Boundedness and stabilization in a multi-dimensional chemotaxis—haptotaxis model

Abstract

This paper deals with the coupled chemotaxis-haptotaxis model of cancer invasion given bywhereχ, ξandμare positive parameters andΩ ⊂ ℝn(n≥ 1) is a bounded domain with smooth boundary. Under zero-flux boundary conditions, it is shown that, for anyμ>χand any sufficiently smooth initial data (u0,w0) satisfyingu0≥ 0 andw0> 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 ofμ. In particular, this will imply that wheneveru0> 0 and 0 <w0< 1 inthere exists a positive constantμ* depending only onχ, ξ, Ω, u0andw0such that for anyμ<μ* the above global solution (u, v, w) approaches the spatially uniform state (1, 1, 0) as time goes to infinity.