Boundedness in a multi-dimensional chemotaxis–haptotaxis model with nonlinear diffusion. (September 2016) - Wang, Yifu
Abstract: We consider the chemotaxis–haptotaxis model { u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ), x ∈ Ω, t > 0, v t = Δ v − v + u, x ∈ Ω, t > 0, w t = − v w, x ∈ Ω, t > 0 in a bounded smooth domain Ω ⊂ R n ( n ≥ 2 ), where χ, ξ and μ are positive parameters, and the diffusivity D ( u ) is assumed to generalize the prototype D ( u ) = δ ( u + 1 ) − α with α ∈ R . Under zero-flux boundary conditions, it is shown that for sufficiently smooth initial data ( u 0, v 0, w 0 ) and α < 2 − n n + 2, the corresponding initial–boundary problem possesses a unique global-in-time classical solution which is uniformly bounded, which improves the previous results.