEM OF THE FLUX-LIMITED KELLER-SEGEL SYSTEM WITH LOGISTIC SOURCE

This paper was concerned with Cauchy problem of the parabolic-parabolic flux-limited Keller-Segel system with logistic source. We discussed the global existence and global boundedness of the classical solution. By constructing auxiliary functions with quasi-linear structures, we can directly obtained the persistence and the asymptotic stability of the positive constant equilibria for strictly positive initial datum. Moreover, for any initial datum satisfying ^R_B(x,δ₎ ln u0(s)ds ∈ L^∞(R^N) for some δ > 0, the hair-trigger effect was detected by constructing the localized Lyapunov type functional.