Boundedness and asymptotic behavior of solutions to a chemotaxis–haptotaxis model in high dimensions

We consider the Neumann value problem for the chemotaxis system {ut=∇⋅(∇u−u(α1+v∇v+ρ∇w))+λu(1−u),x∈Ω,t>0,vt=Δv−v−μuv,x∈Ω,t>0,wt=γu(1−w),x∈Ω,t>0, in a bounded domain Ω⊂Rn(n≥1)Ω⊂Rn(n≥1) with smooth boundary, where α,ρ,λ,μα,ρ,λ,μ and γγ are positive coefficients. It is shown that for any choice of reasonably regular initial data (u0,v0,w0)(u0,v0,w0), there exists a constant λ∗λ∗ depending on α,ρ,μ,γ,n,v0α,ρ,μ,γ,n,v0 and w0w0 such that for any λ>λ∗λ>λ∗, the associated initial–boundary system possesses a global classical solution which is uniformly bounded. Moreover, building on this boundedness property, it is proved that as time tends to infinity, all the solution approaches the homogeneous steady state (1,0,1)(1,0,1) in an appropriate sense.