Global boundedness of weak solution in an attraction–repulsion chemotaxis system with [formula omitted]-Laplacian diffusion

Abstract

In this paper, an attraction–repulsion chemotaxis system with p-Laplacian diffusion ut=∇⋅(|∇u|p−2∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,0=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,u(x,0)=u0(x),x∈Ωis considered associated with homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn, n≥2. χ,ξ,α,β,γ,δ are positive parameters. Global bounded weak solution is constructed for any p>1 under the following cases:⋅ Case I: ξγ−χα>0 with p>1;⋅ Case II: ξγ−χα≤0 with p>3nn+1;⋅ Case III: ξγ−χα≤0 with 1<p≤3nn+1 and ‖u0‖L(3−p)np(Ω) is small.