Global bounded solutions to a Keller–Segel system with singular sensitivity

Abstract

This paper deals with the Neumann initial–boundary value problem for the Keller–Segel chemotaxis system with singular sensitivity ut=d1Δu−χ∇⋅(uv∇v),vt=d2Δv−v+uin a smooth bounded domain Ω⊂Rn (n≥1), where d1>0, d2>0 and χ∈R. When d1=d2+χ, for all initial data 0≤u0∈C0(Ω̄) and 0<v0∈W1,∞(Ω), we prove that the problem possesses a unique global classical solution which is uniformly bounded in Ω×(0,∞).