Extensibility criterion ruling out gradient blow-up in a quasilinear degenerate chemotaxis system with flux limitation

Abstract

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation,{ut=∇⋅(up∇uu2+|∇u|2)−χ∇⋅(uq∇v1+|∇v|2),0=Δv−μ+u, under no-flux boundary conditions in balls Ω⊂Rn, and the initial condition u|t=0=u0 for a radially symmetric and positive initial data u0∈C3(Ω‾), where χ>0 and μ:=1|Ω|∫Ωu0. Bellomo–Winkler [3] proved local existence of unique classical solutions and extensibility criterion ruling out gradient blow-up as well as global existence and boundedness of solutions when p=q=1 under some conditions for χ and ∫Ωu0. This paper derives local existence and extensibility criterion ruling out gradient blow-up when p,q≥1, and moreover shows global existence and boundedness of solutions when p>q+1−1n.