Finite-time blow-up of weak solutions to a chemotaxis system with gradient dependent chemotactic sensitivity

This paper deals with the parabolic–elliptic chemotaxis system with gradient dependent chemotactic sensitivity,{ut=Δu−χ∇⋅(u|∇v|p−2∇v),x∈Ω,t>0,0=Δv−μ+u,x∈Ω,t>0, where Ω:=BR(0)⊂Rn (n≥2) is a ball with some R>0, χ>0, p∈(nn−1,∞), μ:=1|Ω|∫Ωu0 and u0 is an initial datum of arbitrary size. In the case that p∈(1,nn−1), Negreanu and Tello (J. Differential Equations; 2018; 265; 733–751) established global existence and uniform boundedness of solutions, whereas when p∈(nn−1,2), Tello (Comm. Partial Differential Equations; 2022; 47; 307–345) showed that solutions blow up in finite time under the condition that μ>6 and χ is large enough. These works imply that the number p=nn−1 certainly plays the role of a critical blow-up exponent, and it is expected that when p>nn−1, for arbitrary μ>0 the system admits at least one solution which blows up in finite time. The purpose of this paper is to prove that this conjecture is true within a framework of weak solutions with a moment inequality.