Large time behavior of solutions to a fully parabolic chemotaxis–haptotaxis model in N dimensions.

Abstract This paper deals with the Neumann problem for a fully parabolic chemotaxis–haptotaxis model of cancer invasion given by { u t = Δ u − χ ∇ ⋅ (u ∇ v) − ξ ∇ ⋅ (u ∇ w) + u (a − μ u r − 1 − λ w) , x ∈ Ω , t > 0 , τ v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0. Here, Ω ⊂ R N (N ≥ 1) is a bounded domain with smooth boundary and τ > 0 , r > 1 , λ ≥ 0 , a ∈ R , μ , ξ and χ are positive constants. It is shown that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in the cases r > 2 or r = 2 , with μ > μ ⁎ = (N − 2) + N (χ + C β) C N 2 + 1 1 N 2 + 1 for some positive constants C β and C N 2 + 1. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to ((a μ) 1 r − 1 , (a μ) 1 r − 1 , 0) if μ is large enough. This result improves or extends previous results of several authors. [ABSTRACT FROM AUTHOR]