Large time behavior of solution to a fully parabolic chemotaxis–haptotaxis model in higher dimensions

This paper deals with the chemotaxis–haptotaxis model of cancer invasion{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+u(1−μu−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 in a bounded smooth domain Ω⊂RnΩ⊂Rn with zero-flux boundary conditions, where χ, ξ and μ   are positive parameters. It is shown that if μ/χμ/χ is suitably large then for all sufficiently smooth initial data, the associated initial–boundary–value problem possesses a unique global-in-time classical solution that is bounded in Ω×(0,∞)Ω×(0,∞), and if the initial data w0w0 is small, w   becomes asymptotically negligible. Moreover, we prove that when domain Ω is convex, (1μ,1μ,0) is globally asymptotically stable provided that u0≢0u0≢0 and thereby extends the result of Hillen et al. (2013) [18] to the higher space dimensions.