Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term

The following degenerate parabolic system modelling chemotaxis is considered:equation(KS){ut=∇⋅(∇um−uq−1∇v),x∈RN,t>0,τvt=Δv−v+u,x∈RN,t>0,u(x,0)=u0(x),τv(x,0)=τv0(x),x∈RN, where m⩾1m⩾1, q⩾2q⩾2, τ=0τ=0 or 1, and N⩾1N⩾1. The aim of this paper is to prove the existence of a time global weak solution (u,vu,v) of (KS) with the L∞(0,∞;L∞(RN))L∞(0,∞;L∞(RN)) bound. Such a global bound is obtained in the case of (i) m>q−2N for large initial data and (ii) 1⩽m⩽q−2N for small initial data. In the case of (ii), the decay properties of the solution (u,v)(u,v) are also discussed.