Boundedness in a chemotaxis–haptotaxis model with gradient-dependent flux limitation

Abstract

This paper deals with a chemotaxis–haptotaxis system with gradient-dependent flux-limitation ut=Δu−χ∇⋅(uf(|∇v|2)∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0,under a smooth bounded domain Ω⊂Rn,n∈{2,3}, where χ, ξ and μ are positive parameters, f∈C2([0,∞)) satisfies the condition f(|∇v|2)≤(1+(|∇v|2))p−22,with 1<p<nn−1. It is proved that for sufficiently smooth initial data (u0,v0,w0), the corresponding initial–boundary problem possesses a unique classical solution, which is uniformly bounded in time.