Global boundedness in a three-dimensional chemotaxis–haptotaxis model

Abstract

This paper studies the chemotaxis–haptotaxis model of cancer invasion ut=Δu−χ∇⋅(u∇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>0in a bounded smooth domain Ω⊂R3 with zero-flux boundary conditions, where the parameters χ, ξ and μ are positive. It is shown that the corresponding initial–boundary problem possesses a unique classical solution which is global in time and bounded under the explicit condition μ≥(112+ξ2‖w0‖L∞(Ω)2)χ2+372+4ξ2‖w0‖L∞(Ω)2 and suitable regularity assumptions on the initial data (u0,v0,w0).