Boundedness and asymptotic behavior of solutions to a chemotaxis–haptotaxis model in high dimensions. (December 2015) - Li, Yuhuan; Lin, Ke; Mu, Chunlai
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.