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

Abstract

We consider the Neumann value problem for the chemotaxis system { u t = ∇ ⋅ ( ∇ u − u ( α 1 + v ∇ v + ρ ∇ w ) ) + λ u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v − v − μ u v , x ∈ Ω , t > 0 , w t = γ u ( 1 − w ) , x ∈ Ω , t > 0 , in a bounded domain Ω ⊂ R n ( n ≥ 1 ) with smooth boundary, where α , ρ , λ , μ and γ are positive coefficients. It is shown that for any choice of reasonably regular initial data ( u 0 , v 0 , w 0 ) , there exists a constant λ ∗ depending on α , ρ , μ , γ , n , v 0 and w 0 such that for any 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 ) in an appropriate sense.