2014/10/01 by Youshan Tao, Michael Winkler
paper · doi:10.1017/s0308210512000571
This paper deals with the coupled chemotaxis-haptotaxis model of cancer invasion given by where χ, ξ and μ are positive parameters and Ω ⊂ ℝ n ( n ≥ 1) is a bounded domain with smooth boundary. Under zero-flux boundary conditions, it is shown that, for any μ > χ and any sufficiently smooth initial data ( u 0 , w 0 ) satisfying u 0 ≥ 0 and w 0 > 0, the associated initial–boundary-value problem possesses a unique global smooth solution that is uniformly bounded. Moreover, we analyse the stability and attractivity properties of the non-trivial homogeneous equilibrium ( u, v, w ) ≡ (1,1, 0) and establish a quantitative result relating the domain of attraction of this steady state to the size of μ . In particular, this will imply that whenever u 0 > 0 and 0 w 0 there exists a positive constant μ * depending only on χ, ξ, Ω, u 0 and w 0 such that for any μ μ * the above global solution ( u, v, w ) approaches the spatially uniform state (1, 1, 0) as time goes to infinity.