Boundedness and exponential stabilization for time–space fractional parabolic–elliptic Keller–Segel model in higher dimensions

Abstract

We consider initial boundary value problem for the time–space fractional parabolic–elliptic Keller–Segel model 0CDtβu=−(−Δ)α2(ρ(v)u),(t,x)∈(0,T]×Ω(−Δ)α2v+v=u,(t,x)∈(0,T]×Ωin a bounded domain Ω⊂Rn(n≥3) with smooth boundary, where β∈(0,1),α∈(1,2) and ρ stands for a signal-dependent motility. It is shown that for some special initial datum, there exists the uniform-in-time upper bound for v such that the associated initial–boundary system possesses a global classical solution which is uniformly bounded. Moreover, building on this boundedness property, it is proved that the exponential stabilization of the classical solution.