Existence and global asymptotic stability in a fractional double parabolic chemotaxis system with logistic source

Abstract

This paper studies a double parabolic chemotaxis system with logistic source and a fractional diffusion of order α∈(0,2)ut=−Λαu−χ∇⋅(u∇v)+au−bu2,vt=Δv−v+uon two dimensional periodic torus T2. In contrast to the well-known Neumann heat semigroup {etΔ}t≥0 estimates in a smoothly bounded domain Ω⊂Rn introduced by Winkler (2010), we obtain the spatio-temporal estimates of the analytic semigroup {Ttα(x)}t≥0 and {Tt(x)}t≥0 which are generated by −(−Δ)α2−I and Δ−I respectively over periodic torus T2. With the help of these conclusions, we can use the semigroup method to study the global existence and the asymptotic behavior of the above fractional chemotaxis model, which has not been studied yet. It is proved that for any nonnegative initial data (u0,v0)∈H4(T2)×H5(T2), if 1<α<2, there admits a unique globally classical solution. Furthermore, if a=0, we can obtain that the solution components u and v converge to zero with respect to the norm in L∞(T2) as t→∞.